Global Regularity Results for Non-homogeneous Growth Fractional Problems

Abstract

The main goal of this article is to show the global Hölder regularity of weak solutions to a class of problems involving the fractional (p, q)-Laplacian, denoted by $(-\Delta )^{s_1}_{p}+(-\Delta )^{s_2}_{q}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$. We use a suitable Caccioppoli inequality and a local boundedness result in order to prove the weak Harnack inequality. Consequently, by employing a suitable iteration process, we establish the interior Hölder continuity result for local weak solutions. The global Hölder regularity result we prove expands and improves the regularity results of Giacomoni et al. (Adv Calc Var, arXiv:2102.06080) to the subquadratic case (that is, $q<2$) and to the more general right-hand side, which requires a different and new approach. Moreover, we establish a non-local Harnack-type inequality for weak solutions and a strong maximum principle for weak super-solutions, which are of independent interest.

1 Introduction

In this paper, we establish the Hölder continuity results for the weak solutions to the following non-local problem:

$$\begin{aligned} (-\Delta )^{s_1}_{p}u+ (-\Delta )^{s_2}_{q}u = f \quad \text {in } \; \Omega , \end{aligned}$$

(1.1)

where $\Omega$ is a bounded domain in $\mathbb R^N$ with $C^{1,1}$ boundary, $1< q, p<\infty$, $0<s_2\le s_1<1$ and $f\in L^\gamma _{\mathrm{loc}}(\Omega )$ with $\gamma {\left\{ \begin{array}{ll}> N/(ps_1)&{} \quad \text{ if } N>ps_1,\\ \ge 1&{} \quad \text{ otherwise }. \end{array}\right. }$

The fractional p-Laplacian $(-\Delta )^{s}_{p}$ is defined as

$$\begin{aligned} {(-\Delta )^{s}_pu(x)}= 2\lim _{\varepsilon \rightarrow 0}\int _{{\mathbb {R}}^N\setminus B_{\varepsilon }(x)} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy. \end{aligned}$$

These kinds of non-local operators have their applications in real world problems, such as obstacle problems, the study of American options in finance, game theory, image processing, and anomalous diffusion phenomena (see [19] for more details). Due to this reason, elliptic problems involving the fractional Laplacian have been extensively studied in the last two decades.

The leading operator, $(-\Delta )^{s_1}_{p}+(-\Delta )^{s_2}_{q}$, in problem (1.1) is known as the fractional (p, q)-Laplacian, which is non-homogeneous (unless $p=q$) in the sense that for any $t>0$, there does not exist any $\sigma \in \mathbb R$ such that $((-\Delta )^{s_1}_{p}+(-\Delta )^{s_2}_{q})(tu)=t^\sigma ((-\Delta )^{s_1}_{p}u+(-\Delta )^{s_2}_{q}u)$ holds for all $u\in W^{s_1,p}(\Omega )\cap W^{s_2,q}(\Omega )$. The operator is the fractional analogue of the (p, q)-Laplacian ($-\Delta _p -\Delta _q$), which arises from the study of general reaction–diffusion equations with non-homogeneous diffusion and transport aspects. The problems involving these kinds of operators have applications in biophysics, plasma physics, and chemical reactions, with double-phase features, where the function u corresponds to the concentration term, and the differential operator represents the diffusion coefficient, for details, see [36] and references therein.

Regarding the regularity results for weak solutions to quasilinear elliptic equations, Lieberman in [33] proved $C^{1,\alpha }(\overline{\Omega })$ regularity results for problems containing more general operators than the classical p-Laplacian. Subsequently, in [34], the same author established the interior Hölder continuity of the gradient of weak solutions to problems where the leading operator has Orlicz type of growth condition. See [35] for more details on the regularity theory of quasilinear problems.

Concerning the regularity results for problems involving non-local operators, the case of the fractional Laplacian is well understood. In particular, in [12], Caffarelli and Silvestre obtained the interior regularity results, while Ros-Oton and Serra [38] proved the optimal boundary regularity. Precisely, in the latter work, the authors proved that weak solutions of the fractional Laplacian problem (i.e., $p=q=2$ and $s_1=s_2$) with a bounded right-hand side and homogeneous Dirichlet boundary conditions are in $C^{0,s}(\mathbb R^N)$. One can rely on [20] for regularity results of linear fractional problems involving a more general kernel. For the non-linear and homogeneous operator case ($p=q\ne 2$ and $s_1=s_2$), in [17], Di Castro et al. established interior Hölder regularity results for non-local equations whose prototypes include the fractional p-Laplacian. Here, they established the Caccioppoli inequality and the Logarithmic lemma, and following De Giorgi’s iteration technique, they proved the local Hölder continuity result. In [31], Korvenpää et al. further obtained the interior and boundary Hölder regularity results for obstacle problems (also covering the Dirichlet problem) with zero right-hand side. We also mention the work of M. Cozzi [14], for Hölder regularity results for minimizers of functional involving the fractional p-Laplacian energy in the fractional De Giorgi’s class. Using a slightly different approach, Iannizzotto et al. in [26] proved that the weak solutions of problem (1.1) with bounded right-hand side belong to the space $C^{0,\alpha }(\overline{\Omega })$, for some $\alpha \in (0,s_1]$. By constructing the barrier functions as a solution to some non-local equation on $B_1$ (fractional torsion problem) and scaling it appropriately to get a sub-solution, they established a weak Harnack-type inequality to prove the interior regularity results. Moreover, using the barrier arguments, in the spirit of Krylov’s approach (as in [38]), they obtained the boundary behavior of the weak solutions. Subsequently, in [10], Brasco et al. established the optimal Hölder interior regularity result and proved that the local weak solution $u\in W^{s,p}_{\mathrm{loc}}(\Omega )$ of problem (1.1) for the superquadratic and homogeneous case (i.e., $2\le p=q$ and $s_1=s_2=s$) with $f\in L^r_{\mathrm{loc}}(\Omega )$, for suitable $r>0$, is in $C^{0,\alpha }_{\mathrm{loc}}(\Omega )$, for all $\alpha <\min \{1,ps/(p-1)\}$. Consequently, with the help of the boundary behavior of the solution from [26], we get the $C^{s}$ regularity result up to the boundary. We additionally refer to [32], for regularity results for non-local problems involving measure data. In [27], Iannizzotto et al. extended the results of [38] to the non-linear setting and proved that the weak solution u of problem (1.1), again for the case $2\le p=q$, $s_1=s_2=s\in (0,1)$ and bounded right-hand side, satisfies $\frac{u}{d^s} \in C^{0,\alpha }(\overline{\Omega })$, for some $\alpha \in (0,1)$, where $d(x):=\mathrm{dist}(x,\partial \Omega )$.

Due to the non-homogeneous and non-linear nature of the fractional (p, q)-operators, the corresponding non-local problems have attracted the attention of many researchers in recent years. For instance, see [1,2,3,4,5,6, 24] for the existence and multiplicity results for problems involving the fractional (p, q)-Laplacian, and we refer [37] for a survey of recent developments on non-standard growth problems. On the other hand, there is not much literature available regarding the regularity results. Particularly, in [24], Goel et al. have obtained the $C^{0,\alpha }_{\mathrm{loc}}$ regularity result, with some unspecified $\alpha \in (0,1)$, for weak-bounded solutions in the superquadratic case. Further, we mention the recent work of [8], where the global Hölder continuity results (in the spirit of [26]) for weak solutions to problems involving the fractional $(-\Delta )_g^s$-Laplacian, where g is a convex Young’s function, is proved. However, this does not include our problem, even for the case $s_1=s_2$, because of the power type of growth conditions (2.2) and (2.4). In [15], the authors have obtained the interior Hölder regularity of viscosity solutions to a class of fractional double-phase problems. Recently, in [21], the authors of the present work have obtained almost optimal global Hölder continuity results for weak solutions to fractional (p, q)-problems in the superquadratic case ($p\ge q\ge 2$). To establish the optimal interior regularity result for local weak solutions (without the boundedness assumption), the authors proved a local boundedness result, which uses a new Caccioppoli-type inequality for non-homogeneous operators. Then, employing Moser’s iteration technique on the discrete differential of the solution and exploiting the local boundedness of the function f, they proved suitable Besov space inclusion (which in turn gives Hölder continuity). However, the approach fails when either of the exponent is less than 2. By establishing the control over the barrier functions involving the distance function, the authors further obtained almost optimal $d^{s_1}$ boundary behavior of the weak solution. This coupled with the interior Hölder regularity result proves the almost optimal $s_1$-Hölder continuity result globally in $\mathbb R^N$.

Concerning the Harnack-type inequality, in [39], Trudinger established Harnack- and weak Harnack-type estimates for weak solutions to general second-order quasilinear problems. As an application to these, the author obtains Hölder continuity for weak solutions.

However, in the non-local case, the classical Harnack inequality fails, see for instance [29]. Subsequently, in [28], the same author proved the Harnack inequality (where the non-negativity in the whole $\mathbb R^N$ is not assumed) with non-local tails for problems involving the fractional Laplacian. For general p, we refer to [18] for Harnack- and weak Harnack-type inequality for minimizers as well as weak solutions. In this paper, the authors have used suitable Caccioppoli-type estimates and a local boundedness result to achieve these aims.

Inspired from the above discussion, in this work, we answer the open question of Hölder regularity results for weak solutions to fractional (p, q)-problems, for the case $1<q< 2$ and $1<p<\infty$. We prove the interior regularity result for local weak solutions, as in Theorem 2.1, by establishing a weak Harnack-type inequality (see Proposition 4.1) and then we complete the proof of Hölder continuity in the spirit of [24, Theorem 2.10]. As noted earlier, for the homogeneous fractional p-Laplacian case with $p<2$, to prove the weak Harnack inequality, in [26], a suitably scaled version of the solution to the fractional torsion problem is used for appropriate subsolutions. Nevertheless, due to the lack of the scaling property, we can not rely on this method for our case. Moreover, we observe that by establishing the Logarithmic lemma (in accordance with [17, Lemma 1.3] and Lemma 3.1, below), we can not prove the Hölder continuity result by applying De Giorgi’s iteration technique as in the proof of [17, Theorem 1.2]. This is due to the presence of non-homogeneous power of the parameter $\lambda$ (taken in the place of d, there), which prevents from choosing the suitable constant k involved in the proof of [17, Lemma 5.1]. So, we follow the idea of [18] to prove our weak Harnack inequality. However, because of the non-homogeneous nature of the operator, we obtain two non-local tails with different behaviors, which require a technical care at several places. We overcome the difficulty raised by these non-local tails by further improving the parameters involved in the proofs (by adding an additional quantity of the form $r^{s_1p-s_2q}$). Our interior regularity result complements that of [17] to the case of non-homogeneous fractional (p, q)-problems with non-homogeneous right-hand side in $L^\gamma _{\mathrm{loc}}(\Omega )$, whereas in [21] in addition to $q\ge 2$, $f\in L^\infty _{\mathrm{loc}}(\Omega )$ is assumed. We strongly believe that this approach, while currently focused on a specific form of the operator, can be further applied for a wider class of non-local and non-homogeneous operators, see Remarks 2.2 and 2.4 in this regard. It is also worth noting that we do not assume any boundedness condition on the solution in our interior regularity result, thanks to Proposition 3.1, which extends that of [21, Proposition 3.2]. The proof of Proposition 3.1 relies on a Caccioppoli-type inequality for non-homogeneous operators (see [21, Lemma 3.1]) and De Giorgi’s type iteration argument. The main obstacle in the proof is due to $f\in L^\gamma _{\mathrm{loc}}(\Omega )$, for $\gamma <\infty$, as we can not use the Moser-type iteration argument available for the homogeneous case in [11]. In this paper, clever convexity arguments proved that $u_+$ is a sub-solution with right-hand side |f|, whenever u is a solution to (1.1) (with $p=q$ and $s_1=s_2$), but seem to be inefficient in the presence of non-homogeneous operators. Subsequently, using the boundary behavior of [21, Proposition 3.11], we obtain the global Hölder continuity result for weak solutions. Additionally, employing the approach of [18, Theorem 1.1] to our case, we obtain the Harnack inequality, as in Theorem 2.2. Similar to the homogeneous case, our Harnack inequality is valid for sign changing solutions also and the sign changing behavior is displayed in terms of the non-local tails. As an application to our global Hölder regularity, we obtain a strong maximum principle for fractional (p, q)-problems, valid for all $p\ge q>1$. We observe that the proof of [21, Theorem 2.6] can not be generalized to this case, as our weak Harnack-type inequality involves an additional constant term ($r^\frac{ps_1-qs_2}{p-q}$). So, we follow the approach of [16] and prove that continuous weak super-solutions are viscosity super-solutions of fractional (p, q)-problem with right-hand side zero. Subsequently, we apply a suitable barrier function and the weak comparison principle to extend this result for problems involving more general non-linearities (see Theorem 2.3).

2 Function Spaces and Main Results

We first fix some notations which will be used through out the paper. We set $t_\pm =\max \{\pm t,0\}$. We denote $[t]^{p-1}:=|t|^{p-2}t$, for all $p>1$ and $t\in \mathbb R$. Next, for $x_0\in \mathbb R^N$ and $v\in L^1(B_r(x_0))$, we set

The order pair $(\ell ,s)$ should always be considered as $(\ell ,s)\in \{(p,s_1), (q,s_2)\}$, unless otherwise mentioned.

The constants c and C may vary line to line.

For any $E\subset {\mathbb {R}}^N$, $1\le p<\infty$ and $0<s<1$, the fractional Sobolev space $W^{s,p}(E)$ is defined as

$$\begin{aligned} W^{s,p}(E):= \left\{ u \in L^p(E): [u]_{W^{s,p}(E)} < \infty \right\} \end{aligned}$$

endowed with the norm $\Vert u\Vert _{W^{s,p}(E)}:= \Vert u\Vert _{L^p(E)}+ [u]_{W^{s,p}(E)}$, where

$$\begin{aligned}{}[u]_{W^{s,p}(E)}:= \left( \int _{E}\int _{E} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}~dxdy \right) ^{1/p}. \end{aligned}$$

Next, for $1<p<\infty$ and for any (proper) open and bounded subset E of $\mathbb R^N$, we have

$$\begin{aligned} W^{s,p}_0(E):=\{ u\in W^{s,p}(\mathbb R^N) \ : \ u=0 \quad \text{ a.e. } \text{ in } \mathbb R^N\setminus E \} \end{aligned}$$

which is a uniformly convex Banach space when equipped with the norm $[\cdot ]_{W^{s,p}(\mathbb R^N)}$ (equivalent to $\Vert \cdot \Vert _{W^{s,p}(\mathbb R^N)}$, and we will denote it by $\Vert \cdot \Vert _{W^{s,p}_0(E)}$). When the domain E has Lipschitz boundary, $W^{s,p}_0(E)$ coincides with $X_{p,s}$ (as defined in [24]), and the following inclusion holds:

Lemma 2.1

[24, Lemma 2.1] Let $E\subset \mathbb R^N$ be a bounded domain with Lipschitz boundary. Let $1<q\le p<\infty$ and $0<s_2<s_1 <1$, then there exists a positive constant $C=C(|E|,\;N,\; p,\;q,\;s_1,\;s_2)$ such that

$$\begin{aligned} \Vert u\Vert _{W^{s_2,q}_0(E)}\le C \Vert u\Vert _{W^{s_1,p}_0(E)}, \quad \text {for all } \; u \in W^{s_1,p}_0(E). \end{aligned}$$

For any bounded open set $E\subset \mathbb R^N$ and $0<s_2<s_1<1$, an easy consequence of the Hölder inequality yields

$$\begin{aligned} \Vert u\Vert _{W^{s_2,q}(E)}\le C \Vert u\Vert _{W^{s_1,p}(E)}, \quad \text {for all } \; u \in W^{s_1,p}(E). \end{aligned}$$

In what follows, we will focus only on the case $s_1\ne s_2$ and work with the space $W^{s_1,p}$. The case $s_1=s_2=s$ can be treated similarly by considering the space $\mathcal W:= W^{s,p}(E)\cap W^{s,q}(E)$ (in place of $W^{s_1,p}(E)$), equipped with the norm $\Vert \cdot \Vert _{W^{s,p}(E)}+\Vert \cdot \Vert _{W^{s,q}(E)}$.

Definition 2.1

Let $u:\mathbb R^N\rightarrow \mathbb R$ be a measurable function, $0< m<\infty$ and $\theta >0$. We define the tail space as below:

$$\begin{aligned} L^{m}_{\theta }(\mathbb R^N) = \bigg \{ u\in L^{m}_{\mathrm{loc}}(\mathbb R^N) : \int _{\mathbb R^N} \frac{|u(x)|^{m}dx}{(1+|x|)^{N+\theta }} <\infty \bigg \}. \end{aligned}$$

The non-local tail of radius R and center $x_0\in \mathbb R^N$ is defined as

$$\begin{aligned} T_{m,\theta }(u;x_0,R)=\left( R^{\theta }\int _{B_R(x_0)^c} \frac{|u(y)|^{m}}{|x_0-y|^{N+\theta }} dy\right) ^{1/m}. \end{aligned}$$

For brevity, we set $T_{p-1}(u;x,R):=T_{p-1,s_1p}(u;x,R)$ and $T_{q-1}(u;x,R):=T_{q-1,s_2q}(u;x,R)$.

Now, we define the notion of a local weak solution to problem (1.1).

Definition 2.2

A function $u\in W^{s_1,p}_{\mathrm{loc}}(\Omega )\cap L^{p-1}_{s_1p}(\mathbb R^N) \cap L^{q-1}_{s_2q}(\mathbb R^N)$ is said to be a local weak solution of problem (1.1) if

$$\begin{aligned}&\int _{\mathbb R^N}\int _{\mathbb R^N}\frac{[u(x)-u(y)]^{p-1}}{|x-y|^{N+ps_1}}(\psi (x)-\psi (y))dxdy\\&\qquad + \int _{\mathbb R^N}\int _{\mathbb R^N}\frac{[u(x)-u(y)]^{q-1}}{|x-y|^{N+qs_2}}(\psi (x)-\psi (y))dxdy \\&\quad = \int _{\Omega } f\psi dx, \end{aligned}$$

for all $\psi \in W^{s_1,p}(\Omega )$ compactly supported in $\Omega$.

Let $\Omega \Subset \Omega '\subset \mathbb R^N$. For $g\in W^{s_1,p}(\Omega ')\cap L^{p-1}_{s_1p}(\mathbb R^N)\cap L^{q-1}_{s_2q}(\mathbb R^N)$, we consider the following prototype problem:

$$\begin{aligned} \left\{ \begin{array}{rlll} (-\Delta )^{s_1}_{p}u+ (-\Delta )^{s_2}_{q}u &{}= f \quad \text {in} \; \Omega , \\ u &{}=g \quad \text {in} \; \mathbb R^N\setminus \Omega . \end{array} \right. \qquad \qquad \qquad \qquad (\mathcal G_{f,g}(\Omega )) \end{aligned}$$

To study the weak solutions of ($\mathcal G_{f,g}(\Omega )$), we state the following:

Definition 2.3

Let $\Omega \Subset \Omega '\subset \mathbb R^N$ and $1<p<\infty$ with $s_1\in (0,1)$, then we define

$$\begin{aligned} X^{s_1,p}_g(\Omega ,\Omega '):= \{ v\in W^{s_1,p}(\Omega ')\cap L^{p-1}_{s_1p}(\mathbb R^N) : v=g \quad \text{ a.e. } \text{ in } \mathbb R^N\setminus \Omega \}, \end{aligned}$$

equipped with the norm of $W^{s_1,p}(\Omega ')$.

Definition 2.4

A function $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ is said to be a weak super-solution (resp. sub-solution) of ($\mathcal G_{f,g}(\Omega )$) if $(g-u)_+\in X^{s_1,p}_{0}(\Omega ,\Omega ') \cap X^{s_2,q}_{0}(\Omega ,\Omega ')$ (resp. $(u-g)_+\in X^{s_1,p}_{0}(\Omega ,\Omega ') \cap X^{s_2,q}_{0}(\Omega ,\Omega ')$) and the following holds:

$$\begin{aligned}&\int _{\mathbb R^N}\int _{\mathbb R^N}\frac{[u(x)-u(y)]^{p-1}}{|x-y|^{N+ps_1}}(\phi (x)-\phi (y))dxdy \nonumber \\&\qquad + \int _{\mathbb R^N}\int _{\mathbb R^N}\frac{[u(x)-u(y)]^{q-1}}{|x-y|^{N+qs_2}}(\phi (x)-\phi (y))dxdy \nonumber \\&\quad \ge (\mathrm{resp. }\le ) \int _{\Omega } f(x)\phi (x) dx, \end{aligned}$$

(2.1)

for all non-negative $\phi \in X^{s_1,p}_{0}(\Omega ,\Omega ') \cap X^{s_2,q}_{0}(\Omega ,\Omega ')$.

Our first main theorem is the local Hölder continuity result for local weak solutions to problem (1.1). Precisely, we have:

Theorem 2.1

Suppose $1<q<p<\infty$. Let $u\in W^{s_1,p}_{\mathrm{loc}}(\Omega )\cap L^{p-1}_{s_1p}(\mathbb R^N)\cap L^{q-1}_{s_2q}(\mathbb R^N)$ be a local weak solution to problem (1.1). Then, $u\in C^{0,\alpha }_{\mathrm{loc}}(\Omega )$, for some $\alpha \in (0,1)$ satisfying $\alpha <\min \big \{ \frac{\gamma s_1p-N}{\gamma (p-1)}, \frac{qs_2}{q-1}, \frac{ps_1-qs_2}{p-q} \big \}$. Moreover, for any $R_0\in (0,1)$ such that $B_{2R_0}\equiv B_{2R_0}(x_0)\Subset \Omega$, the following holds:

$$\begin{aligned} {[}u]_{C^{0,\alpha }(B_r)} \le {\frac{c}{R_0^\alpha }} \Big [&\Vert u \Vert _{L^\infty (B_{R_0})}+R_0^\frac{ps_1-qs_2}{p-q}+ R_0^\frac{\gamma s_1p-N}{\gamma (p-1)} \Vert f \Vert _{L^\gamma (B_{R_0})}^\frac{1}{p-1}+ T_{p-1}(u;x_0,R_0) \\&+ T_{q-1}(u;x_0,R_0)\Big ], \end{aligned}$$

for all $r\in (0, R_0)$, where $c=c(N,s_1,p,s_2,q)>0$ is a constant.

It is clear from the statement of Theorem 2.1 that the Hölder exponent $\alpha$ is not optimal. However, for the case $q\ge 2$ and $f\in L^\infty _{\mathrm{loc}}(\Omega )$, optimal value of $\alpha$ is mentioned in [21], whereas the optimality of $\alpha$ in the subquadratic case is still unknown, even for the homogeneous operator case, that is, $p=q$ and $s_1=s_2$.

Remark 2.1

• (a) For the case $q=p$ and $s_2< s_1$, the term $R_0^\frac{ps_1-qs_2}{p-q}$ does not appear in Theorem 2.1, and the proof runs analogously. Indeed, in all of the technical results of Section 3, the term $\lambda ^{q-p}r^{s_1p-s_2q}$ disappears because of the p-homogeneity.

• (b) For the case $1<p<q<\infty$ and $s_1p\le s_2 q$, we can proceed analogously by interchanging the role of $(p,s_1)$ with $(q,s_2)$, to get a similar result as in Theorem 2.1. However, for the case $s_1p>s_2 q$, we need to assume the boundedness of the weak solution in whole $\mathbb R^N$. Then, employing the Caccioppoli-type inequality [21, Lemma 3.1] and an analogous inequality of (3.3), we can proceed as in [17, Theorem 1.2] to get our interior Hölder continuity result.

Remark 2.2

As a matter of fact, the result of Theorem 2.1 is valid for equations of the type (1.1) involving a more general class of operators, for instance,

$$\begin{aligned} \mathcal L_{K_{\ell ,s}} u(x)= 2\displaystyle \lim _{\epsilon \rightarrow 0}\int _{\mathbb R^N\setminus B_\epsilon (x)} [u(x)-u(y)]^{\ell -1}K_{\ell ,s}(x,y)dy, \end{aligned}$$

where $(\ell ,s)\in \{(p,s_1),(q,s_2)\}$ with $1<q<p<\infty$ and $0<s_2\le s_1<1$. Here, the singular kernel $K_{\ell ,s}: \mathbb R^N\times \mathbb R^N\rightarrow [0,\infty )$ is such that

• (i) there exists $1\le c_p\le C_p$ satisfying $c_p \le K_{p,s_1}(x,y) |x-y|^{N+ps_1} \le C_p$, for a.a. $x,y\in \mathbb R^N$,

• (ii) there exists $0\le c_q\le C_q$ satisfying $c_q \le K_{q,s_2}(x,y)|x-y|^{N+qs_2} \le C_q$, for a.a. $x,y\in \mathbb R^N$.

For example, one can take $K_{\ell ,s}(x,y)= a_\ell (x,y) |x-y|^{-(N+\ell s)}$, where $a_\ell :\mathbb R^N\times \mathbb R^N\rightarrow \mathbb R$ are non-negative bounded functions, for $\ell \in \{p,q\}$, with $\inf _{\mathbb R^{N}\times \mathbb R^N}a_p\ge 1$.

Next, we establish the following boundary regularity result.

Corollary 2.1

(Boundary regularity) Let $u\in W^{s_1,p}_0(\Omega )$ be a weak solution to ($\mathcal G_{f,g}(\Omega )$) with $f\in L^\infty (\Omega )$ and $g\equiv 0$. Then, there exists $\alpha \in (0,1)$ such that $u\in C^{0,\alpha }(\overline{\Omega })$. Moreover,

$$\begin{aligned} \Vert u \Vert _{C^{0,\alpha }(\overline{\Omega })} \le C, \end{aligned}$$

(2.2)

where $C=C(\Omega ,N,s_1,p,s_2,q, \Vert f\Vert _{L^\infty (\Omega )})>0$ is a constant (which depends as a non-decreasing function of $\Vert f\Vert _{L^\infty (\Omega )}$).

Remark 2.3

As a consequence of Corollary 2.1, we have the boundary regularity result for the critical exponent problem. Precisely, let $u\in W^{s_1,p}_0(\Omega )$ be a solution to problem ($\mathcal G_{f,g}(\Omega )$) with $f(x):=f(x,u)$ and $g\equiv 0$, where f is a Carathéodory function such that $|f(x,t)|\le C_0 (1+|t|^{p^*_{s_1}-1})$, with $C_0 >0$, and $p^*_{s_1}:=Np/(N-ps_1)$ if $N>ps_1$, otherwise an arbitrarily large number. Then, $u\in C^{0,\alpha }(\overline{\Omega })$, for some $\alpha \in (0,s_1]$, and (2.2) holds. The proof follows by noting the fact that $u\in L^\infty (\Omega )$, and hence $f\in L^\infty (\Omega )$.

Remark 2.4

We further observe that for the case $0<s_2<s_1=1$, the problem (1.1) exhibits a non-homogeneous local–non-local behavior, see for instance [7, 13] for corresponding semilinear homogeneous counterparts. In this case, proceeding similarly we can prove an analogous result to Theorem 2.1. Subsequently, following the proofs of [22, Lemma 2.2] (without the approximation argument) and [21, Proposition 3.11], we obtain the boundary behavior of the weak solution. Consequently, we get the global Hölder continuity result as in Corollary 2.1.

As of independent interest, we have the following Harnack inequality for weak solutions.

Theorem 2.2

(Harnack inequality) Let $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ be a weak solution to problem ($\mathcal G_{f,g}(\Omega )$) such that $u\ge 0$ in $B_R\equiv B_R(x_0)\Subset \Omega$, for some $R\in (0,1)$. Then, for all $0<r<R$, the following holds:

$$\begin{aligned} \sup _{B_{r/2}} u&\le C \inf _{B_{r/2}} u+ C\Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1} T_{p-1}(u_-;x_0,R) + C r^\frac{s_1p-s_2q}{p-1} \Big (\frac{r}{R}\Big )^\frac{s_2q}{p-1} T_{q-1}(u_-;x_0,R)^\frac{q-1}{p-1} \\&\quad + C r^\frac{s_1p-s_2q}{p-q} +C\Vert f\Vert _{L^\gamma (B_R)}^{1/(p-1)}, \end{aligned}$$

where $C=C(N,p,q,s_1,s_2)>0$ is a constant.

Next, we have the following strong maximum principle.

Theorem 2.3

Suppose $1<q\le p<\infty$. Let $g\in C(\mathbb R)\cap BV_{\mathrm{loc}}(\mathbb R)$ and let $u\in W^{s_1,p}_0(\Omega )\cap C(\overline{\Omega })$ be such that

$$\begin{aligned} (-\Delta )_p^{s_1} u+ (-\Delta )_q^{s_2} u +g(u)\ge g(0) \quad \text{ weakly } \text{ in } \Omega . \end{aligned}$$

Further, assume that $u\not \equiv 0$ with $u\ge 0$ in $\Omega$. Then, there exists $c_1>0$ such that $u\ge c_1 \mathrm{dist}(\cdot ,\partial \Omega )^{s_1}$ in $\Omega$.

3 Some Technical Results

In this section, we prove some preliminary results such as Caccioppoli-type inequality, local boundedness result, and expansion of positivity result, which are required to prove our main theorems.

Lemma 3.1

Let $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ be a super-solution to problem ($\mathcal G_{f,g}(\Omega )$) such that $u\ge 0$ in $B_R\equiv B_R(x_0)\subset \Omega$, for $R\in (0,1)$. For $k\ge 0$, suppose that there exists $\nu \in (0,1]$ such that

$$\begin{aligned} \frac{|B_r \cap \{ u \ge k \}|}{|B_r|} \ge \nu \quad \text{ for } \text{ all } r\in (0,R/16). \end{aligned}$$

(3.1)

Then, there exists a constant $c=c(N,s_1,p,s_2,q)>0$ such that, for all $\delta \in (0,1/4)$:

$$\begin{aligned} \bigg |B_{6r} \cap&\left\{ u \le 2\delta k-\frac{1}{2} \Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1}T_{p-1}(u_-;x_0,R) -\frac{1}{2}\Big (\frac{r}{R}\Big )^\frac{s_2q}{q-1}T_{q-1}(u_-;x_0,R) -\frac{1}{2} r^\frac{s_1p-s_2q}{p-q} \right. \\&\left. -\frac{1}{2} r^\frac{\gamma s_1p-N}{\gamma (p-1)} \Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1} \right\} \bigg | \le \frac{c \; |B_{6r}|}{\nu \log (\frac{1}{2\delta })}. \end{aligned}$$

Proof

Let $\lambda >0$ be a parameter (to be chosen later) and set ${\bar{u}}= u+\lambda$. Then, ${\bar{u}}$ is still a super-solution to ($\mathcal G_{f,g}(\Omega )$). Let $\phi \in C_c^\infty (B_{7r})$ be such that $0\le \phi \le 1$, $|\nabla \phi |\le c/r$ in $B_{7r}$ and $\phi \equiv 1$ in $B_{6r}$. We take ${\bar{u}}^{1-p}\phi ^p$ as a test function in the weak formulation (2.1), to get

$$\begin{aligned} 0&\le \sum _{(\ell ,s)}\int _{B_{8r}} \int _{B_{8r}} \frac{[{{\bar{u}}}(x)-{{\bar{u}}}(y)]^{\ell -1}}{|x-y|^{N+s\ell }}\left[ \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{p-1}} -\frac{\phi ^p(y)}{{{\bar{u}}}(y)^{p-1}}\right] dxdy \\&\quad + 2 \sum _{(\ell ,s)}\int _{{\mathbb R^N}\setminus B_{8r}}\int _{B_{8r}} \frac{[{{\bar{u}}}(x)-{{\bar{u}}}(y)]^{\ell -1}}{|x-y|^{N+s\ell }} \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{p-1}}dxdy - \int _{\Omega } f(x) {\bar{u}}^{1-p}(x)\phi ^p(x)dx \\&=:J_1(\ell )+J_2(\ell )-J_3(f). \end{aligned}$$

We estimate the quantities $J_1$, $J_2$, and $J_3$ in the following steps:

Step I: Estimate of $J_1(p)$.

Following the proof of [17, Lemma 1.3, (3.12) and (3.17)], we have

$$\begin{aligned} J_1(p)&\le -c \int _{B_{8r}} \int _{B_{6r}} |x-y|^{-N-s_1p} \bigg |\log \left( \frac{{{\bar{u}}}(x)}{{{\bar{u}}}(y)}\right) \bigg |^p dxdy + c \int _{B_{8r}} \int _{B_{8r}} \frac{|\phi (x)-\phi (y)|^p}{|x-y|^{N+s_1p}} dxdy \\&\le -c \int _{B_{8r}} \int _{B_{6r}} |x-y|^{-N-s_1p} \bigg |\log \left( \frac{{{\bar{u}}}(x)}{{{\bar{u}}}(y)}\right) \bigg |^p dxdy + c r^{N-s_1p}. \end{aligned}$$

Step II: Estimate of $J_1(q)$.

Without loss of generality, we assume that $u(x)> u(y)$. Recall the following inequality [17, Lemma 3.1]: For $p\ge 1$, $\epsilon \in (0,1]$ and for all $a,b\in \mathbb R^N$, we have

$$\begin{aligned} |a|^p\le (1+c_p\epsilon ) |b|^p + (1+c_p\epsilon ) \epsilon ^{1-p} |a-b|^p, \quad \text{ where } c_p:=(p-1)\Gamma (\max \{1,p-2\}). \end{aligned}$$

(3.2)

For the choice $p=q$, $a=\phi ^{p/q}(x)$, $b=\phi ^{p/q}(y)$ together with

$$\begin{aligned} \epsilon = t\frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} \in (0,1) \quad \text{ with } t\in (0,1), \end{aligned}$$

in (3.2) (note that $u\ge 0$ in $B_{8r}$), we obtain

$$\begin{aligned} \phi ^p(x)&\le \left( 1+c t\frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)}\right) \phi ^{p}(y) \\&\quad + (1+c \epsilon )t^{1-q} \left( \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)}\right) ^{1-q} |\phi ^{p/q}(x)-\phi ^{p/q}(y)|^q. \end{aligned}$$

This implies that

$$\begin{aligned} ({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1} \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{p-1}} \le&\left( 1+c t \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} \right) \frac{({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1}}{{{\bar{u}}}(x)^{p-1}} \phi ^p(y) \\&+(1+c) t^{1-q}{{\bar{u}}}(x)^{q-p}|\phi ^{p/q}(x)-\phi ^{p/q}(y)|^q. \end{aligned}$$

Now, using the relation $|\phi (x)^{p/q}-\phi (y)^{p/q}| \le \frac{p}{q}(\phi (x)^{p/q}+\phi (y)^{p/q})^{(p-q)/p} |\phi (x)-\phi (y)|$, we get

$$\begin{aligned}&({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1} \left[ \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{p-1}} - \frac{\phi ^p(y)}{{{\bar{u}}}(y)^{p-1}}\right] \\&\quad \le \phi ^p(y) \frac{({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1}}{{{\bar{u}}}(x)^{p-1}} \left[ 1+c t \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} -\frac{{{\bar{u}}}(x)^{p-1}}{{{\bar{u}}}(y)^{p-1}} \right] \\&\qquad +c t^{1-q} \lambda ^{q-p} |\phi (x)-\phi (y)|^q, \end{aligned}$$

where in the last inequality we have used the relation ${{\bar{u}}}\ge \lambda$ in $B_{R}$. The terms inside the bracket on the right-hand side of the above expression is similar to the one of [17, (3.6)]. Therefore, (see [17, (3.9)] and the expression for $g(z)\le -p+1$ there)

$$\begin{aligned} \left[ 1+c t \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} -\frac{{{\bar{u}}}(x)^{p-1}}{{{\bar{u}}}(y)^{p-1}} \right] \le 0. \end{aligned}$$

And an analogous result holds for the case $u(x)<u(y)$. Thus

$$\begin{aligned} J_1(q) \le c \lambda ^{q-p} \int _{B_{8r}} \int _{B_{8r}} |x-y|^{-N-s_2q} |\phi (x)-\phi (y)|^q dxdy \le c \lambda ^{q-p} r^{N-s_2q}. \end{aligned}$$

Step III: Estimate of $J_2(p)$.

From [18, Lemma 3.1, p.1818], we have

$$\begin{aligned} J_2(p)&\le c r^{N-s_1p}+ c \lambda ^{1-p} r^N R^{-s_1p} T_{p-1}(u_-;x_0,R)^{p-1}. \end{aligned}$$

Step IV: Estimate of $J_2(q)$.

Noting $u\ge 0$ in $B_R$, we have

$$\begin{aligned}&\frac{u(x)-u(y)}{u(x)+\lambda } \le 1 \text{ for } \text{ all } x\in B_{8r}, \ y\in B_R, \\&(u(x)-u(y))_+^{q-1} \le 2^{q-1} [ u(x)^{q-1}+(u(y))_-^{q-1} ] \ \text{ for } \text{ all } x\in B_{8r}, \ \ y\in \mathbb R^N\setminus B_{R}. \end{aligned}$$

Therefore,

$$\begin{aligned}&J_2(q)\\&\le 2 \left( \int _{B_R\setminus B_{8r}} \int _{B_{8r}} + \int _{\mathbb R^N\setminus B_{R}} \int _{B_{8r}} \right) |x-y|^{-N-s_2q} \frac{(u(x)-u(y))_+^{q-1}}{(u(x)+\lambda )^{p-1}}\phi ^p(x) dxdy \\&\le \lambda ^{q-p} \int _{\mathbb R^N\setminus B_{8r}} \int _{B_{8r}} \frac{\phi ^p(x)}{|x-y|^{N+s_2q}}dxdy + c \int _{\mathbb R^N\setminus B_{R}} \int _{B_{8r}} \frac{u(x)^{q-1}+(u(y))_-^{q-1}}{(u(x)+\lambda )^{p-1}}\frac{\phi ^p(x)dxdy}{|x-y|^{N+s_2q}} \\&\le \lambda ^{q-p} \int _{\mathbb R^N\setminus B_{8r}} \int _{B_{7r}} \frac{\phi ^p(x)}{|x-y|^{N+s_2q}}dxdy + c \lambda ^{1-p} \int _{\mathbb R^N\setminus B_{R}} \int _{B_{7r}} \frac{(u(y))_-^{q-1}\phi ^p(x)}{|x-y|^{N+s_2q}} dxdy \\&\le cr^{N}\lambda ^{q-p} \int _{\mathbb R^N\setminus B_{8r}} \frac{1}{|x_0-y|^{N+s_2q}}dy + c r^N \lambda ^{1-p} \int _{\mathbb R^N\setminus B_{R}} \frac{(u(y))_-^{q-1}}{|x_0-y|^{N+s_2q}} dy, \end{aligned}$$

that is,

$$\begin{aligned} J_2(q)\le \lambda ^{q-p} c r^{N-s_2q} + c r^N \lambda ^{1-p} R^{-s_2q} T_{q-1}(u_-;x_0,R)^{q-1}. \end{aligned}$$

Step V: Estimate of $J_3(f)$.

Noting the bounds on ${{\bar{u}}}$ and $\phi$, we have

$$\begin{aligned} |-J_3(f)|\le \int _{\Omega } |f(x)| {\bar{u}}^{1-p}(x)\phi ^p(x)dx \le \lambda ^{1-p} \int _{B_{7r}} |f|\phi ^p&\le c \lambda ^{1-p}\Vert f \Vert _{L^\gamma (B_R)} r^{N/\gamma '}. \end{aligned}$$

Combining the estimates of steps I through V, we obtain

$$\begin{aligned}&\int _{B_{6r}} \int _{B_{6r}} \bigg |\log \left( \frac{{{\bar{u}}}(x)}{{{\bar{u}}}(y)}\right) \bigg |^p\nonumber \\&\quad \frac{dxdy}{|x-y|^{N+s_1p} }\nonumber \\&\quad \le c r^{N-s_1p} + c \lambda ^{q-p} r^{N-s_2q}+c \lambda ^{1-p} r^N R^{-s_1p} T_{p-1}(u_-;x_0,R)^{p-1} \nonumber \\&\qquad +c r^N \lambda ^{1-p} R^{-s_2q} T_{q-1}(u_-;x_0,R)^{q-1}+c \lambda ^{1-p}\Vert f \Vert _{L^\gamma (B_R)} r^{N/\gamma '}. \end{aligned}$$

(3.3)

Next, we take

$$\begin{aligned} \lambda := \Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1}T_{p-1}(u_-;x_0,R) +\Big (\frac{r}{R}\Big )^\frac{s_2q}{q-1}T_{q-1}(u_-;x_0,R) +r^\frac{s_1p-s_2q}{p-q}+\Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1} r^\frac{\gamma s_1p-N}{\gamma (p-1)} \end{aligned}$$

(3.4)

and note that $(\gamma s_1p-N)/\gamma =N/\gamma '-N+s_1p > 0$ (thanks to the assumption on $\gamma$). Moreover, by noting $\lambda \ge r^\frac{s_1p-s_2q}{p-q}$, we observe that

$$\begin{aligned} \lambda ^{1-p} r^N R^{-s_2q} T_{q-1}(u_-;x_0,R)^{q-1}&= r^{N-s_1p}\lambda ^{1-q} \Big (\frac{r}{R}\Big )^{s_2q} T_{q-1}(u_-;x_0,R)^{q-1} \lambda ^{q-p} r^{s_1p-s_2q} \\&\le r^{N-s_1p}. \end{aligned}$$

Therefore, from (3.3), taking into account (3.4), we get

$$\begin{aligned} \int _{B_{6r}} \int _{B_{6r}} |x-y|^{-N-s_1p} \bigg |\log \left( \frac{{{\bar{u}}}(x)}{{{\bar{u}}}(y)}\right) \bigg |^p dxdy \le c r^{N-s_1p}. \end{aligned}$$

(3.5)

Next, for any $\delta \in (0,1/4)$, we set

$$\begin{aligned} v:= \bigg [\min \bigg \{ \log \frac{1}{2\delta }, \log \frac{k+\lambda }{{{\bar{u}}}} \bigg \}\bigg ]_+. \end{aligned}$$

Noting that v is a truncation of the sum of a constant and $\log {{\bar{u}}}$, and using the fractional Poincaré-type inequality, we have

where in the last inequality, we have used (3.5). This, on using Hölder’s inequality, yields

(3.6)

By the definition of v, we observe that

$$\begin{aligned} \{v=0\}=\{{{\bar{u}}} \ge k+\lambda \}=\{ u\ge k\}. \end{aligned}$$

Hence, from the assumption of the lemma (see (3.1)), we have

$$\begin{aligned} \frac{|B_{6r} \cap \{ v=0 \}|}{|B_{6r}|} \ge \frac{\nu }{6^N}. \end{aligned}$$

Therefore,

$$\begin{aligned} \log \frac{1}{2\delta }=\frac{1}{| B_{6r}\cap \{ v=0 \}|}\int _{B_{6r}\cap \{ v=0 \}}\log \frac{1}{2\delta } dx&\le \frac{6^N}{\nu }\frac{1}{ |B_{6r}|} \int _{ B_{6r}} \Big (\log \frac{1}{2\delta }-v(x)\Big )dx \\&=\frac{6^N}{\nu } \bigg [\log \frac{1}{2\delta }-(v)_{B_{6r}}\bigg ], \end{aligned}$$

which upon integration and using (3.6) implies that

$$\begin{aligned} \frac{|B_{6r}\cap \{ v=\log \frac{1}{2\delta } \}|}{|B_{6r}|}\log \frac{1}{2\delta }&\le \frac{6^N}{\nu |B_{6r}|}\int _{B_{6r}\cap \{ v=\log \frac{1}{2\delta } \}} [\log \frac{1}{2\delta }-(v)_{B_{6r}}] dx \\&\le \frac{2}{|B_{6r}|}\int _{B_{6r}}|v(x)-(v)_{B_{6r}}|dx \le C. \end{aligned}$$

Again, in view of the definition of v, for all $\delta \in (0,1/4)$, we get

$$\begin{aligned} \frac{|B_{6r}\cap \{ {{\bar{u}}} \le 2\delta (k+\lambda ) \}|}{|B_{6r}|} \le \frac{c}{\nu }\frac{1}{\log \frac{1}{2\delta }}. \end{aligned}$$

This yields the required result of the lemma upon using the fact that ${{\bar{u}}}= u+\lambda$, where $\lambda$ is given by (3.4). $\square$

By slightly modifying the proof of Lemma 3.1, we have the following result.

Lemma 3.2

Suppose $1<q\le p<\infty$. Let $u\in W^{s_1,p}(\Omega )\cap L^\infty _{\mathrm{loc}}(\Omega )$ be a weak super-solution to $(-\Delta )_p^{s_1} u+ (-\Delta )_q^{s_2} u\ge 0$ in $\Omega$ such that $u\ge 0$ in $B_R\equiv B_R(x_0)\subset \Omega$, for some $R\in (0,1)$. Then, for all $r\in (0,R/4)$ and $\lambda >0$, the following holds:

(3.7)

Proof

Set $\overline{u}=u+\lambda$ and take $\overline{u}(x)^{1-q}\phi (x)^p$ as a test function, where $\phi \in C_c^\infty (B_{3r/2})$ is such that $0\le \phi \le 1$, $|\nabla \phi |\le c/r$ in $B_{3r/2}$ and $\phi \equiv 1$ in $B_r$. Thus,

$$\begin{aligned} 0&\le \sum _{(\ell ,s)}\int _{B_{2r}} \int _{B_{2r}} \frac{[{{\bar{u}}}(x)-{{\bar{u}}}(y)]^{\ell -1}}{|x-y|^{N+s\ell }}\left[ \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{q-1}}-\frac{\phi ^p(y)}{{{\bar{u}}}(y)^{q-1}}\right] dxdy \nonumber \\&\quad + 2 \sum _{(\ell ,s)}\int _{\mathbb R^N\setminus B_{2r}}\int _{B_{2r}} \frac{[{{\bar{u}}}(x)-{{\bar{u}}}(y)]^{\ell -1}}{|x-y|^{N+s\ell }} \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{q-1}}dxdy \nonumber \\&=:I_1(\ell )+I_2(\ell ). \end{aligned}$$

(3.8)

We estimate the quantities $I_1$ and $I_2$ in the following steps:

Step I: Estimate of $I_1(p)$.

Without loss of generality, we assume $u(x)> u(y)$. Employing (3.2), for the choice $a=\phi (x)$, $b=\phi (y)$ together with

$$\begin{aligned} \epsilon = t\frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} \in (0,1) \quad \text{ with } t\in (0,1), \end{aligned}$$

(note that $u\ge 0$ in $B_{2r}$), we obtain (upon simplification)

$$\begin{aligned}&({{\bar{u}}}(x)-{{\bar{u}}}(y))^{p-1} \left[ \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{q-1}} - \frac{\phi ^p(y)}{{{\bar{u}}}(y)^{q-1}}\right] \\&\quad \le \phi ^p(y) \frac{({{\bar{u}}}(x)-{{\bar{u}}}(y))^{p-1}}{{{\bar{u}}}(x)^{q-1}} \left[ 1+c t \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} -\frac{{{\bar{u}}}(x)^{q-1}}{{{\bar{u}}}(y)^{q-1}} \right] \\&\quad +c t^{1-p} (u(x)+\lambda )^{p-q} |\phi (x)-\phi (y)|^p. \end{aligned}$$

Therefore, as in Lemma 3.1 (the first term on the r.h.s. is non-positive), we get

$$\begin{aligned} I_1(p)\le c \int _{B_{2r}}\int _{B_{2r}} (u(x)+\lambda )^{p-q} \frac{|\phi (x)-\phi (y)|^p}{|x-y|^{N+s_1p}}dxdy\le c(\Vert u\Vert _{L^\infty (B_R)}+\lambda )^{p-q} r^{N-s_1p}. \end{aligned}$$

(3.9)

Step II: Estimate of $I_1(q)$.

From (3.2) for the choice $p=q$, $a=\phi ^{p/q}(x)$, $b=\phi ^{p/q}(y)$ together with $\epsilon$ as in step I, we get

$$\begin{aligned}&({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1} \left[ \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{q-1}} - \frac{\phi ^p(y)}{{{\bar{u}}}(y)^{q-1}}\right] \\&\quad \le \phi ^p(y) \frac{({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1}}{{{\bar{u}}}(x)^{q-1}} \left[ 1+c t \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} -\frac{{{\bar{u}}}(x)^{q-1}}{{{\bar{u}}}(y)^{q-1}} \right] \\&\quad +c t^{1-q} |\phi ^{p/q}(x)-\phi ^{p/q}(y)|^q. \end{aligned}$$

Following the proof of [17, Lemma 1.3, (3.12) and (3.17)], we have

$$\begin{aligned} I_1(q)\le -c_2 \int _{B_{2r}}\int _{B_{2r}} \bigg |\log \frac{u(x)+\lambda }{u(y)+\lambda }\bigg |^q\frac{\phi ^{p}(y)dxdy}{|x-y|^{N+s_2q}}+ c \int _{B_{2r}}\int _{B_{2r}} \frac{|\phi (x)-\phi (y)|^q}{|x-y|^{N+s_2q}}dxdy. \end{aligned}$$

(3.10)

Step III: Estimate of $I_2(\ell )$.

On a similar note to Lemma 3.1, we can prove the following:

$$\begin{aligned} \begin{aligned}&I_2(p)\le c r^{N-s_1p} (\Vert u\Vert _{L^\infty (B_R)}+\lambda )^{p-q}+ c \lambda ^{1-q} r^N R^{-s_1p} T_{p-1}(u_-;R)^{p-1} \quad \text{ and } \\&I_2(q)\le c r^{N-s_2q} + c \lambda ^{1-q} r^N R^{-s_2q}T_{q-1}(u_-;R)^{q-1}. \end{aligned} \end{aligned}$$

(3.11)

Combining (3.9), (3.10), and (3.11) with (3.8), and noting $\phi \equiv 1$ in $B_r$, we get the required result of the lemma. $\square$

Now, we have the expansion of positivity result as below.

Lemma 3.3

Suppose that the hypotheses of Lemma 3.1 hold true. Then, there exists a constant $\delta =\delta (N,s_1,p,s_2,q)\in (0,1/4)$ such that

$$\begin{aligned} \inf _{B_{4r}} u \ge&\delta k - r^\frac{\gamma s_1p-N}{\gamma (p-1)} \Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1}- r^\frac{s_1p-s_2q}{p-q}-\sum _{(\ell ,s)} \Big (\frac{r}{R}\Big )^\frac{s\ell }{\ell -1}T_{\ell -1}(u_-;x_0,R). \end{aligned}$$

Proof

Due to the fact $u\ge 0$ in $B_R$, we may assume that (otherwise there is noting to prove)

$$\begin{aligned} \Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1}T_{p-1}(u_-;x_0,R) +\Big (\frac{r}{R}\Big )^\frac{s_2q}{q-1}T_{q-1}(u_-;x_0,R)+ r^\frac{s_1p-s_2q}{p-q}+ r^\frac{\gamma s_1p-N}{\gamma (p-1)} \Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1}\le \delta k. \end{aligned}$$

(3.12)

Set $w_-=(b-u)_+$, for $b>0$. For any $r \le \rho \le 6r$ and fixed $\psi \in C_c^\infty (B_\rho )$ with $0\le \psi \le 1$, we take $w_-\psi ^p$ as a test function in the weak formulation, to get

$$\begin{aligned} 0&\le \sum _{(\ell ,s)} \int _{B_\rho }\int _{B_\rho } \frac{[u(x)-u(y)]^{\ell -1} (w_-(x)\psi (x)^p-w_-(y)\psi (y)^p) }{|x-y|^{N+\ell s}} dxdy \\&\quad +2\sum _{(\ell ,s)} \int _{\mathbb R^N\setminus B_\rho }\int _{B_\rho } \frac{[u(x)-u(y)]^{\ell -1}w_-(x)\psi (x)^p}{|x-y|^{N+\ell s}} dxdy-\int _{\Omega } f(x)w_-(x)\psi ^p(x)dx \\&=:I_1(\ell )+I_2(\ell )-I_3(f). \end{aligned}$$

To estimate $I_1(\ell )$, we first observe that

$$\begin{aligned}&[u(x)-u(y)]^{\ell -1} (w_-(x)\psi (x)^p-w_-(y)\psi (y)^p)\\&\quad \le -[w_-(x)-w_-(y)]^{\ell -1} (w_-(x)\psi (x)^p-w_-(y)\psi (y)^p),\end{aligned}$$

for all $x,y\in B_\rho$. Therefore, proceeding similarly to the proof of [21, Lemma 3.1], we have

$$\begin{aligned} I_1(\ell )&\le -\frac{1}{4}\int _{B_\rho }\int _{B_\rho } \frac{|w_-(x)-w_-(y)|^l}{|x-y|^{N+s\ell }} \big (\psi (x)^p+\psi (y)^p\big ) dxdy \\&\quad +C \int _{B_\rho }\int _{B_\rho } \frac{|\psi (x)-\psi (y)|^\ell }{|x-y|^{N+s\ell }} \big (w_-(x)^p+w_-(y)^p\big )dxdy. \end{aligned}$$

Now, using

$$\begin{aligned} |w_-(x)\psi (x) - w_-(y)\psi (y)|^p&\le 2^{p-1} |w_-(x) - w_-(y)|^p (\psi (x)^p+\psi (y)^p) \\&\quad + 2^{p-1} |\psi (x)-\psi (y)|^{p}(w_-(x)+w_-(y))^p, \end{aligned}$$

we obtain

$$\begin{aligned} I_1(p)+I_1(q)&\le -c \int _{B_\rho }\int _{B_\rho } \frac{|w_-(x)\psi (x) - w_-(y)\psi (y)|^p}{|x-y|^{N+s_1p}} dxdy \nonumber \\&\quad + C\sum _{(\ell ,s)} \int _{B_\rho }\int _{B_\rho } \frac{|\psi (x)-\psi (y)|^\ell }{|x-y|^{N+s\ell }} \big (w_-(x)^p+w_-(y)^p\big )dxdy. \end{aligned}$$

(3.13)

Next, we estimate $I_2$, similarly to [18, Lemma 3.2], as below

$$\begin{aligned} I_2(\ell )&= \left( \int _{(\mathbb R^N\setminus B_\rho )\cap \{u(y)<0\} }\int _{B_\rho } + \int _{(\mathbb R^N\setminus B_\rho )\cap \{u(y)\ge 0\} }\int _{B_\rho } \right) \frac{[u(x)-u(y)]^{\ell -1}w_-(x)\psi (x)^p}{|x-y|^{N+\ell s}} dxdy \\&=: I_{2,1}+ I_{2,2}. \end{aligned}$$

First we note that

$$\begin{aligned} \frac{[u(x)-u(y)]^{\ell -1}}{|x-y|^{N+s\ell }} w_-(x)\psi (x)^p \le (b+u(y)_-)^{\ell -1} b \Big (\sup _{x\in \mathrm{supp}\psi } |x-y|^{-N-s\ell } \Big ) \chi _{B_\rho \cap \{u<b\}}(x). \end{aligned}$$

Therefore,

$$\begin{aligned} I_{2,1}(\ell ) \le b \Big (\sup _{x\in \mathrm{supp}\psi } \int _{\mathbb R^N\setminus B_\rho } (b+u(y)_-)^{\ell -1}|x-y|^{-N-s\ell }dy \Big ) |B_\rho \cap \{u<b\}|. \end{aligned}$$

For $I_{2,2}$, we observe that $u\ge 0$ in $B_\rho$, thus proceeding similarly, we obtain

$$\begin{aligned} I_2(\ell )=I_{2,1}(\ell )+I_{2,2}(\ell ) \le c b \Big (\sup _{x\in \mathrm{supp}\psi } \int _{\mathbb R^N\setminus B_\rho } (b+u(y)_-)^{\ell -1}|x-y|^{-N-s\ell }dy \Big ) |B_\rho \cap \{u<b\}|. \end{aligned}$$

(3.14)

For $I_3(f)$, by observing that $w_-\le b$ and applying Hölder’s inequality, we have

$$\begin{aligned} |-I_3(f)| \le b \Vert f \Vert _{L^\gamma (B_R)} |B_\rho \cap \{u<b\}|^{1/\gamma '}. \end{aligned}$$

(3.15)

Therefore, taking into account (3.12), (3.13), (3.14), and (3.15), we deduce that

$$\begin{aligned}&\int _{B_\rho }\int _{B_\rho } \frac{|w_-(x)\psi (x) - w_-(y)\psi (y)|^p}{|x-y|^{N+s_1p}}dxdy \nonumber \\&\le C\sum _{(\ell ,s)} \int _{B_\rho }\int _{B_\rho } \frac{|\psi (x)-\psi (y)|^\ell }{|x-y|^{N+s\ell }} \big (w_-(x)^p+w_-(y)^p\big )dxdy \nonumber \\&\quad + Cb \sum _{(\ell ,s)} \Big (\sup _{x\in \mathrm{supp}\psi }\nonumber \\&\quad \int _{\mathbb R^N\setminus B_\rho } \frac{(b+u(y)_-)^{\ell -1}}{|x-y|^{N+s\ell }}dy \Big )|B_\rho \cap \{u<b\}| \nonumber \\&\quad +b \Vert f \Vert _{L^\gamma (B_R)} |B_\rho \cap \{u<b\}|^\frac{1}{\gamma '}. \end{aligned}$$

(3.16)

Now, we start the iteration process to conclude the proof. Let

$$\begin{aligned} b\equiv b_j:= \delta k + 2^{-j-1}\delta k, \ \rho \equiv \rho _j:= 4r+2^{1-j}r \ \text{ and } {\bar{\rho }}_j=\frac{\rho _j+\rho _{j+1}}{2}, \end{aligned}$$

for all $j\in \mathbb N\cup \{0\}$. Observe that $\rho _j$ and ${\bar{\rho }}_j\in (4r,6r)$ and

$$\begin{aligned} b_j - b_{j+1} = 2^{-j-2}\delta k \ge 2^{-j-3}b_j \quad \text{ for } \text{ all } j\in \mathbb N\cup \{0\}. \end{aligned}$$

On account of (3.12), we have

$$\begin{aligned} b_0=\frac{3}{2}\delta k\le&2\delta k -\frac{1}{2} \Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1}T_{p-1}(u_-;x_0,R) -\frac{1}{2}\Big (\frac{r}{R}\Big )^\frac{s_2q}{q-1}T_{q-1}(u_-;x_0,R)-\frac{1}{2} r^\frac{s_1p-s_2q}{p-q} \\&-\frac{1}{2} \Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1} r^\frac{\gamma s_1p-N}{\gamma (p-1)} , \end{aligned}$$

which implies that

$$\begin{aligned} \{ u<b_0\} \subset&\left\{ u < 2\delta k-\frac{1}{2} \Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1}T_{p-1}(u_-;x_0,R) -\frac{1}{2}\Big (\frac{r}{R}\Big )^\frac{s_2q}{q-1}T_{q-1}(u_-;x_0,R)\right. \\&\left. \quad -\frac{1}{2} r^\frac{s_1p-s_2q}{p-q} -\frac{1}{2} \Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1} r^\frac{\gamma s_1p-N}{\gamma (p-1)} \right\} . \end{aligned}$$

Therefore, Lemma 3.1 gives us

$$\begin{aligned} \frac{|B_{6r} \cap \{ u <b_0 \}|}{|B_{6r}|}\le \frac{\tilde{c}}{\nu \log (\frac{1}{2\delta })}. \end{aligned}$$

(3.17)

Set $w_-\equiv w_j=(b_j-u)_+$ and $B_j:= B_{\rho _j}(x_0)$. Then, we have

$$\begin{aligned} w_j \ge (b_j-b_{j+1}) \chi _{\{u<b_{j+1}\}} \ge 2^{-j-3} b_j \chi _{\{u<b_{j+1}\}}. \end{aligned}$$

Let $\psi _j\in C_c^\infty (B_{{\bar{\rho }}_j})$ be such that $0\le \psi _j \le 1$, $|\nabla \psi _j|\le 2^{j+3}/r$ in $B_{{\bar{\rho }}_j}$ and $\psi _j\equiv 1$ in $B_{j+1}$. Using the fractional Poincaré inequality to the function $w_j\psi _j$, we get

(3.18)

Now, we estimate the right-hand side quantity of (3.18) with the help of (3.16). For the first term in (3.16), we see that

$$\begin{aligned}&\int _{B_j}\int _{B_j} \frac{|\psi _j(x)-\psi _j(y)|^\ell }{|x-y|^{N+s\ell }} \big (w_j(x)^p+w_j(y)^p\big )dxdy \nonumber \\&\quad \le cb_j^\ell \Vert \nabla \psi _j\Vert _{L^\infty }^\ell \int _{B_j}\int _{B_j\cap \{u<b_j\}} |x-y|^{\ell -N-s\ell }dxdy \nonumber \\&\quad \le c 2^{j\ell } b_j^{\ell }r^{-s\ell }|B_j\cap \{u<b_j\}|. \end{aligned}$$

(3.19)

For the second term, we observe that $\mathrm{supp}\psi _j \subset B_{{\bar{\rho }}_j}$. Therefore, for all $y\in \mathbb R^N\setminus B_j$, we have

$$\begin{aligned} \sup _{x\in \mathrm{supp}\psi _j} |x-y|^{-N-s\ell } \le c2^{j(N+s\ell )}|y-x_0|^{N+s\ell }. \end{aligned}$$

This implies that

$$\begin{aligned} \sup _{x\in \mathrm{supp}\psi _j} \int _{\mathbb R^N\setminus B_j} \frac{(b_j+u(y)_-)^{\ell -1}}{|x-y|^{N+s\ell }}dy&\le c 2^{j(N+s\ell )} \int _{\mathbb R^N\setminus B_j} \frac{(b_j+u(y)_-)^{\ell -1}}{|x_0-y|^{N+s\ell }}dy \\&\le c 2^{j(N+s\ell )}\left[ b_j^{\ell -1} r^{-s\ell }+ \int _{\mathbb R^N\setminus B_R} \frac{(u(y)_-)^{\ell -1}}{|x_0-y|^{N+s\ell }}dy\right] \\&= c 2^{j(N+s\ell )} r^{-s\ell } \left[ b_j^{\ell -1}+ \Big (\frac{r}{R}\Big )^{s\ell } T_{l-1}(u_-;x_0,R)^{\ell -1}\right] , \end{aligned}$$

where we have used the fact that $u\ge 0$ in $B_R$. For $(\ell ,s)=(p,s_1)$, we have $b_j>\delta k\ge \big (\frac{r}{R}\big )^{s_1p/(p-1)} T_{p-1}(u_-;x_0,R)$ (thanks to (3.12)), which yields

$$\begin{aligned} \sup _{x\in \mathrm{supp}\psi _j} \int _{\mathbb R^N\setminus B_j} \frac{(b_j+u(y)_-)^{p-1}}{|x-y|^{N+s_1p}}dy \le c 2^{j(N+s_1p)} r^{-s_1p} b_j^{p-1}. \end{aligned}$$

(3.20)

For $(\ell ,s)=(q,s_2)$, on account of (3.12), we have $b_j>\delta k\ge \big (\frac{r}{R}\big )^{s_2q/(q-1)} T_{q-1}(u_-;x_0,R)$. Thus,

$$\begin{aligned} \sup _{x\in \mathrm{supp}\psi _j} \int _{\mathbb R^N\setminus B_j} \frac{(b_j+u(y)_-)^{q-1}}{|x-y|^{N+s_2q}}dy \le c 2^{j(N+s_2q)} r^{-s_2q} b_j^{q-1}. \end{aligned}$$

(3.21)

For the third term in (3.16), using the relation $b_j>\delta \ge r^\frac{\gamma s_1p-N}{\gamma (p-1)}\Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1}$, we obtain

$$\begin{aligned}&b_j \Vert f \Vert _{L^\gamma (B_R)} |B_j\cap \{u<b_j\}|^\frac{1}{\gamma '} \le c b_j \Vert f \Vert _{L^\gamma (B_R)} r^\frac{-N}{\gamma } |B_j\cap \{u<b_j\}| \le c b_j^p r^{-s_1p} | \nonumber \\&\quad \quad \quad \quad \quad \quad \quad B_j\cap \{u<b_j\}|. \end{aligned}$$

(3.22)

Therefore, using (3.19), (3.20), (3.21), (3.22), and (3.16) in (3.18), we deduce that

$$\begin{aligned} (b_j-b_{j+1})^p \Big ( \frac{|B_{j+1} \cap \{ u<b_{j+1} \}|}{|B_{j+1}|} \Big )^\frac{p}{p^*_{s_1}}&\le c 2^{j(N+p+ps_1)} \frac{|B_j\cap \{u<b_j\}|}{|B_j|} \big ( b_j^{p}+b_j^{q}r^{s_1p-s_2q} \big ) \\&\le c 2^{j(N+p+ps_1)} b_j^p \frac{|B_j\cap \{u<b_j\}|}{|B_j|}, \end{aligned}$$

where in the last line, we have used the fact that $b_j>\delta k\ge r^\frac{s_1p-s_2q}{p-q}$ (consequently, $b_j^{p} \ge r^{s_1p-s_2q}b_j^q$). Setting

$$\begin{aligned} A_j:= \frac{|B_{j} \cap \{ u <b_{j} \}|}{|B_{j}|}, \end{aligned}$$

and using the relation $b_j - b_{j+1} \ge 2^{-j-3}b_j$, the above inequality implies that

$$\begin{aligned} A_{j+1}^{\frac{p}{p^*_{s_1}}} \le c 2^{j(N+p+ps_1)} A_j. \end{aligned}$$

Taking into account (3.17), a standard iteration lemma (see for instance [18, Lemma 2.6 and proof of Lemma 3.2, pp. 1823-1824]) yields

$$\begin{aligned} \lim _{j\rightarrow \infty } A_j=0, \end{aligned}$$

that is, $\inf _{B_{4r}} u \ge \delta k$, and noting (3.12), we get the required result of the lemma. $\square$

Lemma 3.4

Let $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ be a weak super-solution to problem ($\mathcal G_{f,g}(\Omega )$). Let $R\in (0,1)$ and $u\ge 0$ in $B_R(x_0)\subset \Omega$. Then, there exist constants $\tau \in (0,1)$ and $c\ge 1$ (both depending only on $N,s_1,p,s_2,q$) such that, for all $r\in (0,R)$,

Proof

The proof of the lemma follows exactly on the similar lines of [18, Proof of Lemma 4.1] by taking $T:=\Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1}T_{p-1}(u_-;x_0,R) +\Big (\frac{r}{R}\Big )^\frac{s_2q}{q-1}T_{q-1}(u_-;x_0,R)+ r^\frac{s_1p-s_2q}{p-q}+ r^\frac{\gamma s_1p-N}{\gamma (p-1)} \Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1}$, there and using Lemma 3.3 instead of [18, Lemma 3.2]. $\square$

Proposition 3.1

(Local boundedness) Let $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ be a weak solution to problem ($\mathcal G_{f,g}(\Omega )$). Let $r\in (0,1)$ and $x_0\in \mathbb R^N$ be such that $B_r(x_0)\Subset \Omega$. Then, there exists a constant $C>0$ depending only on $N,s_1,p,s_2,q$, and $\gamma$ (if $\gamma <\infty$) such that, for all $\varepsilon \in (0,1]$:

where $\theta =\frac{(p^*_{s_1})'}{\gamma }$ and $\sigma =\frac{p-\theta }{p(1-\theta )}$.

Proof

We fix $\tilde{k}\in \mathbb R^+$, $k\in \mathbb R$ and define the following:

$$\begin{aligned} r_i= (1+2^{-i})\frac{r}{2}, \ \ \tilde{r}_i=\frac{r_i+r_{i+1}}{2}; \ \ k_i=k+(1-2^{-i})\tilde{k}, \ \ \tilde{k}_i= \frac{k_{i}+k_{i+1}}{2} \quad \text{ for } \text{ all } i\in \mathbb N. \end{aligned}$$

Clearly, $r_{i+1}\le \tilde{r}_i \le r_i$ and $k_i\le \tilde{k}_i$. Further, we set

$$\begin{aligned}&B_i= B_{r_i}(x_0), \ \ \tilde{B}_{i}= B_{\tilde{r}_i}(x_0); \quad \tilde{w}_i:= (u-\tilde{k}_i)_+ \quad \text{ and } w_i:= (u-k_i)_+,\\&\quad \psi _i\in C^\infty _c(\tilde{B}_i), \ 0\le \psi _i \le 1, \ \psi _i\equiv 1 \text{ in } B_{i+1}, \ \ |\nabla \psi _i| < 2^{i+3}/r. \end{aligned}$$

Applying the fractional Poincaré inequality to the function $\tilde{w}_i\psi _i$ (for the case $ps_1<N$, otherwise taking an arbitrarily large number in place of $p^*_{s_1})$, we have

(3.23)

The first term on the right-hand side is estimated by means of the following Caccioppoli-type inequality (see [21, Lemma 3.1])

$$\begin{aligned}&\int _{B_i}\int _{B_i} \frac{|\tilde{w}_i(x)\psi _i(x)-\tilde{w}_i(y)\psi _i(y)|^p}{|x-y|^{N+ps_1}}dxdy \nonumber \\&\le C \sum _{(\ell ,s)} \int _{B_i}\int _{B_i} \frac{|\psi _i(x)-\psi _i(y)|^\ell }{|x-y|^{N+s\ell }}(\tilde{w}_i(x)^\ell +\tilde{w}_i(y)^\ell )dxdy \nonumber \\&\quad + C \sum _{(\ell ,s)} \Big (\sup _{y\in \mathrm{supp}\psi _i} \int _{\mathbb R^N\setminus B_i} \frac{\tilde{w}_i(x)^{\ell -1}dx}{|x-y|^{N+s\ell }}\Big ) \int _{B_i}\tilde{w}_i\psi _i^p+ \int _{\Omega } |f| \tilde{w}_i\psi _i^p, \end{aligned}$$

(3.24)

where $C=C(p,q)>0$ is a constant. Moreover, from the proof of [21, Proposition 3.2], we have

(3.25)

and

(3.26)

Now, we estimate the integral involving f. First, we note that if $\gamma =\infty$, then

Therefore, we assume that $\gamma <\infty$ (consequently, $\gamma '>1$). Applying Hölder’s inequality and interpolation identity for $L^p$-spaces (with 1 and $p^*_{s_1}$), we get

$$\begin{aligned} \int _{\Omega }|f(x)|\tilde{w}_i(x)\psi _i^p(x)dx&\le \Vert f \psi _i^{p-1} \Vert _{L^\gamma (B_r)} \Vert \tilde{w}_i\psi _i \Vert _{L^{\gamma '}(B_i)} \\&\le \Vert f \Vert _{L^\gamma (B_r)} \Vert \tilde{w}_i\psi _i \Vert _{L^{p^*_{s_1}}(B_i)}^{\theta } \Vert \tilde{w}_i\psi _i \Vert _{L^{1}(B_i)}^{1-\theta }, \end{aligned}$$

where $\theta =\frac{(p^*_{s_1})'}{\gamma } \in (0,1)$ (thanks to the assumption on $\gamma$). Using Young’s inequality in the above expression (with the exponents $p/\theta$ and $p/(p-\theta )$), for $\epsilon >0$, we deduce that

$$\begin{aligned} \int _{\Omega }|f(x)|\tilde{w}_i(x)\psi _i^p(x)dx \le \epsilon \Vert f \Vert _{L^\gamma (B_r)} \Vert \tilde{w}_i\psi _i \Vert _{L^{p^*_{s_1}}(B_i)}^{p} + \epsilon ^{\frac{-\theta }{p-\theta }} \Vert f \Vert _{L^\gamma (B_r)} \Vert \tilde{w}_i\psi _i \Vert _{L^{1}(B_i)}^{\frac{p(1-\theta )}{p-\theta }}. \end{aligned}$$

(3.27)

For convenience in writing, we denote $\sigma =\frac{p-\theta }{p(1-\theta )}>1$, and observe that $p\sigma \in (1,p^*_{s_1})$ (this is because of $\gamma >N/(ps_1)$). Thus on account of the relation $\tilde{w}_i\le (\tilde{k}_i - k_i)^{1-p\sigma } w_i^{p\sigma }$, we get

$$\begin{aligned} \Big (\int _{B_i} | \tilde{w}_i\psi _i |dx\Big )^{1/\sigma } \le \Big ( \frac{1}{(\tilde{k}_i - k_i)^{p\sigma -1}} \int _{B_i} (\tilde{w}_i)^{p\sigma }dx\Big )^{1/\sigma } \le \frac{1}{\tilde{k}^{p-1/\sigma }} \Big (\int _{B_i} (\tilde{w}_i)^{p\sigma }dx\Big )^{1/\sigma }. \end{aligned}$$

Taking $\epsilon = \frac{1}{2cC \Vert f \Vert _{L^\gamma (B_r)}}$ in the above expression, where c and C are as in (3.23) and (3.24), respectively, from (3.27), we get

(3.28)

where $C_1=C_1(c,C,N,\sigma )>0$ is a constant. Therefore, combining (3.25), (3.26), and (3.28) with (3.24), from (3.23), we obtain

where we have used Hölder’s inequality in the integrals involving $w_i^p$. For $\varepsilon \in (0,1]$, we take

$$\begin{aligned} \tilde{k} \ge \varepsilon T_{p-1}(w_0;x_0,\frac{r}{2})+ \varepsilon T_{q-1}(w_0;x_0,\frac{r}{2})+ \varepsilon r^\frac{s_1p-s_2q}{p-q}+ \Vert f\Vert _{L^\gamma (B_r)}^\frac{1}{p-1} \varepsilon ^\frac{p-\theta }{p} r^{\frac{p-\theta }{p(p-1)}(s_1p-N+\frac{N}{\sigma })}, \end{aligned}$$

(3.29)

(note that $s_1p-N+\frac{N}{\sigma }>0$). Thus, the above expression yields

(3.30)

where we have used the relation $\big (r^{s_1p-s_2q} \tilde{k}^{q-p}\big ) \big (\tilde{k}^{1-q} T_{q-1}(w_0;x_0.r/2)^{q-1} \big ) \le \varepsilon ^{1-p}$. Now, we estimate the left-hand side term as [17, Proof of Theorem 1.1, p. 1292]:

Then, setting , from (3.30), we have

$$\begin{aligned} \tilde{k}^\frac{(p^*_{s_1}-p\sigma )p}{p^*_{s_1}} A_{i+1}^\frac{p^2\sigma }{p^*_{s_1}} \le c 2^{i\big (N+s_1p+p-1+\frac{(p^*_{s_1}-p\sigma )p}{p^*_{s_1}}\big )} \varepsilon ^{1-p} A_i^p, \end{aligned}$$

that is,

$$\begin{aligned} \frac{A_{i+1}}{\tilde{k}} \le c \varepsilon ^{\frac{p^*_{s_1}(1-p)}{p^2\sigma }} C_2^{i} \left( \frac{A_{i}}{\tilde{k}} \right) ^{1+\vartheta }, \end{aligned}$$

where $\vartheta =\frac{p^*_{s_1}}{p\sigma }-1>0$ and $C_2 := 2^{\big (\frac{N+s_1p+p-1}{p}+\frac{p^*_{s_1}-p\sigma }{p^*_{s_1}}\big )\frac{p^*_{s_1}}{p\sigma }}>1$. We note that $\frac{p^*_{s_1}(p-1)}{p^2\sigma \vartheta }= \frac{p-1}{p}\frac{p^*_{s_1}}{p^*_{s_1}-p\sigma }$ and $\frac{p-\theta }{p(p-1)}=\frac{\sigma }{p\sigma -1}$. Next, we choose

$$\begin{aligned} \tilde{k}&= \varepsilon T_{p-1}(w_0;x_0,\frac{r}{2}) + \varepsilon T_{q-1}(w_0;x_0,\frac{r}{2})+ \varepsilon r^\frac{s_1p-s_2q}{p-q}+ \Vert f\Vert _{L^\gamma (B_r)}^\frac{1}{p-1} \varepsilon ^\frac{p-\theta }{p} r^{\frac{\sigma }{p\sigma -1}(s_1p-N+\frac{N}{\sigma })} \\&\quad + \varepsilon ^\frac{(1-p)p^*_{s_1}}{p(p^*_{s_1}-p\sigma )} c^{1/\vartheta } C_2^{1/\vartheta ^2} A_0, \end{aligned}$$

which clearly satisfies (3.29) and

$$\begin{aligned} \frac{A_{0}}{\tilde{k}} \le \varepsilon ^{\frac{p^*_{s_1}(p-1)}{p^2\sigma \vartheta }} c^{-1/\vartheta } C_2^{-1/\vartheta ^2}. \end{aligned}$$

Therefore, a well-known iteration argument implies that

$$\begin{aligned} A_i \rightarrow 0 \quad \text{ as } i\rightarrow \infty . \end{aligned}$$

Thus, we obtain

Now, the proof of the proposition, for $u_+$, follows by taking $k=0$. The proof for the case $u_-$ runs analogously.$\square$

By slightly modifying the choice of $\tilde{k}$ in the above proof, we have:

Corollary 3.1

Under the hypothesis of Proposition 3.1, the following holds:

where $C=C(N,p,q,s_1,s_2,\gamma )>0$ is a constant.

Next, we prove the following version of Caccioppoli-type inequality.

Lemma 3.5

Let $m\in (1,p)$ and $\lambda >0$. Let $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ be a weak super-solution to problem ($\mathcal G_{f,g}(\Omega )$) such that $u\ge 0$ in $B_R(x_0)\subset \Omega$, for $R\in (0,1)$. Then, for any $r\in (0,3R/4)$ and $\phi \in C_c^\infty (B_r)$, with $0\le \phi \le 1$, there holds

$$\begin{aligned}&\int _{B_r} \int _{B_r} \frac{|w_p(x)\phi (x)-w_p(y)\phi (y)|^p}{|x-y|^{N+s_1p}}dxdy\\&\le \int _{\Omega }|f(x)|{{\bar{u}}}^{1-m}(x)\phi ^p(x)dx + c \sum _{(\ell ,s)}\lambda ^{\ell -p} \int _{ B_{r}}\int _{ B_{r}}\frac{|\phi (x)-\phi (y)|^\ell }{|x-y|^{N+s\ell }}(w_p(x)^p+w_p(y)^p)dxdy \\&\quad + c \sum _{(\ell ,s)}\Big ( \sup _{x\in \mathrm{supp}\phi } \int _{{\mathbb R^N}\setminus B_{r}} \frac{dy}{|x-y|^{N+s\ell }} + \lambda ^{1-\ell } \int _{{\mathbb R^N}\setminus B_{R}} \frac{u(y)^{\ell -1}_-dy}{|y-x_0|^{N+s\ell }}\Big ) \int _{B_r} w_\ell (x)^\ell \phi ^p(x)dx , \end{aligned}$$

where $w_\ell :=(u+\lambda )^\frac{\ell -m}{\ell }$, and $c=c(p,q,m)>0$ is a constant.

Proof

Set ${\bar{u}}:= u+\lambda$ and let $\eta := {\bar{u}}^{1-m}\phi ^p$, where $m\in [1+\epsilon ,p-\epsilon ]$, for $\epsilon >0$ small. Similarly to Lemma 3.1, we note that ${{\bar{u}}}$ is also a weak super-solution, therefore taking $\eta$ as a test function, we get

$$\begin{aligned} 0&\le \sum _{(\ell ,s)}\int _{B_{r}} \int _{B_{r}} \frac{[{{\bar{u}}}(x)-{{\bar{u}}}(y)]^{\ell -1}}{|x-y|^{N+s\ell }}(\eta (x)-\eta (y))dxdy \nonumber \\&\quad +2 \sum _{(\ell ,s)}\int _{{\mathbb R^N}\setminus B_{r}}\int _{B_{r}} \frac{[{{\bar{u}}}(x)-{{\bar{u}}}(y)]^{\ell -1}}{|x-y|^{N+s\ell }}\eta (x)dxdy -\int _{\Omega }f(x)\eta (x)dx \nonumber \\&=:I_1(\ell )+I_2(\ell )-I_3(f). \end{aligned}$$

(3.31)

We observe that, for all $x\in B_r$ and $y\in \mathbb R^N$ (note that $u\ge 0$ in $B_R$),

$$\begin{aligned}{}[{{\bar{u}}}(x)-{{\bar{u}}}(y)]^{\ell -1} \le c \big ({\bar{u}}(x)^{\ell -1}+u(y)^{\ell -1}_- \big ) \quad \text{ and } {{\bar{u}}}^{1-m}(x) \le \lambda ^{1-\ell }{{\bar{u}}}^{\ell -m}(x). \end{aligned}$$

Therefore, setting $d\mu _\ell =|x-y|^{-N-s\ell }dxdy$, we deduce that

$$\begin{aligned} I_2(\ell )&\le c \int _{{\mathbb R^N}\setminus B_{r}}\int _{B_{r}} {{\bar{u}}}^{\ell -1}(x){\bar{u}}^{1-m}(x)\phi ^p(x) d\mu _\ell + c\int _{{\mathbb R^N}\setminus B_{r}}\int _{ B_{r}} u(y)^{\ell -1}_-{\bar{u}}^{1-m}(x)\phi ^p(x) d\mu _\ell \nonumber \\&\le c \Big ( \sup _{x\in \mathrm{supp}\phi } \int _{{\mathbb R^N}\setminus B_{r}} |x-y|^{-N-s\ell }dy + \lambda ^{1-\ell } \nonumber \\&\quad \int _{{\mathbb R^N}\setminus B_{R}} \frac{u(y)^{\ell -1}_-}{|y-x_0|^{N+s\ell }}dy\Big ) \int _{B_r} w_\ell (x)^\ell \phi ^p(x)dx, \end{aligned}$$

(3.32)

where we have used $u\ge 0$ in $B_R$ and $w_\ell := {{\bar{u}}}^\frac{\ell -m}{\ell }$. Next, for the first integral $I_1(p)$, from [18, proof of Lemma 5.1, pp.1830-1833], we have

$$\begin{aligned}&I_1(p)\le -c \int _{ B_{r}}\int _{ B_{r}} |w_p(x)-w_p(y)|^p\phi ^p(y) d\mu _p+ c\int _{ B_{r}}\nonumber \\&\quad \int _{ B_{r}}|\phi (x)-\phi (y)|^p(w_p(x)^p+w_p(y)^p)d\mu _p. \end{aligned}$$

(3.33)

To estimate $I_1(q)$, without loss of generality, we assume ${{\bar{u}}}(x)>{{\bar{u}}}(y)$. We proceed exactly as in Step II of Lemma 3.1 to get

$$\begin{aligned}&({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1} \left[ \frac{\phi ^p(x)}{{{\bar{u}}}(x)^{m-1}} - \frac{\phi ^p(y)}{{{\bar{u}}}(y)^{m-1}}\right] \\&\quad \le \phi ^p(y) \frac{({{\bar{u}}}(x)-{{\bar{u}}}(y))^{q-1}}{{{\bar{u}}}(x)^{m-1}} \left[ 1+c\delta \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} -\frac{{{\bar{u}}}(x)^{m-1}}{{{\bar{u}}}(y)^{m-1}} \right] \\&\qquad +c\delta ^{1-q} {{\bar{u}}}(x)^{q-m} |\phi (x)-\phi (y)|^q. \end{aligned}$$

For suitable choice of $\delta \in (0,1)$, from [18, Proof of Lemma 5.1, p.1831] (see the expression for g(t) there), we have

$$\begin{aligned} \left[ 1+c\delta \frac{{{\bar{u}}}(x)-{{\bar{u}}}(y)}{{{\bar{u}}}(x)} -\frac{{{\bar{u}}}(x)^{m-1}}{{{\bar{u}}}(y)^{m-1}} \right] \le 0, \end{aligned}$$

and an analogous result holds for the case ${{\bar{u}}}(x)< {{\bar{u}}}(y)$. Thus,

$$\begin{aligned} I_1(q) \le c \lambda ^{q-p}\int _{ B_{r}}\int _{ B_{r}} {{\bar{u}}}(x)^{p-m} |\phi (x)-\phi (y)|^q d\mu _q. \end{aligned}$$

(3.34)

Therefore, combining (3.32), (3.33), (3.34) and using (3.31), we obtain

$$\begin{aligned}&\int _{ B_{r}}\int _{ B_{r}} |w_p(x)-w_p(y)|^p\phi ^p(y) d\mu _p \\&\quad \le \int _{\Omega }|f(x)|{{\bar{u}}}^{1-m}(x)\phi ^p(x)dx+ c \sum _{(\ell ,s)}\lambda ^{\ell -p} \int _{ B_{r}}\int _{ B_{r}} |\phi (x)-\phi (y)|^\ell (w_p(x)^p+w_p(y)^p)d\mu _\ell \\&\qquad + c \sum _{(\ell ,s)}\Big ( \sup _{x\in \mathrm{supp}\phi } \int _{{\mathbb R^N}\setminus B_{r}} \frac{dy}{|x-y|^{N+s\ell }} + \lambda ^{1-\ell } \int _{{\mathbb R^N}\setminus B_{R}} \frac{u(y)^{\ell -1}_-dy}{|y-x_0|^{N+s\ell }}\Big ) \int _{B_r} w_\ell (x)^\ell \phi ^p(x)dx. \end{aligned}$$

By observing

$$\begin{aligned} |w_p(x)\phi (x)&-w_p(y)\phi (y)|^p \le c (w_p(x)^p+w_p(y)^p) |\phi (x)-\phi (y)|^p + c|w_p(x)\\&-w_p(y)|^p \phi (y)^p, \end{aligned}$$

we complete the proof of the lemma. $\square$

Lemma 3.6

Let $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ be a solution to problem ($\mathcal G_{f,g}(\Omega )$) such that $u\ge 0$ in $B_R(x_0)\Subset \Omega$, for some $R\in (0,1)$. Then, for all $0<r<R$, the following holds:

$$\begin{aligned}&T_{p-1}(u_+;x_0,r)^{p-1}+ r^{s_1p-s_2q} T_{q-1}(u_+;x_0,r)^{q-1} \\&\le C (\sup _{B_r} u\big )^{p-1}+ C\Big (\frac{r}{R}\Big )^{s_1p} T_{p-1}(u_-;x_0,R)^{p-1} + C r^{s_1p-s_2q} \Big (\frac{r}{R}\Big )^{s_2q} T_{q-1}(u_-;x_0,R)^{q-1}\\&\quad + C r^\frac{(s_1p-s_2q)(p-1)}{p-q} +C \Vert f\Vert _{L^\gamma (B_R)}r^\frac{\gamma s_1p-N}{\gamma (p-1)}, \end{aligned}$$

where $C=C(N,p,q,s_1,s_2)>0$ is a constant.

Proof

Let $k:=\sup _{B_r} u$ and $\phi \in C_c^\infty (B_r)$ such that $0\le \phi \le 1$ and $|\nabla \phi |\le 8/r$ in $B_r$, and $\phi \equiv 1$ in $B_{r/2}$. Then, testing the equation with $\eta := (u-2k)\phi$ (note that $\eta \le 0$), we get

$$\begin{aligned} 0&= \sum _{(\ell ,s)} \int _{B_{r}} \int _{B_{r}} \frac{[u(x)- u(y)]^{\ell -1}}{|x-y|^{N+s\ell }}(\eta (x)-\eta (y))dxdy \nonumber \\&\quad +2\sum _{(\ell ,s)} \int _{{\mathbb R^N}\setminus B_{r}}\int _{B_{r}} \frac{[u(x)- u(y)]^{\ell -1}}{|x-y|^{N+s\ell }}\eta (x) dxdy -\int _{\Omega }f(x)\eta (x)dx \nonumber \\&=:I_1(\ell )+I_2(\ell )-I_3(f). \end{aligned}$$

(3.35)

Proceeding similarly to [18, Lemma 4.2], we have

$$\begin{aligned}&I_2(\ell ) \ge -c k^{\ell }r^{-s\ell }|B_r|-ck|B_r|R^{-s\ell }T_{\ell -1}(u_-;x_0,R)^{\ell -1}+ck|B_r| r^{-s\ell } \\&T_{\ell -1}(u_+;x_0,r)^{\ell -1} \text{ and } I_1(p) \ge -c k^{p}r^{-s_1p}|B_r|. \end{aligned}$$

To estimate $I_1(q)$, following the proof of [21, Lemma 3.1], with the notation $\tilde{u}=(u-2k)$, we get

$$\begin{aligned} {[}\tilde{u}(x)- \tilde{u}(y)]^{q-1} (\tilde{u}(x)\phi (x)-\tilde{u}(y)\phi (y))&\ge \frac{1}{2} |\tilde{u}(x)-\tilde{u}(y)|^p \phi (x)^p - c|\tilde{u}(y)|^q |\phi (x)-\phi (y)|^q \\&\ge -ck^{q}|\phi (x)-\phi (y)|^q, \end{aligned}$$

where in the last line, we have used $|\tilde{u}|\le 3k$ in $B_r$. Thus,

$$\begin{aligned} I_1(q) \ge - ck^q \int _{ B_{r}}\int _{ B_{r}} \frac{|\phi (x)-\phi (y)|^q}{|x-y|^{N+s_2q}}dxdy&\ge r^{-s_2q}|B_r|- ck^q \int _{ B_{r}}\int _{ B_{r}} |x-y|^{q-N-s_2q}dxdy\\&\ge -ck^{q}r^{-s_2q}|B_r|. \end{aligned}$$

For $I_3(f)$, noting $|u-k|\le 3k$ in $B_r$, we have

$$\begin{aligned} I_3(f) \le 3k \int _{ B_{r}} |f|\phi dx \le 3 k \Vert f \Vert _{L^\gamma (B_r)} |B_r|^{1/\gamma '}. \end{aligned}$$

Collecting all information in (3.35), we obtain (upon multiplication with $r^{s_1p}(k|B_r|)^{-1}$)

$$\begin{aligned}&T_{p-1}(u_+;x_0,r)^{p-1}+ r^{s_1p-s_2q} T_{q-1}(u_+;x_0,r)^{q-1} \\&\le C\left[ k^{p-1}+ r^{s_1p-s_2q} k^{q-1}+ \Vert f\Vert _{L^\gamma (B_r)}r^\frac{\gamma s_1p-N}{\gamma (p-1)} \right] +C\Big (\frac{r}{R}\Big )^{s_1p} T_{p-1}(u_-;x_0,R)^{p-1} \\&\quad + C r^{s_1p-s_2q} \Big (\frac{r}{R}\Big )^{s_2q} T_{q-1}(u_-;x_0,R)^{q-1}. \end{aligned}$$

This proves the lemma on account of the relation $cr^{s_1p-s_2q}k^{q-1} \le ck^{p-1}+c r^\frac{(s_1p-s_2q)(p-1)}{p-q}$ (by Young’s inequality). $\square$

4 Harnack and Weak Harnack Inequalities

In this section, we prove the Harnack- and weak Harnack-type inequalities for fractional (p, q)-problems. We start with the following result.

Proposition 4.1

(Weak Harnack inequality) Let $u\in X^{s_1,p}_{g}(\Omega ,\Omega ') \cap X^{s_2,q}_{g}(\Omega ,\Omega ')$ be a weak super-solution to problem ($\mathcal G_{f,g}(\Omega )$) such that $u\ge 0$ in $B_R(x_0)\subset \Omega$, for $R\in (0,1)$. Then, for any $r\in (0,R)$ and for any $t<\frac{N(p-1)}{N-s_1p}$ (for $ps_1<N$), there holds:

where $c=c(N,p,s_1,q,s_2)>0$ is a constant.

Proof

Let $1/2< \kappa '<\kappa \le 3/4$ and let $\psi \in C_c^\infty (B_{\kappa r})$ be such that $\phi \equiv 1$ in $B_{\kappa 'r}$, $0\le \phi \le 1$ and $|\nabla \phi |\le \frac{4}{(\kappa -\kappa ')r}$. Set $w\equiv w_p={{\bar{u}}}^\frac{p-m}{p}:=(u+\lambda )^\frac{p-m}{p}$, for $\lambda >0$. Applying the fractional Poincaré inequality to the function $w\phi$, we get

(4.1)

To estimate the right-hand side quantity, we will use Lemma 3.5. Using the bound on $|\nabla \phi |$, we have

$$\begin{aligned} \int _{ B_{r}}\int _{ B_{r}} \frac{|\phi (x)-\phi (y)|^\ell }{|x-y|^{N+s\ell }}(w_p(x)^p+w_p(y)^p)dxdy \le \frac{cr^{-s\ell }}{(\kappa -\kappa ')^\ell } \int _{ B_{\kappa r}} w_p(x)^p dx. \end{aligned}$$

Moreover, using $\mathrm{supp}\phi \subset B_{\kappa r}$, we see that

$$\begin{aligned} \sup _{x\in \mathrm{supp}\phi } \int _{\mathbb R^N\setminus B_r} |x-y|^{-N-s\ell }dy \le c r^{-s\ell }. \end{aligned}$$

Furthermore, employing Hölder’s inequality and interpolation result for $L^p$-spaces (observe that ${{\bar{u}}}\ge \lambda$ and $p<p\gamma '<p^*_{s_1}$), we deduce that

$$\begin{aligned} \int _{\Omega }|f|{{\bar{u}}}^{1-m}\phi ^pdx \le \lambda ^{1-p} \int _{\Omega }|f|{{\bar{u}}}^{p-m} \phi ^p dx&\le \lambda ^{1-p} \Vert f \Vert _{L^\gamma (B_R)} \Vert w\phi \Vert _{L^{p\gamma '}(B_R)}^p. \\&\le \lambda ^{1-p} \Vert f \Vert _{L^\gamma (B_R)} \Vert w\phi \Vert _{L^{p^*_{s_1}}(B_R)}^\frac{N}{\gamma s_1} \Vert w\phi \Vert _{L^{p}(B_R)}^\frac{\gamma s_1p-N}{\gamma s_1}, \end{aligned}$$

which on using Young’s inequality (with the exponents $\frac{\gamma s_1p}{N}$ and $\frac{\gamma s_1p}{\gamma s_1p-N}$), for $\epsilon >0$, yields

where $C_1>0$ is a constant which depends only on $N,s_1,p$. Choosing $\epsilon =\frac{\lambda ^{p-1}}{2c C_1}\frac{1}{\Vert f \Vert _{L^\gamma (B_R)}}$, with c as in (4.1), we get

where $c_2=c_2(N,s_1,p,c)>0$ is a constant. Collecting all these information in (4.1) and using Lemma 3.5, we obtain

Noting the fact that $w_q(x)^q={{\bar{u}}}(x)^{q-m} \le \lambda ^{q-p}{{\bar{u}}}(x)^{p-m}= \lambda ^{q-p}w_p(x)^p$, we have

(4.2)

We take

$$\begin{aligned} \lambda =\Big (\frac{r}{R}\Big )^\frac{s_1p}{p-1}T_{p-1}(u_-;x_0,R) +\Big (\frac{r}{R}\Big )^\frac{s_2q}{q-1}T_{q-1}(u_-;x_0,R)+ r^\frac{s_1p-s_2q}{p-q}+ \Vert f \Vert _{L^\gamma (B_R)}^\frac{1}{p-1} r^\frac{\gamma s_1p-N}{\gamma (p-1)}. \end{aligned}$$

(4.3)

Moreover, as in the proof of Lemma 3.1, observe that

$$\begin{aligned} \lambda ^{1-p}&r^{s_1p-s_2q}\Big (\frac{r}{R}\Big )^{s_2q}T_{q-1}(u_-;x_0,R)^{q-1} \\&= \lambda ^{1-q} \Big (\frac{r}{R}\Big )^{s_2q}T_{q-1}(u_-;x_0,R)^{q-1} \lambda ^{q-p} r^{s_1p-s_2q} \le 1 \end{aligned}$$

and

$$\begin{aligned} \Big (\frac{\Vert f \Vert _{L^\gamma (B_R)}}{\lambda ^{p-1}}\Big )^\frac{\gamma s_1p}{\gamma s_1p-N} \le r^{-\frac{\gamma s_1 p-N}{\gamma }\times \frac{\gamma s_1 p}{\gamma s_1p-N}} = r^{-s_1p}. \end{aligned}$$

Therefore, from (4.2) and noting that $\phi \equiv 1$ in $B_{\kappa ' r}$, we get

that is,

Thus, proceeding as in [18, (5.14), p.1834] (or one can use a standard finite Moser iteration argument similar to [23, Theorem 8.18]), we have

Applying Lemma 3.4, for $t_1=\tau$, we obtain

(4.4)

Then, noticing ${{\bar{u}}}=u+\lambda$, we observe that

This combined with (4.4) and the definition of $\lambda$ (given by (4.3)) proves the proposition. $\square$

By replacing the constant c appearing in Proposition 4.1 with ${{\bar{c}}}=\max \{c,2\}$, we obtain the following result.

Corollary 4.1

Under the hypothesis of Proposition 4.1, for all $0<t<\frac{N(p-1)}{N-ps_1}$, we have

for some $\varsigma \in (0,1)$.

Proof of Theorem 2.2

From Corollary 3.1, for all $\rho \in (0,1)$, we have

where $\theta =\frac{(p^*_{s_1})'}{\gamma }$ and $\sigma =\frac{p-\theta }{p(1-\theta )}$. Using Lemma 3.6, the above inequality reduces to

Let $1/2\le \kappa '<\kappa \le 3/4$ and let $\phi \in C_c^\infty (B_{\kappa r})$ be such that $\phi \equiv 1$ in $B_{\kappa 'r}$ and $|\nabla \phi | \le 4/[(\kappa -\kappa ')r]$. Then, by a covering argument, setting $\rho =(\kappa -\kappa ')r$, we get

Next, upon using Young’s inequality and taking $\varepsilon =1/(4C)$, for all $t\in (0,p\sigma )$, we obtain

Therefore, using Hölder’s inequality, for all $t\in (0,p^*_{s_1})$, we deduce that

Now, the rest of the proof follows using a standard iteration technique, e.g., see [18, Proof of Theorem 1.1, p.1829]. $\square$

5 Hölder Continuity Results

In this section, we prove our main Hölder regularity results.

Proof of Theorem 2.1

We first observe that

$$\begin{aligned} u\in X^{s_1,p}_{u}(B_{3R_0/2}(x_0),B_{2R_0}(x_0))\cap X^{s_2,q}_{u}(B_{3R_0/2}(x_0),B_{2R_0}(x_0)). \end{aligned}$$

Furthermore, u solves

$$\begin{aligned} (-\Delta )^{s_1}_{p}u+ (-\Delta )^{s_2}_{q}u = f \quad \text {in} \; B_{3R_0/2}(x_0) \end{aligned}$$

and $u\in L^\infty (B_R(x_0))$ (thanks to Proposition 3.1). For all $j\in \mathbb N\cup \{0\}$, define sequences $R_j=\frac{R_0}{4^j}$, $B_j=B_{R_j}$, $\frac{1}{2}B_j=B_{R_j/2}$. We claim that there exists $\alpha >0$ (a generic constant), $\omega >0$, a non-decreasing sequence $\{m_j\}$ and a non-increasing sequence $\{M_j\}$ such that

$$\begin{aligned} m_j\le \inf _{B_j}\; u\le \sup _{B_j}\; u\le M_j ,\qquad M_j-m_j=\omega R_j^\alpha . \end{aligned}$$

We will proceed by induction. For $j=0$, we set $M_0= \Vert u\Vert _{L^\infty (B_{R_0})}$ and $m_0= M_0-\omega R_0^\alpha$, where $\omega$ satisfies

$$\begin{aligned} \omega \ge \frac{2\Vert u\Vert _{L^\infty (B_{R_0})}}{R_0^\alpha }>0. \end{aligned}$$

(5.1)

Hence, $m_0\le \displaystyle \inf _{B_0}\; u\le \displaystyle \sup _{B_0}\; u\le M_0.$ Suppose that the claim holds for all $i\in \{0,\dots , j\}$ for some $j\in \mathbb N$. To prove the claim for $j+1$, we first consider the case $1<\frac{N(p-1)}{N-s_1p}$. Then

(5.2)

for some $1<t<\frac{N(p-1)}{N-s_1p}$. Employing Corollary 4.1, for the choice $r=R_{j+1}$ and $R=R_j$, we obtain

$$\begin{aligned} \varsigma (M_j-m_j)&\le \inf _{B_{j+1}}(M_j-u)+ \inf _{B_{j+1}}(u-m_j) + C\sum _{(\ell ,s)} T_{\ell -1}((M_j-u)_-;x_0,R_j) \\&\quad + C\sum _{(\ell ,s)} T_{\ell -1}((u-m_j)_-;x_0,R_j) + C R_j^\frac{ps_1-qs_2}{p-q} + C \Vert f \Vert _{L^\gamma (B_{R_0})}^\frac{1}{p-1} R_j^\frac{\gamma s_1p-N}{\gamma (p-1)}, \end{aligned}$$

that is,

$$\begin{aligned} \underset{B_{j+1}}{\mathrm{osc}}\, u&\le \left( 1-\varsigma \right) (M_j-m_j) +C\sum _{(\ell ,s)} T_{\ell -1}((M_j-u)_-;x_0,R_j)\nonumber \\&\quad + C\sum _{(\ell ,s)} T_{\ell -1}((u-m_j)_-;x_0,R_j) \nonumber \\&\quad + C R_j^\frac{ps_1-qs_2}{p-q} +C \Vert f \Vert _{L^\gamma (B_{R_0})}^\frac{1}{p-1} R_j^\frac{\gamma s_1p-N}{\gamma (p-1)}. \end{aligned}$$

(5.3)

Now, we estimate the different tail terms appearing in the above expression:

$$\begin{aligned}&T_{\ell -1}((u-m_j)_-;x_0,R_j)^{\ell -1}\\ {}&\quad = R_j^{s\ell } \left[ \sum _{k=0}^{j-1} \int _{B_k\setminus B_{k+1}}\frac{(u(y)-m_j)^{\ell -1}}{|x_0-y|^{N+s\ell }}dy + \int _{B_0^c} \frac{(u(y)-m_j)^{\ell -1}}{|x_0-y|^{N+s\ell }}dy \right] , \end{aligned}$$

where $B_0^c:=\mathbb R^N\setminus B_0$. By the induction hypothesis and [26, Proof of Theorem 5.4], for $\alpha \in (0,1)$, we have

$$\begin{aligned} \sum _{k=0}^{j-1} \int _{B_k\setminus B_{k+1}}\frac{(u(y)-m_j)^{\ell -1}}{|x_0-y|^{N+s\ell }}dy \le c \omega ^{\ell -1}R_j^{\alpha (\ell -1)-\ell s}S_\ell (\alpha ), \end{aligned}$$

where $S_\ell (\alpha ):= \displaystyle \sum _{k=1}^\infty \frac{(4^{\alpha k}-1)^{\ell -1}}{4^{\ell sk}}\rightarrow 0$ as $\alpha \rightarrow 0^+$. For the second term, we set $Q(u;x_0,R_0):=\Vert u\Vert _{L^\infty (B_0)}+T_{p-1}(u;x_0,R_0)+T_{q-1}(u;x_0,R_0)$. By observing that $m_j \le \inf _{B_j} u\le \sup _{B_j} u \le \Vert u\Vert _{L^\infty (B_0)}$, we deduce that

$$\begin{aligned} \int _{\mathbb R^N\setminus B_0} \frac{(u(y)-m_j)^{\ell -1}}{|x_0-y|^{N+s\ell }}dy&\le \int _{\mathbb R^N\setminus B_0} \frac{(|u(y)|+\Vert u\Vert _{L^\infty (B_0)})^{\ell -1}}{|x_0-y|^{N+s\ell }}dy \\&\le C\frac{\Vert u\Vert _{L^\infty (B_0)}^{\ell -1}+T_{\ell -1}(u;x_0,R_0)^{\ell -1}}{R_0^{sl}}. \end{aligned}$$

Therefore,

$$\begin{aligned} T_{\ell -1}((u-m_j)_-;x_0,R_j)^{\ell -1}\le c \omega ^{\ell -1}R_j^{\alpha (\ell -1)}S_\ell (\alpha ) + c R_j^{s\ell } \frac{\big (\Vert u\Vert _{L^\infty (B_0)}+T_{\ell -1}(u;x_0,R_0)\big )^{\ell -1}}{R_0^{s\ell }}, \end{aligned}$$

and an analogous estimate holds for $T_{\ell -1}((M_j-u)_-;x_0,R_j)^{\ell -1}$ also. Thus, from (5.3) and the inductive hypothesis, we get

$$\begin{aligned} \underset{B_{j+1}}{\mathrm{osc}}\, u&\le \left( 1-\varsigma \right) \omega R_j^\alpha +c \omega R_j^{\alpha } S_p(\alpha )^{1/(p-1)} + c R_j^\frac{s_1p}{p-1} \frac{Q(u;x_0,R_0)}{R_0^{s_1p/(p-1)}} \\&\quad +c \omega R_j^{\alpha }S_q(\alpha )^{1/(q-1)}+c R_j^\frac{s_2q}{q-1} \frac{Q(u;x_0,R_0)}{R_0^{s_2q/(q-1)}} + C R_j^\frac{ps_1-qs_2}{p-q}+ C \Vert f \Vert _{L^\gamma (B_{R_0})}^\frac{1}{p-1} R_j^\frac{\gamma s_1p-N}{\gamma (p-1)}. \end{aligned}$$

Then, for $\alpha <\min \big \{ \frac{ps_1}{p-1}, \frac{qs_2}{q-1}, \frac{ps_1-qs_2}{p-q},\frac{\gamma s_1p-N}{\gamma (p-1)} \big \}$, we have

$$\begin{aligned} \underset{B_{j+1}}{\mathrm{osc}}\, u&\le 4^{\alpha }\left[ 1-\varsigma + c S_p(\alpha )^\frac{1}{p-1}+ c S_q(\alpha )^\frac{1}{q-1} \right] \omega R_{j+1}^\alpha \nonumber \\&\quad + \frac{c4^\alpha }{R_0^\alpha } \left[ Q(u;x_0,R_0) + R_0^\frac{ps_1-qs_2}{p-q}+ \Vert f \Vert _{L^\gamma (B_{R_0})}^\frac{1}{p-1} R_0^\frac{\gamma s_1p-N}{\gamma (p-1)} \right] R_{j+1}^\alpha , \end{aligned}$$

(5.4)

where in the last line, we have used $R_{j}= 4^{\alpha }R_{j+1} \le R_0$. On account of $S_\ell (\alpha )\rightarrow 0$ as $\alpha \rightarrow 0^+$, we can choose $\alpha <\min \big \{ \frac{ps_1}{p-1}, \frac{qs_2}{q-1}, \frac{ps_1-qs_2}{p-q},\frac{\gamma s_1p-N}{\gamma (p-1)} \big \}$ small enough such that

$$\begin{aligned} 4^\alpha \left( 1-\varsigma + c S_p(\alpha )^\frac{1}{p-1}+ c S_q(\alpha )^\frac{1}{q-1} \right) \le 1-\frac{\varsigma }{4}\end{aligned}$$

and set

$$\begin{aligned} \omega =\frac{4^{\alpha +1}\; c}{\varsigma \;R_0^\alpha }\Bigg ( Q(u;x_0,R_0)+R_0^\frac{ps_1-qs_2}{p-q}+ \Vert f \Vert _{L^\gamma (B_{R_0})}^\frac{1}{p-1} R_0^\frac{\gamma s_1p-N}{\gamma (p-1)} \Bigg ). \end{aligned}$$

(5.5)

Note that with the above choice of $\omega$, (5.1) is satisfied if the constant c, appearing in (5.4), is replaced by a bigger constant such that $4c/\varsigma \ge 2$. Thus, from (5.4), we have

$$\begin{aligned} \underset{B_{j+1}}{\mathrm{osc}}\, u \le \omega R_{j+1}^\alpha . \end{aligned}$$

Therefore, we pick $m_{j+1},\; M_{j+1}$ such that

$$\begin{aligned} m_j\le m_{j+1}\le \inf _{B_j}\; u\le \sup _{B_j}\; u\le M_{j+1}\le M_j \quad \text{ and } M_{j+1}-m_{j+1}= \omega R_{j+1}^\alpha . \end{aligned}$$

To finish the proof of the theorem, we fix $r\in (0,R_0)$. Let $j\in \mathbb N\cup \{0\}$ be such that $R_{j+1}< r\le R_j$, then taking into account (5.5) and $R_j\le 4r$, we have

$$\begin{aligned} \underset{B_r}{\mathrm{osc}}\, u \le \underset{B_j}{\mathrm{osc}} \, u \le \omega R_j^\alpha \le C\Bigg (Q(u;x_0,R_0)+R_0^\frac{ps_1-qs_2}{p-q}+ \Vert f \Vert _{L^\gamma (B_{R_0})}^\frac{1}{p-1} R_0^\frac{\gamma s_1p-N}{\gamma (p-1)} \Bigg )\frac{r^\alpha }{R_0^\alpha }. \end{aligned}$$

For the case $\frac{N(p-1)}{N-s_1p}\le 1$ (this forces $p<2$), using the induction hypothesis, we observe that

$$\begin{aligned} M_j-u \le (M_j-m_j)^{2-p} (M_j-u)^{p-1}, \quad \text{ in } B_j, \end{aligned}$$

and this still holds for $(u-m_j)$. Hence,

This implies that

which is similar to (5.2), with $t=p-1$. Multiplying both the sides of the above expression by ${\bar{\varsigma }}:=\varsigma /2^\frac{2-p}{p-1}$, we can employ Corollary 4.1. Then, the rest of the proof follows similarly as before with ${\bar{\varsigma }}$ in place of $\varsigma$. This completes the proof of the theorem. $\square$

Proof of Corollary 2.1

The proof follows by using the boundary behavior of the solution, given by [21, Proposition 3.11], and the interior regularity result of Theorem 2.1. For details, see the proof of [26, Theorem 1.1]. $\square$

6 Strong Maximum Principle

The main goal of this section is to prove our strong maximum principle. We first recall the notion of viscosity solution (see e.g., [30]). To this end, for $u:\mathbb R^N\rightarrow \mathbb R$ and $D\subset \Omega$, set the following:

$$\begin{aligned}&N_u:=\{x\in \Omega : \ \nabla u(x)=0\}, \quad d_u(x):=\mathrm {dist}(x,N_u) \quad \text{ and }\\&C^2_\beta (D):=\biggl \{ u\in C^2(D): \ \sup _{x\in D} \Big ( \frac{\min \{d_u(x),1\}^{\beta -1}}{|\nabla u(x)|}+\frac{|D^2u(x)|}{d_u(x)^{\beta -2}} \Big )<\infty \biggr \}. \end{aligned}$$

Definition 6.1

A function $u:\mathbb R^N\rightarrow [-\infty ,\infty ]$ is said to be a viscosity super-solution to $(-\Delta )_p^{s_1} u+ (-\Delta )_q^{s_2} u\ge 0$ in $\Omega$, if the following hold:

• (i) u is lower semi-continuous in $\Omega$ such that $u <\infty$ a.e. in $\mathbb R^N$ and $u>-\infty$ everywhere in $\Omega$;

• (ii) $u_-\in L_{s_1p}^{p-1}(\mathbb R^N)\cap L_{s_2q}^{q-1}(\mathbb R^N)$;

• (iii) if whenever $B_r(x_0)\subset \Omega$ and $\phi \in C^2(B_r(x_0))$ are such that

  $$\begin{aligned} \phi (x_0)=u(x_0) \quad \text{ and } u(x) \ge \phi (x) \ \ \text{ in } B_r(x_0), \end{aligned}$$

  and one of the following holds

  • (a) $p>\frac{2}{2-s_1}$ and $q>\frac{2}{2-s_2}$ or $\nabla \phi (x_0)\ne 0$,

  • (b) either $p\le \frac{2}{2-s_1}$ or $q\le \frac{2}{2-s_2}$; $\nabla \phi (x_0)=0$ such that $x_0$ is an isolated critical point of $\phi$ and $\phi \in C^2_{\beta }(B_r(x_0))$, for some $\beta >\max \{\frac{s_1p}{p-1},\frac{s_2q}{q-1}\}$;

  then $(-\Delta )_{p}^{s_1} \phi _r(x_0) +(-\Delta )_q^{s_2} \phi _r(x_0) \ge 0$, where

  $$\begin{aligned} \phi _r(x)={\left\{ \begin{array}{ll} \phi (x) &{}\quad \text{ if } x\in B_r(x_0),\\ u(x) &{}\quad \text{ if } x\in \mathbb R^N\setminus B_r(x_0). \end{array}\right. } \end{aligned}$$

  (6.1)

We recall the following weak comparison principle (e.g., see [24]).

Proposition 6.1

Let ${\widetilde{W}}^{s,p}(\Omega ):= \{u\in L^p_{\mathrm{loc}}(\mathbb R^N)\cap L^{p-1}_{sp}(\mathbb R^N): \exists \;U\Supset \Omega \ \text{ with } \ \Vert u\Vert _{W^{s,p}(U)} <\infty \}$. Assume that $u,v\in {\widetilde{W}}^{s_1,p}(\Omega )$ are such that

$$\begin{aligned}&\sum _{(\ell ,s)}{\int _{\mathbb R^{N}}\int _{\mathbb R^N}} \frac{[u(x)-u(y)]^{\ell -1}(\phi (x)-\phi (y))}{|x-y|^{N+s\ell }}dxdy \\&\quad \le \sum _{(\ell ,s)}{\int _{\mathbb R^{N}}\int _{\mathbb R^N}} \frac{[v(x)-v(y)]^{\ell -1}(\phi (x)-\phi (y))}{|x-y|^{N+s\ell }}dxdy \end{aligned}$$

for all $\phi \in W^{s_1,p}_0(\Omega )$ and $u\le v$ in $\Omega ^c$. Then, $u\le v$ in $\Omega$.

Now, we prove that continuous weak super-solutions are viscosity super-solutions.

Lemma 6.1

Let $u\in W^{s_1,p}_0(\Omega )\cap C(\overline{\Omega })$ be a weak super-solution to $(-\Delta )_p^{s_1} u+ (-\Delta )_q^{s_2} u\ge 0$ in $\Omega$. Then, u is also a viscosity super-solution.

Proof

From the assumption on u, it is clear that items (i) and (ii) of Definition 6.1 are satisfied. To prove (iii), on the contrary, assume that there exist $x_0\in \Omega$ and $\phi \in C^2(B_r(x_0))$ such that $\phi (x_0)= u(x_0)$, $u(x)\ge \phi (x)$ in $B_r(x_0)$, either (a) or (b) of Definition 6.1(iii) holds and

$$\begin{aligned} (-\Delta )_{p}^{s_1} \phi _r(x_0) +(-\Delta )_q^{s_2} \phi _r(x_0) <0, \end{aligned}$$

for some $r>0$ and $\phi _r$ given by (6.1). By [30, Lemma 3.8], we have $(-\Delta )_{p}^{s_1} \phi _r(x)$ and $(-\Delta )_q^{s_2} \phi _r(x)$ are continuous at $x_0$. Therefore, there exist $\rho '\in (0,r)$ and $\eta >0$ such that

$$\begin{aligned} (-\Delta )_{p}^{s_1} \phi _r(x) +(-\Delta )_q^{s_2} \phi _r(x) <-\eta \quad \text{ for } \text{ all } x\in B_{\rho '}(x_0). \end{aligned}$$

Following the proof of [30, Lemma 3.9], there exist $\epsilon >0$, $\rho \in (0,\rho '/2)$ and $b\in C_c^2(B_{\rho /2}(x_0))$ such that $b(x_0)=1$ with $0\le b \le 1$, and $\psi _\epsilon (x)=\phi _r(x)+\epsilon b(x)$ satisfies

$$\begin{aligned} \sup _{B_\rho (x_0)}|(-\Delta )_p^{s_1} \psi _\epsilon (x)-(-\Delta )_p^{s_1} \phi _r(x)|<\frac{\eta }{2} \quad \text{ and } \sup _{B_\rho (x_0)}|(-\Delta )_q^{s_2} \psi _\epsilon (x)-(-\Delta )_q^{s_2} \phi _r(x)|<\frac{\eta }{2}. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} (-\Delta )_{p}^{s_1} \psi _\epsilon (x)+(-\Delta )_q^{s_2} \psi _\epsilon (x) \le 0 \quad \text{ for } \text{ all } x\in B_\rho (x_0). \end{aligned}$$

Let $v\in W^{s_1,p}_0(B_\rho (x_0))$ be a non-negative function, then multiplying the above equation with it and upon integration, we get

$$\begin{aligned}&\sum _{(\ell ,s)}{\int _{\mathbb R^{N}}\int _{\mathbb R^N}} \frac{[\psi _\epsilon (x)-\psi _\epsilon (y)]^{\ell -1}(v(x)-v(y))}{|x-y|^{N+s\ell }}d\mu \\&\quad \le 0 \le \sum _{(\ell ,s)}{\int _{\mathbb R^{N}}\int _{\mathbb R^N}} \frac{[u(x)-u(y)]^{\ell -1}(v(x)-v(y))}{|x-y|^{N+s\ell }}d\mu , \end{aligned}$$

where $d\mu =dxdy$. Further, $\psi _\epsilon \le u$ in $B_\rho (x_0)^c$. Therefore, by the weak comparison principle, we obtain $\psi _\epsilon \le u$ in $B_\rho (x_0)$. But this contradicts $\psi _\epsilon (x_0)=\phi (x_0)+\epsilon b(x_0)=u(x_0)+\epsilon >u(x_0)$. Thus, u is a viscosity super-solution. This completes the proof of the lemma. $\square$

Lemma 6.2

Suppose $1<q\le p<\infty$. Let $u\in W^{s_1,p}_0(\Omega )\cap C(\overline{\Omega })$ be a weak super-solution to $(-\Delta )_p^{s_1} u+ (-\Delta )_q^{s_2} u\ge 0$ in $\Omega$ and $u\ge 0$ a.e. in $\mathbb R^N\setminus \Omega$. Then, either $u\equiv 0$ a.e. in $\mathbb R^N$ or $u>0$ in $\Omega$.

Proof

We will show that if $u\not \equiv 0$, then $u>0$ in $\Omega$. By the weak comparison principle, we have $u\ge 0$ a.e. in $\mathbb R^N$.

Case I: If $\Omega$ is connected.

We proceed similarly to [9, Theorem A.1]. Let $K\Subset \Omega$ be any connected compact such that $u\not \equiv 0$ in K. Then, we will show that $u>0$ a.e. in K. Since K is compact, $K\subset \{x\in \Omega : \ \mathrm{dist}(x,\partial \Omega )>2r\}$, for some $r>0$. Moreover, there exists a finite covering $\{B_{r/2}(x_i)\}_{\{i=1,2\dots ,n\}}$ for K such that

$$\begin{aligned} |B_{r/2}(x_i)\cap B_{r/2}(x_{i+1})|>0 \quad \text{ for } \text{ all } i=1,2,\dots ,n-1. \end{aligned}$$

(6.2)

Suppose $u\equiv 0$ on some subset of K with positive measure. Then, for some $i\in \{1,\dots , n-1\}$,

$$\begin{aligned} |E:=\{x\in B_{r/2}(x_i) : \ u(x)=0 \}|>0. \end{aligned}$$

For $\lambda >0$, set

$$\begin{aligned} U_\lambda (x):=\log \Big (1+\frac{u(x)}{\lambda }\Big ), \quad \text{ for } \text{ all } x\in B_{r/2}(x_i). \end{aligned}$$

By observing that $U_\lambda \equiv 0$ on E, for $x\in B_{r/2}(x_i)$ and $y\in E$, we have

$$\begin{aligned} |U_\lambda (x)|^q =\frac{|U_\lambda (x)-U_\lambda (y)|^q}{|x-y|^{N+qs_2}}|x-y|^{N+qs_2}, \end{aligned}$$

which upon integration with respect to $y\in E$ and $x\in B_{r/2}(x_i)$ yields

$$\begin{aligned} |E|\int _{B_{r/2}(x_i)} |U_\lambda (x)|^q dx \le c r^{N+qs_2} \int _{B_{r/2}(x_i)}\int _{B_{r/2}(x_i)} \frac{|U_\lambda (x)-U_\lambda (y)|^q}{|x-y|^{N+qs_2}} dxdy. \end{aligned}$$

(6.3)

On account of

$$\begin{aligned} \bigg |\log \frac{u(x)+\lambda }{u(y)+\lambda }\bigg |^q = |U_\lambda (x)-U_\lambda (y)|^q \end{aligned}$$

and (3.7), we deduce from (6.3) that

$$\begin{aligned} \int _{B_{r/2}(x_i)} \bigg | \log \Big (1+\frac{u(x)}{\lambda }\Big ) \bigg |^q dx \le c\frac{r^{N+s_2q}}{|E|} \big ( r^{N-s_2q}+r^{N-s_1p}(\Vert u\Vert _{L^\infty (B_R)}+\lambda )^{p-q} \big ), \end{aligned}$$

where we have used that $u\ge 0$ in $\mathbb R^N$ (consequently, $u_-=0$). Passing to the limit as $\lambda \rightarrow 0$ in the above expression, we get

$$\begin{aligned} u=0 \quad \text{ a.e. } \text{ in } B_{r/2}(x_i). \end{aligned}$$

By using the property (6.2), we can proceed similarly for balls $B_{r/2}(x_{i-1})$ and $B_{r/2}(x_{i+1})$ (note that $|E:=\{x\in B_{r/2}(x_{i+1}) : \ u(x)=0 \}|\ge |B_{r/2}(x_{i+1})\cap B_{r/2}(x_{i})|>0$) and so on for all $i\in \{1,\dots ,n\}$, that is, $u=0$ a.e. on K. This is a contradiction to our assumption that $u\not \equiv 0$ in K. Thus $u>0$ a.e. in K.

Since $\Omega$ is open and connected (path connected), there exists a sequence of compact connected sets $\Omega _n\subset \Omega$ such that

$$\begin{aligned} \Omega =\cup _{n\in \mathbb N}\Omega _n, \quad \Omega _n\subset \Omega _{n+1} \quad \forall \ n\in \mathbb N\quad \text{ and } u\not \equiv 0 \quad \text{ in } \Omega _n,\quad \text{ for } \text{ all } n\ge n_0. \end{aligned}$$

Then, proceeding as above, we get $u>0$ a.e. in $\Omega _n$ for all $n\ge n_0$. Therefore, $u>0$ a.e. in $\Omega$.

Case II: Let $\Omega$ be any bounded domain.

By the weak comparison principle we have $u\ge 0$ a.e. in $\mathbb R^N$. Next, we will show that if $u\not \equiv 0$ in $\Omega$, then $u\not \equiv 0$ in every connected component of $\Omega$. Suppose there exists a connected component E of $\Omega$ such that $u\equiv 0$ in E. Let $\psi \in W^{s_1,p}_0(E)$ be a test function. Then,

$$\begin{aligned}&0 \le \sum _{(\ell ,s)} \int _{\mathbb R^N} \int _{\mathbb R^N} \frac{[u(x)- u(y)]^{\ell -1}}{|x-y|^{N+s\ell }} \big (\psi (x)-\psi (y)\big )dxdy\\&\quad = -2 \sum _{(\ell ,s)}\int _{E} \int _{E^c} \frac{\big ( u(y)\big )^{\ell -1} \psi (x)}{|x-y|^{N+s\ell }}dxdy. \end{aligned}$$

Thus, $u=0$ in $E^c$, that is, $u=0$ a.e. in $\mathbb R^N$. This is a contradiction to the assumption $u\not \equiv 0$.

To complete the proof of the lemma, we will show that if there exists $x_0\in \Omega$ such that $u(x_0)=0$, then $u(x)=0$ a.e. $x\in \mathbb R^N$. By the above discussion, we have either $u>0$ a.e. in $\Omega$ or $u=0$ a.e. in $\mathbb R^N$. By Lemma 6.1, we have that u is a viscosity super-solution. Thus, proceeding as in [16, Lemma 3.5], we get $u>0$ for all $x\in \Omega$. $\square$

Proceeding similarly to the proof of [21, Theorem 3.12], we have the following result.

Proposition 6.2

Let $1<q\le p<\infty$. Then, for every $\vartheta >0$, there exists a unique solution $w_{\vartheta }\in W^{s_1,p}_0(\Omega )\cap C^{0,\alpha }(\overline{\Omega })$, for some $\alpha \in (0,s_1)$, of the following problem:

$$\begin{aligned} \left\{ \begin{array}{lllll} (-\Delta )^{s_1}_{p}u+ (-\Delta )^{s_2}_{q}u = \vartheta ,&{} \quad u>0 \quad \text{ in } \Omega , \\ u =0 &{} \quad \text{ in } \mathbb R^N\setminus \Omega . \end{array} \right. \qquad \qquad \qquad \qquad (Q_\vartheta ) \end{aligned}$$

Moreover, $w_\vartheta \rightarrow 0$ in $C^{0,\sigma }(\overline{\Omega })$, as $\vartheta \rightarrow 0$, for all $\sigma <\alpha$.

Proof of Theorem 2.3

Without loss of generality, we may assume that g is non-decreasing and $g(0)=0$ (by Jordan’s decomposition). Since $u\not \equiv 0$, there exist $x_0\in \Omega$, $\rho ,\epsilon >0$, and $\vartheta _0\in (0,1)$ such that

$$\begin{aligned} \sup _{\overline{B_\rho (x_0)}} w_{\vartheta _0} \le \inf _{\overline{B_\rho (x_0)}} u -\epsilon , \end{aligned}$$

(6.4)

where $w_{\vartheta _0}$ is the solution to problem $(Q_{\vartheta _0})$. Indeed, there exists $x_0\in \Omega$ such that $u(x_0)>0$. Then by continuity of u and Proposition 6.2 (in particular, $w_\vartheta \rightarrow 0$ in $C_0(\overline{\Omega })$) as $\vartheta \rightarrow 0$, for $\epsilon <u(x_0)/4$, there exist $\rho >0$ and $\vartheta _0\in (0,1)$ such that

$$\begin{aligned} \Vert w_{\vartheta _0}\Vert _{C(\overline{\Omega })} \le u(x_0)/2\le u(x_0)-2\epsilon < u(x)-\epsilon \quad \text{ for } \text{ all } x\in B_\rho (x_0). \end{aligned}$$

For all $\vartheta \in (0,\vartheta _0]$, set the following:

$$\begin{aligned} v_\vartheta :={\left\{ \begin{array}{ll} w_\vartheta &{}\quad \text{ in } \mathbb R^N\setminus \overline{B_{\rho /2}(x_0)}, \\ u &{}\quad \text{ in } \overline{B_{\rho /2}(x_0)}, \end{array}\right. } \end{aligned}$$

where $w_{\vartheta }$ is the solution to problem ($Q_\vartheta$). Since $w_\vartheta \le w_{\vartheta _0}$, on account of (6.4), we have $v_\vartheta \le u$ in $\overline{B_{\rho }(x_0)}$ and $v_\vartheta \in {\widetilde{W}}^{s_1,p}(\Omega \setminus \overline{B_{\rho }(x_0)})$. By the non-local superposition principle [26, Proposition 2.6] and proceeding as in [25, Theorem 2.6], we have weakly in $\Omega \setminus \overline{B_{\rho }(x_0)}$

$$\begin{aligned} (-\Delta )_p^{s_1}v_\vartheta \le (-\Delta )_p^{s_1}w_\vartheta - C_\rho \epsilon ^{p-1} \quad \text{ and } (-\Delta )_q^{s_2}v_\vartheta \le (-\Delta )_q^{s_2}w_\vartheta - C'_\rho \epsilon ^{q-1}. \end{aligned}$$

Choosing $\vartheta \in (0,\vartheta _0]$ small enough, for a positive constant C (independent of $\epsilon$), we obtain

$$\begin{aligned} (-\Delta )_p^{s_1}v_\vartheta + (-\Delta )_q^{s_2}v_\vartheta \le \vartheta -c_\rho \epsilon ^{p-1} \le -C\epsilon ^{p-1}. \end{aligned}$$

Moreover, on account of the fact that $g(v_\vartheta )\rightarrow$ uniformly in $\Omega \setminus \overline{B_{\rho }(x_0)}$, as $\vartheta \rightarrow 0$, for even smaller $\vartheta$ (if necessary), we get

$$\begin{aligned} (-\Delta )_p^{s_1}v_\vartheta&+ (-\Delta )_q^{s_2}v_\vartheta +g(v_\vartheta )\le 0 \le (-\Delta )_p^{s_1}u + (-\Delta )_q^{s_2}u +g(u) \quad \text{ weakly } \text{ in } \\&\quad \Omega \setminus \overline{B_{\rho }(x_0)} \end{aligned}$$

and $v_\vartheta \le u$ in $\mathbb R^N\setminus (\Omega \setminus \overline{B_{\rho }(x_0)})$. Thus, by the weak comparison, we obtain $v_\vartheta \le u$ in $\Omega \setminus \overline{B_{\rho }(x_0)}$. Consequently, using (6.4), we have

$$\begin{aligned} u \ge w_\vartheta \quad \text{ in } \Omega . \end{aligned}$$

Since $0<w_\vartheta \in C(\overline{\Omega })$, it is evident from the proof of [21, Proposition 2.6] that $w_\vartheta \ge c_1 d^{s_1}$, for some positive constant $c_1$. Hence, the result of the theorem follows. $\square$