Eigenvalue for a problem involving the fractional (p, q)-Laplacian operator and nonlinearity with a singular and a supercritical Sobolev growth

Abstract

In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problems for the Dirichlet fractional (p, q)-Laplacian. The nonlinearity considered involves supercritical Sobolev growth. Our approach is variational together with the sub- and supersolution methods, and in this way we can address a wide range of problems not yet contained in the literature. Even when $W^{s_1,p}_0(\Omega ) \hookrightarrow L^{\infty }\left( \Omega \right)$ failing, we establish $\Vert u\Vert _{L^{\infty }\left( \Omega \right) } \le C[u]_{s_1,p}$ (for some $C>0$ ), when u is a solution.

1 Introduction

Let $\Omega \subset {\mathbb {R}}^{N}$ be a bounded domain. In this paper, we study the following singular eigenvalue problem for the Dirichlet fractional (p, q)-Laplacian

with $\lambda >0$, $0<s_2<s_1<1$, $0<\eta <1$ and $1<q<p$.

The fractional p-laplacian operator $(-\Delta _p)^{s}$ is defined as

$$\begin{aligned} (-\Delta _p)^{s}u(x) = C(N,s,p)\lim _{\varepsilon \searrow 0}\int \limits _{\mathrm {I\!R\!}^{N}\setminus B_\varepsilon (x)}\frac{\vert u(x) - u(y) \vert ^{p-2} (u(x)-u(y))}{\vert x-y\vert ^{N+sp}}\; dy\,, \end{aligned}$$

for all $x \in \mathrm {I\!R\!}^{n}$, where C(N, s, p) is a normalization factor. The fractional p-Laplacian is a nonlocal version of the p-Laplacian and is an extension of the fractional Laplacian $(p = 2)$.

In $\left( P_\lambda \right)$, we have the sum of two such operators. So, in problem $\left( P_\lambda \right)$, the differential operator is nonhomogeneous, and this is a source of difficulties in the study of $\left( P_\lambda \right)$. Boundary value problems, driven by a combination of two or more operators of different natures, arise in many mathematical models of physical processes. One of the first such models was introduced by Cahn-Hilliard [5] describing the process of separation of binary alloys. Other applications can be found in Bahrouni-Radulescu-Repovs [1] (on transonic flow problems). Problems with or without singularity involving fractional operators have been considered in different directions, as we can see in [6, 7, 20]. In [8, 19], the authors study singular systems, considering operators of the types (p, q)-Laplacian and fractional (p, q)-Laplacian, respectively. However, none of the works addressed operators of distinct fractional powers or nonlinearities involving supercritical powers.

In the reaction of $\left( P_\lambda \right) , \lambda >0$ is a parameter, $u \mapsto u^{-\eta }$ with $0<\eta <1$ is a singular term and f(z, x) is a Carathéodory perturbation (that is, for all $x \in {\mathbb {R}}, z \mapsto f(z, x)$ is measurable on $\Omega$ and for a.e. $z \in \Omega , x \mapsto f(z, x)$ is continuous). Unlike many authors, we will not assume that for a.e. $z \in \Omega , f(z, \cdot )$ is $(p-1)$-superlinear near $+\infty$. However, this superlinearity of the perturbation $f(z, \cdot )$ is not formulated using the very common in the literature Ambrosetti-Rabinowitz condition (the AR-condition, for short), see Ref. [2]. The main goal of the paper is to explore the existence of a positive solution to $\left( P_\lambda \right)$. Using variational tools from the critical point theory together with truncations and comparison techniques, we show that $\left( P_\lambda \right)$ has a positive solution.

Throughout this paper, to simplify notation, we omit the constant C(N, s, p). From now on, given a subset $\Omega$ of $R^N$ we set $\Omega ^c = R^N\backslash \Omega$ and $\Omega ^2 = \Omega \times \Omega$. The fractional Sobolev spaces $W^{s,p}(\Omega )$ are defined to be the set of functions $u \in L^p(\Omega )$ such that

$$\begin{aligned} \left[ u \right] _{s,p} = \left( \int \limits _{\mathrm {I\!R\!}^{N}}\int \limits _{\mathrm {I\!R\!}^{N}}\frac{\vert u(x) - u(y) \vert ^{p}}{\vert x-y\vert ^{N+sp}}\; dxdy\right) ^{\frac{1}{p}} < \infty . \end{aligned}$$

and we defined the space $W^{s,p}_0(\Omega )$ by

$$\begin{aligned} W^{s,p}_0(\Omega ) = \bigg \{ u \in W^{s,p}(\Omega ); \ \ u = 0 \ \ \text {in} \ \ \Omega ^c\bigg \}. \end{aligned}$$

In [3] the authors showed that,

$$\begin{aligned} W^{s_1,p}_0(\Omega ) \hookrightarrow W^{s_2,q}_0(\Omega ). \end{aligned}$$

Thus, the ideal space to study the problem ($P_\lambda$) is $W^{s_1,p}_0(\Omega )$.

The main spaces that will be used in the analysis of problem ($P_\lambda$) are the Sobolev space $W_0^{s_1,p}(\Omega )$ and the Banach space

$$\begin{aligned} C^0_{s_1}({\overline{\Omega }}) =\big \{ u \in C^0({\overline{\Omega }}); \frac{u}{d_{\Omega }^{s_1}} \ \text {has a continuous extension to } {\overline{\Omega }}\big \}. \end{aligned}$$

where $d_{\Omega }$ is the distance function, $d_{\Omega } = \text {dist}(x,\partial \Omega )$.

On account of the Poincaré inequality, we have that $\left[ .\right] _{s,p}$ is a norm of the Sobolev space $W^{{s_1},p}_0(\Omega )$. Moreover, in [3] the authors show that

$$\begin{aligned} \left[ u \right] _{s_2,p} \le \frac{C}{s_2(s_1-s_2)} \left[ u \right] _{s_1,p}, \ \ \text {for all} \ \ u \in W_0^{s_1,p}(\Omega ), \end{aligned}$$

for $0<s_2<s_1<1$ and $1<p<q<\infty$, in other words, we have $W_{0}^{s_1,p}(\Omega ) \hookrightarrow W_0^{s_2,q}(\Omega )$.

The Banach space $C^0_{s_1}({\overline{\Omega }})$ is ordered with positive (order) cone

$$\begin{aligned} (C^0_{s_1}({\overline{\Omega }}))_+ =\bigg \{f \in C^0_{s_1}({\overline{\Omega }}); \ \ f \ge 0 \ \ \text {in} \ \ \Omega \bigg \} \end{aligned}$$

which is nonempty and has topological interior

$$\begin{aligned} \text {int} \left( C^0_{s_1}({\overline{\Omega }})_+\right) = \bigg \{ v \in C^0_{s_1}({\overline{\Omega }}); \ \ v> 0 \ \ \text {in} \ \ \Omega \ \ \text {and} \ \ \inf \frac{v}{d_{\Omega }^{s_1}} > 0 \bigg \}. \end{aligned}$$

Given $u,v \in W_0^{s_1,p}(\Omega )$ with $u \le v$ we denote

$$\begin{aligned}{}[u,v]&= \{h \in W_0^{s_1,p}(\Omega ); \ u(x) \le h(x) \le v(x) \ \ \text {for a. a.} \ \Omega \}\\ [u)&= \{h \in W_0^{s_1,p}(\Omega ); \ \ u(x) \le h(x) \ \text {for a. a.} \ \Omega \}. \end{aligned}$$

2 The hypotheses

The hypotheses on the perturbation f(x, t) are following:

${{\textbf {H}}}$::

$f: \Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function such that $f(x,0) = 0$ for a. a. $x \in \Omega$ and for each $t >0$ fixed $f(\cdot ,t), \frac{1}{f(\cdot ,t)} \in L^{\infty }(\Omega )$, moreover

• (i) $\displaystyle \lim _{n\rightarrow \infty } \displaystyle \frac{F(x,t)}{t^p} = \infty$ uniformly for a. a. $x \in \Omega$, where $F(x,t) = \displaystyle \int _{0}^{t} f(x,s)\textrm{d}s$;

• (ii) If $e(x,t) = \left[ 1- \displaystyle \frac{p}{1-\eta }\right] t^{1-\eta } + f(x,t).t -pF(x,t)$, then there exists $\beta \in (L^{1}(\Omega ))_+$ such that

  $$\begin{aligned} e(x,t) \le e(x,s) + \beta (x) \ \ \text {for a.e.} \ \ x\in \Omega \ \ \text {all} \ \ 0\le t\le s. \end{aligned}$$

• (iii) There exist $\delta >0$ and $\tau \in (1,q)$ and $c_0 >0$ such that,

  $$\begin{aligned} c_0t^{\tau -1} \le f(x,t) \ \ \text {for a.e.} \ \ x \in \Omega \ \ \text {all} \ \ t \in [0,\delta ] \end{aligned}$$

  and for $s>0$, we have

  $$\begin{aligned} 0<m_s\le f(x,t) \ \ \text {for a.e.} \ \ x \in \Omega \ \ \text {all} \ \ t\ge s. \end{aligned}$$

• (iv) For every $\rho >0$, there exists ${\widehat{E}}_{\rho } >0$ such that for a.e. $x\in \Omega$, the function

  $$\begin{aligned} t\mapsto f(x,t) + {\widehat{E}}_{\rho }t^{p-1} \end{aligned}$$

  is nondecreasing on $[0,\rho ]$.

• (v) We assume that there exists a number $\theta >0$ such that

  $$\begin{aligned} \limsup _{t \rightarrow \infty } \frac{f(x,t)}{t^{p^*_{s_1}-1+\theta }}<+\infty \text{ uniformly } \text{ in } x. \end{aligned}$$

• (vi) At last, we assume that there exists a sequence $\left( M_k\right)$ with $M_k \rightarrow \infty$ and such that, for each $r \in (p, p^*_{s_1})$,

  $$\begin{aligned} t \in \left[ 0, M_k\right] \Longrightarrow \frac{f(x,t)}{t^{r-1}} \le \frac{f\left( x, M_k\right) }{\left( M_k\right) ^{r-1}} \text{ uniformly } \text{ in } x. \end{aligned}$$

The classical AR-condition restricts f(x, .) to have at least $(\mu -1)$-polynomial growth near $\infty$. In contrast, the quasimonotonicity condition that we use in this work (see hypothesis ${{\textbf {H}}} \ (ii)$), does not impose such a restriction on the growth of f(x, .) and permits also the consideration of superlinear nonlinearities with slower growth near $\infty$ (see the examples below). Besides, hypothesis (${{\textbf {H}}} \ (ii)$) is a slight extension of a condition used by Li-Yang [14, condition $(f_4)$].

There are convenient ways to verify (${{\textbf {H}}} \ (ii)$). So, the hypothesis (${{\textbf {H}}} \ (ii)$) holds, if we can find $M>0$ such that for a.e. $x \in \Omega$

• $t \mapsto \displaystyle \frac{t^{-\eta } + f(x,t)}{t^{p-1}} \ \ \text {is nondecreasing on} \ \ [M,\infty )$.

• or $t \mapsto e(x,t) \ \ \text {is nondecreasing on} \ \ [M,\infty )$.

Hypothesis (${{\textbf {H}}} \ (iii)$) implies the presence of a concave term near zero, while hypothesis (${{\textbf {H}}} \ (iv)$) is a one-sided local Hölder condition. It is satisfied if, for a.e. $x \in \Omega$, f(x, .) is differentiable, and for every $\rho >0$, we can find ${\widehat{c}}_\rho$ such that

$$\begin{aligned} -{\widehat{c}}_{\rho }t^{p-1} \le f'_t(x,t)t \ \ \text {for a.e.} \ \ x \in \Omega , \ \ \text {all} \ \ 0\le t \le \rho . \end{aligned}$$

Below we list two examples of functions that satisfy the conditions $({{\textbf {H}}})$

• The function $f_1(x,t) = \left\{ \begin{array}{llll} t^{\tau -1}&{} \textrm{if} \ \ 0\le t \le 1,\\ t^{p^*_{s_1}-1+\theta } &{} \textrm{if} \ t>1, \end{array} \right.$ with $1<\tau<q<p<\theta <p_{s_1}^*$ satisfies the hipotheses $({{\textbf {H}}}$) and also the AR-condition.

• The function $f_2(x,t) = \left\{ \begin{array}{llll} t^{\tau -1}&{} \textrm{if} \ \ 0\le t \le 1,\\ t^{p^*_{s_1}-1+\theta }\ln {t} + t^{s-1} &{} \textrm{if} \ t>1, \end{array} \right.$ with $1<\tau<q<p, \ \ 1< s < p$ satisfies the hipotheses $({{\textbf {H}}}$) but does not satisfy the AR-condition.

3 Preliminary

For any $r>1$ consider the function $J_r:{\mathbb {R}} \rightarrow {\mathbb {R}}$ given by $J_r(t) = \vert t\vert ^{r-2}.t$. Thus, using the arguments of [21], there exists $c_r>0$ and ${\tilde{c}}_r>0$ such that

$$\begin{aligned} \left\langle J_r(z) - J_r(w), z-w \right\rangle&\ge \left\{ \begin{array}{ll} c_r \vert z-w\vert ^{r}, &{} \textrm{if} \ \ r\ge 2,\\ c_r\displaystyle \frac{\vert z-w \vert ^2}{\left( \vert z \vert + \vert w \vert \right) ^{2-r}}, &{} \textrm{if} \ \ r\le 2. \end{array} \right. \end{aligned}$$

(1)

$$\begin{aligned} \vert J_r(t_1) - J_r(t_2) \vert \le&\left\{ \begin{array}{ll} \tilde{c_r} \vert t_1 - t_2\vert ^{r-1}, &{} \textrm{if} \ \ r\le 2,\\ \tilde{c_r}\vert t_1-t_2\vert ^2 . \left( \vert t_1 \vert + \vert t_2 \vert \right) ^{r-2}, &{} \textrm{if} \ \ r\ge 2. \end{array} \right. \end{aligned}$$

(2)

Lemma 1

Let $u, v \in W_0^{s,r}(\Omega )$ and denote $w = u-v$. Then,

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^{2N}} \frac{\big (J_r(u(x) - u(y)) - J_r(v(x) - v(y))\big )\big (w(x) - w(y)\big )}{\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y \\{} & {} \ge \left\{ \begin{array}{ll} c_r \left[ u - v \right] _{s,r}^{r}, &{} \textrm{if} \ \ r\ge 2,\\ c_r\displaystyle \frac{\left[ u-v \right] _{s,r}^2}{\left( \left[ u \right] _{s,r} + \left[ v \right] _{s,r}\right) ^{2-r}}, &{} \textrm{if} \ \ r\le 2. \end{array} \right. \end{aligned}$$

Proof

The case $r \ge 2$, the result is an immediate application of the above inequality.

Case $r\le 2$. Note that, using the Holder inequality we have

$$\begin{aligned}&\int _{{\mathbb {R}}^{2N}} \frac{\vert u(x) - u(y) \vert ^{r}}{\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y = \int _{{\mathbb {R}}^{2N}} \frac{\vert u(x) - u(y) \vert ^{r}}{\vert x-y \vert ^{N+sr}}.\frac{\big (\vert u(x) - u(y) \vert + \vert v(x) - v(y) \vert \big )^{\frac{r(2-r)}{2}}}{\big (\vert u(x) - u(y) \vert + \vert v(x) - v(y) \vert \big )^{\frac{r(2-r)}{2}}}\textrm{d}x \textrm{d}y\\&= \int _{{\mathbb {R}}^{2N}} \left[ \frac{\vert u(x) - u(y) \vert }{\left( \vert u(x) - u(y) \vert + \vert v(x) - v(y) \vert \right) ^{\frac{(2-r)}{2}}\vert x-y \vert ^{\frac{N+sr}{2}}}\right] ^{r}\\&\frac{\big (\vert u(x) - u(y) \vert + \vert v(x) - v(y) \vert \big )^{\frac{r(2-r)}{2}}}{\vert x-y\vert ^{\frac{2-r}{2}}}\textrm{d}x \textrm{d}y\\&\le \left( \int _{{\mathbb {R}}^{2N}} \frac{\vert u(x) - u(y) \vert ^2}{\big (\vert u(x) - u(y) \vert + \vert v(x) - v(y) \vert \big )^{2-r}\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y\right) ^{\frac{r}{2}} \left( \left[ u \right] _{s,r} + \left[ v \right] _{s,r} \right) ^{\frac{r(2-r)}{2}} \end{aligned}$$

Thus, using the inequality (1) we have

$$\begin{aligned}&\left( \frac{\left[ u-v \right] _{s,r}^r}{\left( \left[ u \right] _{s,r} + \left[ v \right] _{s,r}\right) ^{\frac{r(2-r)}{2}}}\right) ^{\frac{2}{r}} \le \int _{\mathbb {R^{2N}}} \frac{\vert u(x) - u(y) \vert ^2}{\left( \vert u(x) - u(y) \vert + \vert v(x) - v(y) \vert \right) ^{2-r}\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y \\&\le \frac{1}{c_r} \int _{{\mathbb {R}}^{2N}} \frac{\big (J_r(u(x) - u(y)) - J_r(v(x) - v(y))\big )\big ((u-v)(x) - (u-v)(y)\big )}{\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y. \\ \end{aligned}$$

$\square$

For every $1< r < \infty$, denote by $A_{s,r}:W_0^{s,r}(\Omega ) \rightarrow \left( W_0^{s,r}(\Omega )\right) ^{*}$ the nonlinear map defined by

$$\begin{aligned} \langle A_{s,r}(u),\varphi \rangle = \int _{{\mathbb {R}}^{2N}} \frac{J_r(u(x)-u(y))(\varphi (x) - \varphi (y))}{\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y, \ \ \text {for all} \ \ u, \varphi \in W_{0}^{s,r}(\Omega ). \end{aligned}$$

An immediate consequence of Lemma 1 is the following proposition

Proposition 1

The map $A_{s,r}:W_0^{s,r}(\Omega ) \rightarrow \left( W_0^{s,r}(\Omega )\right) ^{*}$ maps bounded sets to bounded sets, is continuous, strictly monotone and satisfies,

$$\begin{aligned} u_n \rightharpoonup u \ \ \text {in} \ \ W_0^{s,r}(\Omega ) \ \text {and} \ \limsup _{n\rightarrow \infty } \ \langle A_{s,r}(u_n),(u_n-u) \rangle \le 0 \Rightarrow u_n \rightarrow u \ \ \text {in} \ \ W_0^{s,r}(\Omega ). \end{aligned}$$

Proof

Indeed, using the inequality (2) we have

$$\begin{aligned} \Vert A_{s,r}(u) - A_{s,r}(w) \Vert _{*} \le \left\{ \begin{array}{ll} \tilde{c_r} \left[ u - w \right] _{s,r}^{r-1}, &{} \textrm{if} \ \ r\le 2,\\ \tilde{c_r}\left[ u-w\right] _{s,r}^2. \left( \left[ u \right] _{s,r} + \left[ w \right] _{s,r}\right) ^{r-2}, &{} \textrm{if} \ \ r\ge 2. \end{array} \right. \end{aligned}$$

and thus $A_{s,r}$ maps bounded sets to bounded sets, is continuous.

Moreover, if $p\ge 2$ then using also the Lemma 1 results,

$$\begin{aligned}&\lim _{n \rightarrow \infty } c_r\left[ u_n - u \right] _{s,r}^2\\&\le \lim _{n\rightarrow \infty }\int _{{\mathbb {R}}^{2N}} \frac{\big (J_r(u_n(x) - u_n(y)) - J_r(u(x) - u(y))\big )\big ((u_n-u)(x) - (u_n-u)(y)\big )}{\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y \\&= \limsup _{n\rightarrow \infty }\bigg \langle A_{s,r}(u_n) - A_{s,r}(u), u_n-u \bigg \rangle \le 0, \end{aligned}$$

and if $p\le 2$ let’s use again the Lemma 1 and obtain

$$\begin{aligned}&c_r\frac{\left[ u_n - u \right] _{s,r}^2}{\left( \left[ u_n \right] _{s,r} + \left[ u \right] _{s,r}\right) ^{2-r}}\\&\le \int _{{\mathbb {R}}^{2N}} \frac{\big (J_r(u_n(x) - u_n(y)) - J_r(u(x) - u(y))\big )\big ((u_n-u)(x) - (u_n-u)(y)\big )}{\vert x-y \vert ^{N+sr}} \textrm{d}x \textrm{d}y \\&= \bigg \langle A_{s,r}(u_n) - A_{s,r}(u), u_n-u \bigg \rangle \end{aligned}$$

thus, if $u_n \rightharpoonup u$ in $W^{s,r}_{0}(\Omega )$ and $\displaystyle \limsup _{n\rightarrow \infty } A_{s,r}(u_n).(u_n-u) \le 0$ then, there exists $M>0$ such that $\Vert u_n \Vert _{s,r} \le M$ and thus

$$\begin{aligned}&\lim _{n\rightarrow \infty } c_r\frac{\left[ u_n - u\right] _{s,r}^2}{\left( M + \left[ u \right] _{s,r}\right) ^{2-r}} \le \lim _{n\rightarrow \infty }c_r\frac{\left[ u_n - u\right] _{s,r}^2}{\left( \left[ u_n \right] _{s,r} + \left[ u \right] _{s,r}\right) ^{2-r}} \\&\le \limsup _{n\rightarrow \infty }\bigg \langle A_{s,r}(u_n) - A_{s,r}(u), u_n - u \bigg \rangle \le 0. \end{aligned}$$

Consequently, for all $1<p<\infty$, we have $u_n \rightarrow u$ in $W_{0}^{s,r}(\Omega )$. $\square$

The following result is a natural improvement of [15, Lemma 9] to the Dirichlet fractional (p, q)-Laplacian.

Proposition 2

(Weak comparison principle) Let $0<s_1<s_2<1$, $1<q<p$, $\Omega$ be bounded in ${\mathbb {R}}^{N}$ and $u,v \in W_0^{s_1,p}(\Omega ) \cap C_{s_1}^{0}({\overline{\Omega }})$. Suppose that,

$$\begin{aligned} \bigg \langle A_{s_1,p}(u) + A_{s_2,q}(u), (u-v)^+ \bigg \rangle \le \bigg \langle A_{s_1,p}(v) + A_{s_2,q}(v),(u-v)^+ \bigg \rangle \end{aligned}$$

then $u \le v$.

Proof

The proof is a straightforward calculation, but for convenience of the reader we present a sketch of it. By considering the equations for both p and q, and subtracting them and adjusting the terms, we obtain

$$\begin{aligned} \bigg \langle A_{s_1,p}(u) + A_{s_2,q}(u), (u-v)^+ \bigg \rangle - \bigg \langle A_{s_1,p}(v) + A_{s_2,q}(v),(u-v)^+ \bigg \rangle \le 0. \end{aligned}$$

(3)

Using the identity

$$\begin{aligned} J_{m}(b) - J_m(a) = (m-1)(b-a)\displaystyle \int _{0}^{1} \vert a + t(b-a) \vert ^{m-2} \textrm{d}t \end{aligned}$$

for $a = v(x) - v(y)$ and $b = u(x)- u(y)$, we have

$$\begin{aligned} J_{m}(u(x) - u(y)) - J_{m}(v(x) - v(y)) = (m-1)\left[ (u-v)(x) - (u-v)(y) \right] Q_{m}(x,y), \end{aligned}$$

where $Q_{m}(x,y) = \displaystyle \int _{0}^{1}\left| (v(x) - v(y)) + t[(u-v)(x) - (u-v)(y)]\right| ^{m-2}\textrm{d}t$.

We have $Q_{m}(x,y) \ge 0$ and $Q_{m}(x,y) = 0$ only if $v(x) = v(y)$ and $u(x) = u(y)$. Rewriting the integrands in (3) we obtain

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^{2N}} \left( \frac{(p-1)\left[ (u-v)(x) - (u-v)(y) \right] Q_{p}(x,y)}{\vert x-y\vert ^{N+sp}}\right) ((u-v)^+(x)-(u-v)^+(y))\textrm{d}x\textrm{d}y\\{} & {} + \int _{{\mathbb {R}}^{2N}} \left( \frac{(q-1)\left[ (u-v)(x) - (u-v)(y) \right] Q_{q}(x,y)}{\vert x-y\vert ^{N+sq}}\right) ((u-v)^+(x)\\{} & {} \quad -(u-v)^+(y)) \textrm{d}x\textrm{d}y \le 0. \end{aligned}$$

We now consider

$$\begin{aligned} \psi = u-v = (u-v)^{+} - (u-v)^{-}, \quad \varphi = (u-v)^{+} = \psi ^{+}. \end{aligned}$$

It follows from the last inequality that

$$\begin{aligned} \displaystyle \int _{{\mathbb {R}}^{2N}}&\left( \frac{(p-1)(\psi (x) - \psi (y) )(\psi ^{+}(x)-\psi ^{+}(y))Q_{p}(x,y)}{\vert x-y\vert ^{N+sp}}\right) \textrm{d}x\textrm{d}y \\&+\int _{{\mathbb {R}}^{2N}} \left( \frac{(q-1)(\psi (x) - \psi (y) )(\psi ^{+}(x)-\psi ^{+}(y))Q_{q}(x,y)}{\vert x-y\vert ^{N+sq}}\right) \textrm{d}x\textrm{d}y \le 0. \end{aligned}$$

Applying the inequality $(\xi - \eta )(\xi ^{+} - \eta ^{+}) \ge \vert \xi ^{+} - \eta ^{+} \vert ^{2}$ we obtain

$$\begin{aligned}&\int _{{\mathbb {R}}^{2N}} \frac{(p-1)\vert \psi ^{+}(x)-\psi ^{+}(y)\vert ^{2}Q_{p}(x,y)}{\vert x-y\vert ^{N+sp}} \textrm{d}x\textrm{d}y \\&\quad + \int _{{\mathbb {R}}^{2N}}\frac{(q-1)\vert \psi ^{+}(x)-\psi ^{+}(y)\vert ^{2}Q_{q}(x,y)}{\vert x-y\vert ^{N+sq}} \textrm{d}x\textrm{d}y \le 0. \end{aligned}$$

Thus, at almost every point (x, y) we have $\psi ^{+}(x) = \psi ^{+}(y)$ or

$$\begin{aligned} Q_{p}(x,y) = Q_{q}(x,y) = 0. \end{aligned}$$

Since $Q_{p}(x,y) = Q_{q}(x,y) = 0$ also imply $\psi ^{+}(x) = \psi ^{+}(y)$, we conclude that

$$\begin{aligned} (u-v)^{+}(x) = C \ge 0, \ \ \forall x \in {\mathbb {R}}^{N} \end{aligned}$$

and since, $u,v \in W_{0}^{s_1,p}(\Omega )$, results that $C=0$ and consequently $u\le v$. $\square$

Proposition 3

(Strong comparison principle) Let $0<s_1<s_2<1$, $1<q<p$, $\Omega$ be bounded in ${\mathbb {R}}^{N}$, $g \in C^{0}({\mathbb {R}}) \cap BV_{loc}({\mathbb {R}})$, $u,v \in W_0^{s_1,p}(\Omega ) \cap C_{s_1}^{0}({\overline{\Omega }})$ such that $u \ne v$ and $K>0$ satisfy,

$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta _p)^{s_1}u + (-\Delta _p)^{s_1}u + g(u) \le (-\Delta _p)^{s_1}v + (-\Delta _q)^{s_2}v + g(v) \le K \ \ {} &{} \mathrm{weakly \ in} \ \ \Omega ,\\ 0<u\le v \ \ {} &{}\textrm{in} \ \ \Omega . \end{array} \right. \end{aligned}$$

then $u \le v$ in $\Omega$. In particular, if $u,v \in \text {int}[(C_{s_1}^{0}({\overline{\Omega }})^+)]$ then $v-u \in \text {int}[(C_{s_1}^{0}({\overline{\Omega }})^+)]$.

Proof

Without loss of generality, we may assume that g is nondecreasing and $g(0) = 0$. In fact, by Jordan’s decomposition we can find $g_1,g_2 \in C^{0}({\mathbb {R}})$ nondecreasing such that $g(t) = g_1(t) - g_2(t)$ and $g_1(0) = 0$.

Since, $u \ne v$ by continuity, we can find $x_0 \in \Omega$, $\rho ,\varepsilon > 0$ such that $\overline{B_{\rho }(x_0)} \subset \Omega$ and

$$\begin{aligned} \sup _{\overline{B_\rho (x_0)}} u < \inf _{\overline{B_\rho (x_0)}}v - \varepsilon . \end{aligned}$$

Hence, for all $\eta > 1$ close enough to 1 we have

$$\begin{aligned} \sup _{\overline{B_\rho (x_0)}} \eta u < \inf _{\overline{B_\rho (x_0)}}v - \frac{\varepsilon }{2}. \end{aligned}$$

Define $w_\eta \in W_{0}^{s_1,p}(\Omega \backslash \overline{B_{\rho }(x_0)})$ by

$$\begin{aligned} w_\eta (x) =\left\{ \begin{array}{ll} \eta u(x), &{} \textrm{if} \ \ x \in \overline{B_{\rho }(x_0)}^{c},\\ v(x), &{} \textrm{if} \ \ x \in \overline{B_{\rho }(x_0)}, \end{array} \right. \end{aligned}$$

so $w_\eta \le v(x)$ in $\overline{B_{\rho }(x_0)}$ and by the nonlocal superposition principle ([11], Proposition 2.6) we have weakly in $\Omega \backslash \overline{B_{\rho }(x_0)}$

$$\begin{aligned} (-\Delta _p)^{s_1} w_\eta \le \eta ^{p-1} (-\Delta _p)^{s_1} u - C_{\rho }\varepsilon ^{p-1} \ \ \text {and} \ \ (-\Delta _q)^{s_2} w_\eta \le \eta ^{q-1} (-\Delta _q)^{s_2} u - C_{\rho }\varepsilon ^{q-1} \end{aligned}$$

for some $C_{\rho }>0$ and all $\eta > 1$ close enough to 1. Further, we have weakly in $\Omega \backslash \overline{B_{\rho }(x_0)}$

$$\begin{aligned}&(-\Delta _p)^{s_1} w_\eta + (-\Delta _q)^{s_2} w_\eta + g(w_\eta ) \le \eta ^{p-1} (-\Delta _p)^{s_1} u\\&+ \eta ^{q-1}(-\Delta _q)^{s_2} u + g(w_\eta )- C_{\rho }\varepsilon ^{q-1} - C_{\rho }\varepsilon ^{p-1} \\&\le \eta ^{p-1}\bigg ( (-\Delta _p)^{s_1} u + (-\Delta _q)^{s_2} u + g(u)\bigg ) \\&+ \bigg (g(w_\eta ) - \eta ^{p-1}g(u)\bigg )-C_{\rho }\varepsilon ^{q-1} - C_{\rho }\varepsilon ^{p-1}\\&\le \bigg ( (-\Delta _p)^{s_1} u + (-\Delta _q)^{s_2} u + g(u)\bigg ) + \bigg (g(w_\eta ) - \eta ^{p-1}g(u)\bigg ) \\&+ K\big (\eta ^{p-1} - 1\big ) - C_{\rho }\varepsilon ^{q-1} - C_{\rho }\varepsilon ^{p-1}\\&\le \bigg ( (-\Delta _p)^{s_1} v + (-\Delta _q)^{s_2} v + g(v)\bigg ) + \bigg (g(w_\eta ) - \eta ^{p-1}g(u)\bigg ) + K\big (\eta ^{p-1} - 1\big )\\&- C_{\rho }\varepsilon ^{q-1} - C_{\rho }\varepsilon ^{p-1}. \end{aligned}$$

Since

$$\begin{aligned} \bigg (g(w_\eta ) - \eta ^{p-1}g(u)\bigg ) + K\big (\eta ^{p-1} - 1\big ) \rightarrow 0 \end{aligned}$$

uniformly in $\Omega \backslash \overline{B_{\rho }(x_0)}$ as $\eta \rightarrow 1^{+}$, we have, for all $\eta >1$ close enough to 1,

$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta _p)^{s_1}w_{\eta } + (-\Delta _p)^{s_1}w_{\eta } + g(w_\eta ) \le (-\Delta _p)^{s_1}v + (-\Delta _q)^{s_2}v + g(v) \le K \\ \qquad \qquad \qquad \qquad \mathrm{weakly \ in} \ \ \Omega \backslash \overline{B_{\rho }(x_0)},\\ 0<w_{\eta } \le v \ \ \textrm{in} \ \ \left( \Omega \backslash \overline{B_{\rho }(x_0)}\right) . \end{array} \right. \end{aligned}$$

Testing with $\varphi = (w_{\eta } - v)^{+} \in W_{0}^{s_1}(\Omega )\backslash \overline{B_{\rho }(x_0)}$, recalling the monotonicity of g, and applying Proposition 2 we get $v > w_\eta$ in $\Omega )\backslash \overline{B_{\rho }(x_0)}$. So we have

$$\begin{aligned} v \ge \eta u \ge u. \end{aligned}$$

In particular, if $u, v \in \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }}))_+\right]$ then

$$\begin{aligned} \inf _{\Omega } \frac{v-u}{d^{s_1}_{\Omega }} \le \inf _{\Omega }\frac{(\eta -1)u}{d^{s_1}_{\Omega }} > 0 \end{aligned}$$

and so $v-u \in \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }}))_+\right]$. $\square$

4 An auxiliary problem

Firstly, we will need to define, with the help of the real sequence defined in H(vii), a sequence of auxiliary equations that will be important for our purpose. More specifically, for each $k \in {\mathbb {N}}$, we define the auxiliary truncation functions by choosing $r \in \left( p,p^*_{s_1}\right)$ such that $p^*_{s_1}-r<\theta$ and we set

$$\begin{aligned} f_k(x,t)=\left\{ \begin{array}{ll} 0, &{} \text {if}\ t \le 0 \\ f(x,t), &{} \text {if} \ 0 \le t \le M_k \\ \displaystyle \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r-1}, &{} \ \ \text {if} \ \ t \ge M_k. \end{array}\right. \end{aligned}$$

(4)

Notice that we define $f_k$ to be such that r in its definition is independent of k. We see that we are really truncating our original function, making it subcritical for large arguments. Furthermore, in view of conditions H(vi), H(vii) and the choice of $\theta$, we can prove that, for k big enough, $f_k$ satisfies, for a constant $C>0$,

$$\begin{aligned} \left| f_k(x,t)\right| \le C\left( M_k\right) ^{2 \theta }|t|^{r-1}. \end{aligned}$$

(5)

Indeed, for all $t>0$, condition H(vii) and (4) gives

$$\begin{aligned} f_k(x,t) \le \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r-1} \end{aligned}$$

and, by H(vi), if k is sufficiently large,

$$\begin{aligned} \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} \le C\left( M_k\right) ^{p^*_{s_1}-r+\theta } \le C\left( M_k\right) ^{2 \theta }. \end{aligned}$$

For each $k \in {\mathbb {N}}$, let us consider the following auxiliary problem

with $\lambda >0$, $0<s_1<s_2<1$, $0<\eta <1$ and $1<q<p$.

By the hypotheses $({{\textbf {H}}})$, the hypotheses on the truncation $f_k(x,t)$ are following:

${{\textbf {H}}}_k$::

$f_k: \Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function such that $f_k(x,0) = 0$ for a. a. $x \in \Omega$ and

• (i) $f_k(x,t) \le \alpha _k(x) [1 + t^{r -1}]$ for a. a. $x \in \Omega$ all $t \ge 0$ with $\alpha _k \in L^{\infty }(\Omega )$ and $p< r < p^*_{s_1} = \displaystyle \frac{NP}{N-s_1p}$;

• (ii) $\displaystyle \lim _{t\rightarrow \infty } \displaystyle \frac{F_k(x,t)}{t^p} = \infty$ uniformly for a. a. $x \in \Omega$, where $F_k(x,t) = \displaystyle \int _{0}^{t} f_k(x,s)\textrm{d}s$;

• (iii) If $e_k(x,t) = \left[ 1- \displaystyle \frac{p}{1-\eta }\right] t^{1-\eta } + f_k(x,t).t -pF_k(x,t)$, then there exists $\beta _k \in (L^{1}(\Omega ))_+$ such that

  $$\begin{aligned} e_k(x,t) \le e_k(x,s) + \beta _k(x) \ \ \text {for a.e.} \ \ x\in \Omega \ \ \text {all} \ \ 0\le t\le s. \end{aligned}$$

• (iv) There exist $\delta >0$ and $\tau \in (1,q)$ and $c_0 >0$ such that,

  $$\begin{aligned} c_0t^{\tau -1} \le f_k(x,t) \ \ \text {for a.e.} \ \ x \in \Omega \ \ \text {all} \ \ t \in [0,\delta ] \end{aligned}$$

  and for all $s>0$, we have

  $$\begin{aligned} 0<m_{k,s}\le f_k(x,t) \ \ \text {for a.e.} \ \ x \in \Omega \ \ \text {all} \ \ t\ge s. \end{aligned}$$

• (v) For every $\rho >0$, there exists ${\widehat{E}}_{k,\rho } >0$ such that for a.e. $x\in \Omega$, the function

  $$\begin{aligned} t\mapsto f_k(x,t) + {\widehat{E}}_{k,\rho }t^{p-1} \end{aligned}$$

  is nondecreasing on $[0,\rho ]$.

The hypothesis (${{\textbf {H}}}_k \ (i)$) holds by (5), (${{\textbf {H}}}_k \ (ii)$) holds by (4) and $p<r$. We will prove first that (${{\textbf {H}}}_k \ (iv)$) holds. Since $\delta >0$, $\tau \in (1,q)$ and $c_0 >0$, if $\delta <M_k$, we have

$$\begin{aligned} c_0t^{\tau -1} \le f(x,t)=f_k(x,t) \ \ \text {for a.e.} \ \ x \in \Omega \ \ \text {all} \ \ t \in [0,\delta ]. \end{aligned}$$

For $s>0$, we have

$\bullet$ $0<s\le t \le M_k$,

$$\begin{aligned} f_k(x,t)=f(x,t)\ge m_s>0, \end{aligned}$$

by (${{\textbf {H}}} \ (iii)$).

$\bullet$ $0<s\le M_k <t$,

$$\begin{aligned} f_k(x,t)=\frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r-1}\ge \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} M_k^{r-1} = f\left( x,M_k\right) >0. \end{aligned}$$

$\bullet$ $0< M_k <s\le t$,

$$\begin{aligned} f_k(x,t)=\frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r-1}\ge \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} M_k^{r-1} = f\left( x,M_k\right) >0. \end{aligned}$$

So, for all $s>0$ we have

$$\begin{aligned} f_k(x,t)\ge m_{k,s}>0 \ \ \text {for a.e.} \ \ x \in \Omega \ \ \text {all} \ \ t\ge s, \end{aligned}$$

with $m_{k,s}=\max \left\{ m_s, \displaystyle \inf _{x \in \Omega }f(x,M_k)\right\} >0$.

To prove that (${{\textbf {H}}}_k \ (iii)$) holds it is sufficiently verify that there is a constant $C_k>0$ such that $t \mapsto e_k(x,t) \ \ \text {is nondecreasing on} \ \ [C_k,\infty )$. Since for $t\ge M_k$ we have

$$\begin{aligned} \begin{array}{rcl} e_k(x,t) &{}=&{} \left[ 1- \displaystyle \frac{p}{1-\eta }\right] t^{1-\eta } + f_k(x,t).t -pF_k(x,t)\\ &{}=&{}\displaystyle \left[ 1- \frac{p}{1-\eta }\right] t^{1-\eta } + \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r} - p\int _0^{M_k}f(x,s)ds - \int _{M_k}^t\frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} s^{r-1}ds \\ &{}=&{}\displaystyle \left[ 1- \frac{p}{1-\eta }\right] t^{1-\eta } + \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r} - p\int _0^{M_k}f(x,s)ds - \frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}}\frac{1}{r}[t^r-M_k^r]. \end{array} \end{aligned}$$

Hence

$$\begin{aligned} \begin{array}{rcl} \displaystyle \frac{\partial }{\partial t}e_k(x,t)= & {} \displaystyle \left[ 1-\eta - p\right] t^{-\eta } + (r-1)\frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r-1}. \end{array} \end{aligned}$$

Notice that $\frac{\partial }{\partial t}e_k(x,t)\ge 0$ if

$$\begin{aligned} \left[ 1-\eta - p\right] t^{-\eta } + (r-1)\frac{f\left( x,M_k\right) }{\left( M_k\right) ^{r-1}} t^{r-1}\ge 0, \end{aligned}$$

or equivalently, if

$$\begin{aligned} t\ge \left( -\left[ 1-\eta - p\right] \frac{\left( M_k\right) ^{r-1}}{(r-1)f(x,M_k)}\right) ^{\frac{1}{r+\eta }}. \end{aligned}$$

We can consider

$$\begin{aligned} C_k=\left( -\left[ 1-\eta - p\right] \frac{\left( M_k\right) ^{r-1}}{(r-1)m_{k,s}}\right) ^{\frac{1}{r+\eta }}, \end{aligned}$$

where $m_{k,s}$ is as in (${{\textbf {H}}}_k \ (iv)$). Hence, $t \mapsto e_k(x,t) \ \ \text {is nondecreasing on} \ \ [C_k,\infty )$. The proof of (${{\textbf {H}}}_k \ (v)$) follows from (4) and (${{\textbf {H}}} \ (iv)$).

Definition 1

A function $u \in W^{s_1,p}_0(\Omega )$ is a weak solution of the problem ($P_{k,\lambda }$) if, $u^{-\eta }\varphi \in W_0^{s_1,p}(\Omega )$ for all $\varphi \in W_0^{s_1,p}(\Omega )$ and

$$\begin{aligned} \bigg \langle A_{s_1,p}(u) + A_{s_2,q}(u),\varphi \bigg \rangle = \int _{\Omega } \lambda \left[ u^{-\eta } + f_k(x,u)\right] \varphi \textrm{d}x, \ \ \text {for all} \ \ \varphi \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

The difficulty that we encounter in the analysis of problem ($P_{k,\lambda }$) is that the energy (Euler) function of the problem $I_{\lambda }: W_{0}^{s_1,p}(\Omega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} I_\lambda (u) = \frac{1}{p} \left[ u \right] _{s_1,p}^p + \frac{1}{q} \left[ u \right] _{s_2,p}^{q} - \lambda \int _{\Omega } \left[ \frac{1}{1-\eta }(u^{+})^{1-\eta } + F_k(x,u^+)\right] \textrm{d}x. \end{aligned}$$

(6)

for all $u \in W_{0}^{s_1,p}(\Omega )$, is not $C^{1}$ (due to the singular term). So, we can not use the minimax methods of critical point theory directly on $I_\lambda (.)$. We have to find ways to bypass the singularity and deal with $C^1$-functionals.

The hypotheses ${{\textbf {H}}} \ (i)$ and ${{\textbf {H}}} \ (iv)$ assure us that, there are $c_0 > 0$ and $c_2 > 0$ such that,

$$\begin{aligned} f_k(x,z) \ge c_0z^{\tau - 1} - c_2z^{\theta -1}, \ \ \text {for a. a.} \ \ x \in \Omega \ \ \text {and} \ \ z \ge 0. \end{aligned}$$

(7)

We consider the following auxiliary Dirichilet fractional (p, q)-equation

$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta _p)^{s_1}u + (-\Delta _q)^{s_2}u = \lambda \left[ c_0u(x)^{\tau -1} - c_2u^{\theta -1}\right] &{} \textrm{in} \ \ \Omega ,\\ u= 0 &{} \textrm{in} \ \mathrm {I\!R\!}^N\setminus \Omega ,\\ u> 0 &{} \textrm{in} \ \Omega \end{array} \right. \end{aligned}$$

(8)

with $0<s_2<s_1$, $\lambda > 0$ and $1<\tau<q<p<\theta < p_s^*= \displaystyle \frac{Np}{N-sp}$.

Lemma 2

If ${\underline{u}}_{\lambda }\in W_{0}^{s_1,p}(\Omega )$ be a weak solution of problem (8). Then ${\underline{u}}_{\lambda }\in L^\infty (\Omega )$.

Proof

We denote by $h_{\lambda }(t) = \lambda c_0 t^{\tau - 1} - \lambda c_2t^{\theta -1}$. Thus,

$$\begin{aligned} \begin{aligned}&\langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }),\phi \rangle \\&= \int _{{\mathbb {R}}^{2N}}\left( \frac{J_p({\underline{u}}_{\lambda }(x)-{\underline{u}}_{\lambda }(y))}{|x-y|^{N+s_1p}} + \frac{J_q({\underline{u}}_{\lambda }(x)-{\underline{u}}_{\lambda }(y)) }{|x-y|^{N+s_2q}}\right) (\phi (x) - \phi (y))\textrm{d}x\textrm{d}y \\&=\int _\Omega h_{\lambda }({\underline{u}}_{\lambda })\phi \textrm{d}x \end{aligned} \end{aligned}$$

(9)

for any $\phi \in W_{0}^{s_1,p}(\Omega )$.

For each $k\in {\mathbb {N}}$, set

$$\begin{aligned} \Omega _k:=\{x\in \Omega ~:~u(x)>k\}. \end{aligned}$$

Since ${\underline{u}}_{\lambda }\in W_{0}^{s_1,p}(\Omega )$ and ${\underline{u}}_{\lambda } \ge 0$ in $\Omega$, we have that $({\underline{u}}_{\lambda }-k)^+\in W_{0}^{s_1,p}(\Omega )$. Taking $\phi = ({\underline{u}}_{\lambda } - k)^+$ in (9), we obtain

$$\begin{aligned} \left\langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }),\phi \right\rangle =\int _\Omega h_{\lambda }({\underline{u}}_{\lambda })({\underline{u}}_{\lambda }-k)^+\textrm{d}x. \end{aligned}$$

(10)

Applying the algebraic inequality $|a-b|^{p-2}(a-b)(a^+-b^+)\ge |a^+-b^+|^p$ to estimate the left-hand side of (10), we obtain

$$\begin{aligned} \left( \int _{\Omega _k}({\underline{u}}_{\lambda }-k)^{p^*_s}\textrm{d}x\right) ^{\frac{p}{p^*_s}}&\le C\int _{{\mathbb {R}}^{2N}}\frac{|{\underline{u}}_{\lambda }(x)-{\underline{u}}_{\lambda }(y)|^p}{|x-y|^{N+sp}}\textrm{d}x\textrm{d}y\nonumber \\&\le C \big \langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }),\phi \big \rangle \nonumber \\&= C\int _{\Omega _{k}} h_{\lambda }({\underline{u}}_{\lambda })({\underline{u}}_{\lambda }-k)\textrm{d}x\nonumber \\&= C\int _{\Omega _{k}} \big [\lambda c_0 {\underline{u}}_{\lambda }^{\tau - 1} - \lambda c_2{\underline{u}}_{\lambda }^{\theta -1}\big ]({\underline{u}}_{\lambda }-k)\textrm{d}x \nonumber \\&\le C\int _{\Omega _{k}} \lambda c_0 {\underline{u}}_{\lambda }^{\tau - 1} ({\underline{u}}_{\lambda }-k)\textrm{d}x. \end{aligned}$$

(11)

Since $1<\tau <p$, for $k>1$ in $\Omega _k$ we have

$$\begin{aligned} {\underline{u}}_{\lambda }^{\tau -1}({\underline{u}}_{\lambda }-k)\le {\underline{u}}_{\lambda }^{p-1}({\underline{u}}_{\lambda }-k)\le 2^{p-1}({\underline{u}}_{\lambda }-k)^p+2^{p-1}k^{p-1}({\underline{u}}_{\lambda }-k) \end{aligned}$$

and thus,

$$\begin{aligned} \int _{\Omega }{\underline{u}}_{\lambda }^{\tau -1}({\underline{u}}_{\lambda }-k)\textrm{d}x\le 2^{p-1}\int _{\Omega }({\underline{u}}_{\lambda }-k)^p \textrm{d}x+2^{p-1}k^{p-1}\int _{\Omega _k}({\underline{u}}_{\lambda }-k)\textrm{d}x. \end{aligned}$$

(12)

Applying Hölder’s inequality, we obtain

$$\begin{aligned} \int _{\Omega _k}({\underline{u}}_{\lambda }-k)^p\textrm{d}x\le |\Omega _k|^{\frac{p^*_s-p}{p^*_s}}\left( \int _{\Omega _k}({\underline{u}}_{\lambda }-k)^{p^*_s} \textrm{d}x\right) ^{\frac{p}{p^*_s}}. \end{aligned}$$

(13)

So, using the inequalities (12) and (13) in (11), we have

$$\begin{aligned} \int _{\Omega _k}({\underline{u}}_{\lambda } -k)^p\textrm{d}x \le C_0\left| \Omega _k\right| ^{\frac{p^*_s-p}{p^*_s}}\left[ 2^{p-1}\int _{\Omega _k}({\underline{u}}_{\lambda }-k)^p\textrm{d}x+2^{p-1}k^{p-1}\int _{\Omega _k}({\underline{u}}_{\lambda }-k)\textrm{d}x \right] . \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \left[ 1- 2^{p-1}C_0|\Omega _k|^{\frac{p^*_s-p}{p^*_s}}\right] \int _{\Omega _k}({\underline{u}}_{\lambda } - k)^p\textrm{d}x\le 2^{p-1}k^{p-1}\left| \Omega _k\right| ^{\frac{(p^*_s-p)}{p^*_s}}\int _{\Omega _k}({\underline{u}}_{\lambda } - k)\textrm{d}x. \end{aligned}$$

If $k\rightarrow \infty$, then $|\Omega _k|\rightarrow 0$. Therefore, there exists $k_0> 1$ such that

$$\begin{aligned} 1-2^{p-1}C_0|\Omega _k|^{\frac{p^*_s-p}{p^*_s}}\ge \frac{1}{2}\quad \text {if}~~k\ge k_0>1. \end{aligned}$$

Thus, for such k, we conclude that

$$\begin{aligned} \frac{1}{2}\int _{\Omega _k}({\underline{u}}_{\lambda }-k)^p\textrm{d}x \le 2^{p-1}k^{p-1}C_0|\Omega _k|^{\frac{p^*_s-p}{p^*_s}}\int _{A_k}({\underline{u}}_{\lambda }-k)\textrm{d}x. \end{aligned}$$

(14)

Hölder’s inequality and (14) yield

$$\begin{aligned}{} & {} \left( \int _{\Omega _k}({\underline{u}}_{\lambda }-k)\textrm{d}x \right) ^{p} \le \\{} & {} \quad |\Omega _k|^{p-1}\int _{\Omega _k}({\underline{u}}_{\lambda }-k)^p\textrm{d}x\le |\Omega _k|^{p-1}2^{p-1}k^{p-1}C_0|\Omega _k|^{\frac{p^*_s-p}{p^*_s}}\int _{A_k}({\underline{u}}_{\lambda }-k)\textrm{d}x. \end{aligned}$$

Thus,

$$\begin{aligned} \int _{\Omega _k}(u-k)\textrm{d}x\le 2{\tilde{C}} k|\Omega _k|^{1+\epsilon },\quad \forall \, k\ge k_0, \end{aligned}$$

(15)

where $\epsilon =\displaystyle \frac{p^*_s-p}{p^*_s{(p-1)}}>0$ and ${\tilde{C}} >0$.

The same arguments used in [16] assures us that ${\underline{u}}_{\lambda } \in L^{\infty }(\Omega )$. Then the nonlinear regularity theory, see [9] says that ${\underline{u}}_{\lambda } \in \text {int}(C_{s_1}^0(\Omega ))_+$. $\square$

Proposition 4

For every $\lambda >0$, the problem (8) admits a unique positive solution ${\underline{u}}_\lambda \in$ int$(C^0_{s_1}(\Omega )_+)$ and ${\underline{u}}_{\lambda } \rightarrow 0$ in $C_{s_1}^{0}({\overline{\Omega }})$ as $\lambda \rightarrow 0^+$.

Proof

Existence Note that, the solutions of the problem (8) are critical points of the functional ${\tilde{I}}_{\lambda }: W_{0}^{s_1,p}(\Omega ) \rightarrow W_{0}^{s_2,q}(\Omega )$ given by

$$\begin{aligned} {\tilde{I}}_\lambda (u) = \frac{1}{p} \left[ u \right] _{s_1,p}^p + \frac{1}{q} \left[ u \right] _{s_2,p}^{q} - \frac{\lambda c_0}{\tau } \Vert u^+ \Vert _{\tau }^{\tau } + \frac{\lambda c_2}{\theta } \Vert u^+ \Vert _\theta ^\theta , \ \text {for all} \ u \in W_0^{s_1,p}(\Omega )\nonumber \\ \end{aligned}$$

(16)

where $\Vert .\Vert _{t}$ denote the norm in space $L^{t}(\Omega )$.

Since $1<\tau<q<p<\theta$, then ${\tilde{I}}_\lambda (tu) \rightarrow \infty$ as $t \rightarrow \infty$, is that, $J_\lambda$ is coercive. Also using the Sobolev embedding theorem, we see that ${\tilde{I}}_\lambda$ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find ${\underline{u}}_\lambda \in W_0^{s_1,p}(\Omega )$ such that

$$\begin{aligned} {\tilde{I}}_\lambda ({\underline{u}}_\lambda ) = \min \bigg \{J_\lambda (u); \ \ u \in W_0^{s_1,p}(\Omega )\bigg \}. \end{aligned}$$

Now notice that $1<\tau<q<p<\theta$ and $u\in$ int$(C^0_{s_1}(\Omega )_+)$ results

$$\begin{aligned} {\tilde{I}}_{\lambda }(tu) < 0 \ \ \text {for} \ \ t \in (0,1) \ \ \text {small enough} \end{aligned}$$

(17)

thus ${\tilde{I}}_{\lambda }({\underline{u}}_{\lambda }) < 0 = {\tilde{I}}_{\lambda }(0)$ and therefore ${\underline{u}}_{\lambda } \ne 0$.

Using the (17) we have,

$$\begin{aligned} {\tilde{I}}_{\lambda }'({\underline{u}}_{\lambda }) = 0 \end{aligned}$$

and consequently

$$\begin{aligned}{} & {} \bigg \langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }),\varphi \bigg \rangle = \lambda \int _{\Omega } c_0({\underline{u}}^{+}_{\lambda })^{\tau -1}\varphi \textrm{d}x \nonumber \\{} & {} - \lambda \int _{\Omega } c_2({\underline{u}}^{+}_{\lambda })^{\theta -1}\varphi \textrm{d}x, \ \ \text {for all} \ \ \varphi \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

(18)

Choosing $\varphi = {\underline{u}}_{\lambda }^{-} \in W_{0}^{s_1,p}(\Omega )$ results

$$\begin{aligned}{} & {} \left[ {\underline{u}}_{\lambda }^{-} \right] _{s_1,p}^{p} + \left[ {\underline{u}}_{\lambda }^{-} \right] _{s_2,q} \le \bigg \langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }),{\underline{u}}_{\lambda }^{-} \bigg \rangle \\{} & {} = \lambda \int _{\Omega } c_0({\underline{u}}^{+}_{\lambda })^{\tau -1}{\underline{u}}_{\lambda }^{-} \textrm{d}x - \lambda \int _{\Omega } c_2({\underline{u}}^{+}_{\lambda })^{\theta -1}{\underline{u}}_{\lambda }^{-} \textrm{d}x = 0 \end{aligned}$$

and therefore $\left[ {\underline{u}}_{\lambda }^{-} \right] _{s_1,p}^{p} = 0$, is that, ${\underline{u}}_{\lambda } \ge 0$ and ${\underline{u}}_{\lambda } \ne 0$.

Uniqueness To show the uniqueness of the solution, we will use arguments similar to those used in [12]. Let’s use the following discrete Picone’s inequality from [4]

$$\begin{aligned} J_r(a-b)\left( \frac{c^r}{a^{r-1}} - \frac{d^{r}}{b^{r-1}}\right) \le \vert c-d \vert ^{r}, \ \ \text {for all} \ a,b \in {\mathbb {R}}^{*}_+, c,d \in {\mathbb {R}}^{+}. \end{aligned}$$

(19)

Let ${\underline{u}}_{\lambda },{\underline{v}}_{\lambda } \in W_{0}^{s_1,p}(\Omega )$ positive solutions of the problem (8). As above, we show that ${\underline{u}}_\lambda ,{\underline{v}}_\lambda \in$ int$(C^0_{s_1}(\Omega )_+)$. Thus, using the same arguments as Lemma 2.4 of [12] we have,

$$\begin{aligned} \frac{{\underline{u}}_\lambda ^p}{{\underline{v}}_\lambda ^{p-1}} \in W_0^{s_1,p}(\Omega ). \end{aligned}$$

Consider $w_{\lambda } = ({\underline{u}}_{\lambda }^{p} - {\underline{v}}_{\lambda }^{p})^{+}$, thus,

$$\begin{aligned} \frac{w_{\lambda }}{{\underline{v}}_{\lambda }^{p - 1}} = \left( \frac{{\underline{u}}_{\lambda }^{p}}{{\underline{v}}_{\lambda }^{p-1}} - {\underline{v}}_{\lambda }\right) ^{+} \in W_{0}^{s_1,p}(\Omega ) \ \text {and} \ \frac{w_{\lambda }}{{\underline{u}}_{\lambda }^{p - 1}} = \left( {\underline{u}}_{\lambda } -\frac{{\underline{v}}_{\lambda }^{p}}{{\underline{u}}_{\lambda }^{p-1}}\right) ^{+} \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

We denote by $g_{\lambda }(t) = \lambda c_0 t^{\tau - p} - \lambda c_2t^{\theta -p}$. Thus, g is strictly decreasing in ${\mathbb {R}}^{+}_0$.

Testing (18) with $\frac{w_{\lambda }}{{\underline{u}}_{\lambda }^{p - 1}}$ we have

$$\begin{aligned} \bigg \langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }), \frac{w_{\lambda }}{{\underline{u}}_{\lambda }^{p - 1}} \bigg \rangle&= \lambda \int _{\Omega } c_0{\underline{u}}_{\lambda }^{\tau -1}\frac{w_{\lambda }}{{\underline{u}}_{\lambda }^{p - 1}} \textrm{d}x - \lambda \int _{\Omega } c_2{\underline{u}}_{\lambda }^{\theta -1}\frac{w_{\lambda }}{{\underline{u}}_{\lambda }^{p - 1}} \textrm{d}x \\&= \lambda \int _{\Omega } c_0{\underline{u}}_{\lambda }^{\tau -p} w_{\lambda } \textrm{d}x - \lambda \int _{\Omega } c_2{\underline{u}}_{\lambda }^{\theta -p}w_{\lambda }\textrm{d}x \\&= \int _{\{{\underline{u}}_{\lambda } > {\underline{v}}_{\lambda }\}} g_{\lambda }({\underline{u}}_{\lambda }) ({\underline{u}}_{\lambda }^p - {\underline{v}}_{\lambda }^p) \textrm{d}x \end{aligned}$$

and testing (18) with $\frac{w_{\lambda }}{{\underline{v}}_{\lambda }^{p - 1}}$ we have

$$\begin{aligned} \bigg \langle A_{s_1,p}({\underline{v}}_{\lambda }) + A_{s_2,q}({\underline{v}}_{\lambda }), \frac{w_{\lambda }}{{\underline{v}}_{\lambda }^{p - 1}} \bigg \rangle&= \lambda \int _{\Omega } c_0{\underline{v}}_{\lambda }^{\tau -1}\frac{w_{\lambda }}{{\underline{v}}_{\lambda }^{p - 1}} \textrm{d}x - \lambda \int _{\Omega } c_2{\underline{v}}_{\lambda }^{\theta -1}\frac{w_{\lambda }}{{\underline{v}}_{\lambda }^{p - 1}} \textrm{d}x \\&= \lambda \int _{\Omega } c_0 {\underline{v}}^{\tau - p} w_{\lambda } \textrm{d}x - \lambda \int _{\Omega } c_2{\underline{v}}_{\lambda }^{\theta -\tau }w_{\lambda }\textrm{d}x\\&= \int _{\{{\underline{u}}_{\lambda } > {\underline{v}}_{\lambda }\}} g_{\lambda }({\underline{v}}_{\lambda }) ({\underline{u}}_{\lambda }^p - {\underline{v}}_{\lambda }^p) \textrm{d}x \end{aligned}$$

Thus,

$$\begin{aligned}&\bigg \langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }), \frac{w_{\lambda }}{{\underline{u}}_{\lambda }^{p - 1}} \bigg \rangle - \bigg \langle A_{s_1,p}({\underline{v}}_{\lambda }) + A_{s_2,q}({\underline{v}}_{\lambda }), \frac{w_{\lambda }}{{\underline{v}}_{\lambda }^{p - 1}} \bigg \rangle \\&= \int _{\{{\underline{u}}_{\lambda } > {\underline{v}}_{\lambda }\}} \left[ g_{\lambda }(\underline{u_\lambda }) - g_{\lambda }({\underline{v}}_{\lambda })\right] ({\underline{u}}_{\lambda }^p - {\underline{v}}_{\lambda }^p) \textrm{d}x. \end{aligned}$$

Note that, using the discrete Picone’s inequality (19), see (Proposition 3.1, [12]) we have

$$\begin{aligned} j_p(u(x) - u(y))\left( \frac{w_{\lambda }(x)}{{\underline{u}}_{\lambda }(x)^{p-1}} - \frac{w_{\lambda }(y)}{{\underline{u}}_{\lambda }(y)^{p-1}} \right) \ge j_p(v(x) - v(y))\left( \frac{w_{\lambda }(x)}{{\underline{v}}_{\lambda }(x)^{p-1}} - \frac{w_{\lambda }(y)}{{\underline{v}}_{\lambda }(y)^{p-1}} \right) \end{aligned}$$

and thus,

$$\begin{aligned} \bigg \langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }), \frac{w_{\lambda }}{{\underline{u}}_{\lambda }^{p - 1}} \bigg \rangle \ge \bigg \langle A_{s_1,p}({\underline{v}}_{\lambda }) + A_{s_2,q}({\underline{v}}_{\lambda }), \frac{w_{\lambda }}{{\underline{v}}_{\lambda }^{p - 1}} \bigg \rangle . \end{aligned}$$

Therefore, since $g_{\lambda }$ is strictly decreasing in ${\mathbb {R}}^{+}_0$ results

$$\begin{aligned} 0 \le \int _{\{{\underline{u}}_{\lambda } > {\underline{v}}_{\lambda }\}} \left[ g_{\lambda }(\underline{u_\lambda }) - g_{\lambda }({\underline{v}}_{\lambda })\right] ({\underline{u}}_{\lambda }^p - {\underline{v}}_{\lambda }^p) \textrm{d}x \le 0 \end{aligned}$$

so we deduce that $\{{\underline{u}}_{\lambda } > {\underline{v}}_{\lambda }\}$ has null measure, is that, ${\underline{u}}_{\lambda } \le {\underline{v}}_{\lambda }$ in $\Omega$. Similarly, using the function test $w_\lambda = ({\underline{v}}_{\lambda }^p - {\underline{u}}_{\lambda }^p)^+$ we see that ${\underline{u}}_{\lambda } \ge {\underline{v}}_{\lambda }$ in $\Omega$, and thus ${\underline{u}}_{\lambda } = {\underline{v}}_{\lambda }$.

Moreover, we have

$$\begin{aligned} \left[ {\underline{u}}_{\lambda } \right] _{s_1,p}^{p}&\le \left[ {\underline{u}}_{\lambda } \right] _{s_1,p}^{p} + \left[ {\underline{u}}_{\lambda } \right] _{s_2,q}^{q} \\&= \lambda c_0 \Vert {\underline{u}}_{\lambda } \Vert _{\tau }^{\tau } - \lambda c_2 \Vert {\underline{u}}_{\lambda } \Vert _{\theta }^{\theta }\\&\le \lambda c_0\Vert {\underline{u}}_{\lambda } \Vert _{\tau }^{\tau }\\&\le \lambda {\hat{c}}_{0} \left[ {\underline{u}}_{\lambda } \right] _{s_1,p}^{\tau }, \end{aligned}$$

for some ${\hat{c}}_0 >0$. Thus,

$$\begin{aligned} \left[ {\underline{u}}_{\lambda } \right] _{s_1,p}^{p-\tau } \le \lambda {\hat{c}}_{0} \end{aligned}$$

and therefore, ${\underline{u}}_{\lambda } \rightarrow 0$ in $W^{s_1,p}_0(\Omega )$ as $\lambda \rightarrow 0^+$. Using the nonlinear regularity theorem, see [9], results that

$$\begin{aligned} {\underline{u}}_{\lambda } \rightarrow 0 \ \ \text {in} \ \ C_{s_1}^{0}({\overline{\Omega }}) \ \ \text {as} \ \ \lambda \rightarrow 0^+. \end{aligned}$$

$\square$

We consider another auxiliary problem,

$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta _p)^{s_1}u + (-\Delta _q)^{s_2}u = \lambda {\underline{u}}_{\lambda }^{-\eta } + 1 &{} \textrm{in} \ \ \Omega ,\\ u= 0 &{} \textrm{in} \ \mathrm {I\!R\!}^N\setminus \Omega ,\\ u> 0 &{} \textrm{in} \ \Omega \end{array} \right. \end{aligned}$$

(20)

with $\lambda > 0$, $0<\eta <1$ and $1< q<p$.

Proposition 5

For every $\lambda > 0$, there exists a unique solution ${\overline{u}}_{\lambda } \in \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }}))_+\right]$ of the problem (20) and a $\lambda _0>0$ such that, for all $0 < \lambda \le \lambda _0$ it holds

$$\begin{aligned} {\underline{u}}_{\lambda } \le {\overline{u}}_{\lambda }. \end{aligned}$$

Proof

Note that, the Lemma 14.16 of Gilbarg-Trundiger [10] says that $d^{s_1}_{\Omega } \in C^{2}(\Omega _{\delta _0})$, where $\Omega _{\delta _0} = \{x \in \Omega ; d^{s_1}_{\Omega }(x) < \delta _0\}$. Thus, $d^{s_1}_{\Omega } \in \text {int}\left[ (C_{s_1}^0(\Omega ))_+\right]$ and so by Proposition 4.1.22 of [17], there exists $c_3 = c_3({\underline{u}}_{\lambda }) >0$ and $c_4 = c_4({\underline{u}}_{\lambda }) > 0$ such that,

$$\begin{aligned} c_3 d^{s_1}_{\Omega } \le {\underline{u}}_{\lambda } \le c_4d^{s_1}_{\Omega }. \end{aligned}$$

(21)

Since due to (21), $\lambda {\underline{u}}_{\lambda }^{-\eta } + 1 \in L^1(\Omega )$. The existence of a weak solution of (20) follows from direct minimization in $W_0^{s_1, p}(\Omega )$ of the functional

$$\begin{aligned} \frac{1}{p} \left[ u \right] _{s_1,p}^p + \frac{1}{q} \left[ u \right] _{s_2,p}^{q} - \int _{\Omega }(\lambda {\underline{u}}_{\lambda }^{-\eta } + 1)udx, \end{aligned}$$

whereas the uniqueness comes from, for instance, the comparison principle for the Dirichlet fractional (p, q)-Laplacian, Propossition 2. Using the maximum principle, [9], the solution ${\overline{u}}_{\lambda } \in \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }})_+)\right]$.

For show the existence of $\lambda _0 >0$ such that ${\underline{u}}_{\lambda } \le {\overline{u}}_{\lambda }$ for all $0< \lambda \le \lambda _0$, acting on (20) with ${\overline{u}}_{\lambda }$ and obtain

$$\begin{aligned} \left[ {\overline{u}}_{\lambda } \right] _{s_1,p}^p&\le \left[ {\overline{u}}_{\lambda } \right] _{s_1,p}^p + \left[ {\overline{u}}_{\lambda } \right] _{s_2,q}^q\\&= \lambda \displaystyle \int _{\Omega } {\underline{u}}_{\lambda }^{-\eta }.{\overline{u}}_{\lambda } \textrm{d}x + \displaystyle \int _{\Omega } {\overline{u}}_{\lambda } \textrm{d}x \\&= \lambda \displaystyle \int _{\Omega } {\underline{u}}_{\lambda }^{1-\eta }.\frac{{\overline{u}}_{\lambda }}{{\underline{u}}_{\lambda }} \textrm{d}x + \displaystyle \int _{\Omega } {\overline{u}}_{\lambda } \textrm{d}x \\&\le \lambda c_5 \int _{\Omega } \frac{{\overline{u}}_{\lambda }}{d^{s_1}_{\Omega }} \textrm{d}x + \vert \Omega \vert ^{\frac{p-1}{p}} \left( \int _{\Omega } {\overline{u}}_{\lambda }^{p} \textrm{d}x\right) ^{\frac{1}{p}} \ \ (\text {Holder inequality})\\&\le \left( \lambda c_5 + \frac{1}{\lambda _1(p)}\right) \vert \Omega \vert ^{\frac{p-1}{p}} \left[ {\overline{u}}_{\lambda } \right] _{s_1,p} (\text {Hardy's inequality and first eigenvalue}). \end{aligned}$$

So, we have $\{{\overline{u}}_{\lambda }\}_{\lambda \in (0,1]}$ is uniformly bounded in $W_{0}^{s_1,p}(\Omega )$. Using arguments similar to the Lemma 1, (see also Ladyzhenskaya-Ural’tseva [13] Theorem 7.1) results

$$\begin{aligned} \{{\overline{u}}_{\lambda }\}_{\lambda \in (0,1]} \subset L^{\infty }(\Omega ) \ \ \text {is uniformly bounded in} \ \ \lambda . \end{aligned}$$

The condition ${{\textbf {H}}} \ (i)$ implies that there exists $\lambda _0>0$ such that,

$$\begin{aligned} \lambda f_k(x,{\overline{u}}_{\lambda }) \le \lambda \Vert a \Vert (1 + \Vert {\overline{u}}_{\lambda } \Vert ^{\theta -1}) \le 1 \ \ \text {for all} \ \ \lambda \in (0,\lambda _0] \ \ \text {and} \ \ x \ \ \text {a. a. in} \ \Omega . \end{aligned}$$

For each $\lambda \in (0,\lambda _0]$ consider the Carathéodory function

$$\begin{aligned} \kappa _{\lambda }(x,t) = \left\{ \begin{array}{llll} \lambda [c_0(t^{+})^{\tau -1} - c_2(t^{+})^{\theta -1}] &{} \textrm{if} \ \ t \le {\overline{u}}_{\lambda }(x),\\ \lambda [c_0{\overline{u}}_{\lambda }(x)^{\tau -1} - c_2{\overline{u}}_{\lambda }(x)^{\theta -1}] &{} \textrm{if} \ \ {\overline{u}}_{\lambda }(x) < t. \end{array} \right. \end{aligned}$$

Let $\Psi _{\lambda }: W_{0}^{s_1,p} \rightarrow {\mathbb {R}}$ the $C^1$-functional defined by

$$\begin{aligned} \Psi _{\lambda }(u) = \frac{1}{p} \left[ u \right] _{s_1,p}^{p} + \frac{1}{q} \left[ u \right] _{s_2,p}^{q} - \int _{\Omega } K_{\lambda }(x,u) \textrm{d}x, \ \ \text {for all} \ \ u \in W_{0}^{s_1,p}(\Omega ) \end{aligned}$$

where $K_{\lambda }(x,t) = \displaystyle \int _{0}^{t} \kappa _\lambda (x,s) \textrm{d}s$.

Note that, $\Psi _{\lambda }$ is coercive and sequentially wekly lower semicontinuous. So, there exists ${\tilde{u}}_{\lambda } \in W_0^{s_1,p}(\Omega )$ such that

$$\begin{aligned} \Psi _{\lambda }({\tilde{u}}_{\lambda }) = \text {min}\left[ \Psi _{\lambda }(u); \ u \in W_0^{s_1,p}(\Omega )\right] . \end{aligned}$$

Since $1<\tau<q<p<\theta$ results

$$\begin{aligned} \Psi _{\lambda }(tu) < 0 \ \ \text {for} \ \ t \in (0,1) \ \ \text {small enough} \end{aligned}$$

(22)

thus $\Psi _{\lambda }({\underline{u}}_{\lambda }) < 0 = \Psi _{\lambda }(0)$ and therefore ${\underline{u}}_{\lambda } \ne 0$.

Using the (22) we have,

$$\begin{aligned} \Psi _{\lambda }'({\tilde{u}}_{\lambda }) = 0 \end{aligned}$$

and consequently

$$\begin{aligned} \bigg \langle A_{s_1,p}({\tilde{u}}_{\lambda }) + A_{s_2,q}({\tilde{u}}_{\lambda }),\varphi \bigg \rangle = \int _{\Omega }\kappa _{\lambda }(x,{\tilde{u}}_{\lambda }) \varphi \textrm{d}x, \ \ \text {for all} \ \ \varphi \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

Choosing $\varphi = -{\tilde{u}}_{\lambda } \in W_{0}^{s_1,p}(\Omega )$, we see that ${\tilde{u}}_{\lambda } \ge 0$ and ${\tilde{u}}_{\lambda } \ne 0$. Taking $\varphi = ({\tilde{u}}_{\lambda } - {\overline{u}}_{\lambda })^+ \in W_0^{s_1,p}(\Omega )$ we find,

From (7), we have that there exits $c_0>0$ and $c_2>0$ such that $f_k(x,t) \ge c_0t^{\tau -1} - c_2t^{\theta -1}$ and so

$$\begin{aligned}&\bigg \langle A_{s_1,p}({\tilde{u}}_{\lambda }) + A_{s_2,q}({\tilde{u}}_{\lambda }),({\tilde{u}}_{\lambda } - {\overline{u}}_{\lambda })^+ \bigg \rangle \\&\quad = \int _{\Omega } \kappa _{\lambda }(x,{\tilde{u}}_{\lambda })({\tilde{u}}_{\lambda } - {\overline{u}}_{\lambda })^+ \textrm{d}x\\&\quad = \int _{\Omega } \lambda [c_0{\overline{u}}_{\lambda }^{\tau -1} - c_2{\overline{u}}_{\lambda }^{\theta -1}]({\tilde{u}}_{\lambda } - {\overline{u}}_{\lambda })^+ \textrm{d}x\\&\quad \le \int _{\Omega } \lambda f_k(x,{\overline{u}}_{\lambda })({\tilde{u}}_{\lambda } - {\overline{u}}_{\lambda })^+ \textrm{d}x\\&\quad \le \int _{\Omega } [\lambda {\underline{u}}_{\lambda }^{-\eta } + 1]({\tilde{u}}_{\lambda } - {\overline{u}}_{\lambda })^+ \textrm{d}x \ \ \ (\text {for all} \ \ 0< \lambda \le \lambda _0)\\&\quad = \bigg \langle A_{s_1,p}({\overline{u}}_{\lambda }) + A_{s_2,q}({\overline{u}}_{\lambda }),({\tilde{u}}_{\lambda } - {\overline{u}}_{\lambda })^+ \bigg \rangle \end{aligned}$$

and so, by Proposition 2${\tilde{u}}_{\lambda } \le {\overline{u}}_{\lambda }$. Moreover, note that,

$$\begin{aligned} \Psi _{\lambda }(u) = {\tilde{I}}_{\lambda }(u), \ \ \text {for all} \ \ u \in [0,{\overline{u}}_{\lambda }], \end{aligned}$$

thus

$$\begin{aligned} {\tilde{I}}_\lambda ({\tilde{u}}_{\lambda })&= \Psi _{\lambda }({\tilde{u}}_\lambda ) = \text {min}\left[ \Psi _{\lambda }(u); \ u \in W_0^{s_1,p}(\Omega )\right] \\&= \text {min}\bigg \{\Psi _{\lambda }(u); \ u \in [0,{\overline{u}}_{\lambda }]\bigg \}\\&= \text {min}\bigg \{{\tilde{I}}_{\lambda }(u); \ u \in [0,{\overline{u}}_{\lambda }]\bigg \}\\&= {\tilde{I}}_{\lambda }({\underline{u}}_{\lambda }). \end{aligned}$$

By Proposition 4 we have ${\tilde{u}}_{\lambda } = {\underline{u}}_{\lambda }$ and therefore ${\underline{u}}_{\lambda } \le {\overline{u}}_{\lambda }$ for all $0 < \lambda \le \lambda _0$. $\square$

5 Existence of positive solution for $P_{k,\lambda }$

We consider the set

$$\begin{aligned} {\mathcal {L}} = \bigg \{\lambda >0; \ \ \text {problem}~{P_{k,\lambda }} \text {admits a positive solution}\bigg \} \end{aligned}$$

and the set $S_{\lambda }$ of the positive solutions to the problem $P_{k,\lambda }$.

Proposition 6

Assume the hypotheses $({{\textbf {H}}}_k)$ hold, then

• i) ${\mathcal {L}} \ne \varnothing$;

• ii) If $\lambda \in {\mathcal {L}}$, then ${\underline{u}}_{\lambda } \le u$ for all $u \in S_{\lambda }$ and $S_{\lambda } \subseteq \text {int}[(C_{s_1}^0(\Omega ))_+]$.

Proof

Let $\lambda _0 > 0$ given in the Proposition 4, so for $\lambda \in (0,\lambda _0]$ we have

$$\begin{aligned} {\underline{u}}_{\lambda } \le {\overline{u}}_{\lambda } \ \ \text {and} \ \ \lambda f(x,{\overline{u}}_{\lambda }) \le 1 \ \ \text {for a. a.} \ \ x \in \Omega . \end{aligned}$$

(23)

We consider the function

$$\begin{aligned} g_{\lambda }(x,t) = \left\{ \begin{array}{llll} \lambda [ {\underline{u}}_{\lambda }^{-\eta } + f_k(x, {\underline{u}}_{\lambda })] &{} \textrm{if} \ \ t< {\underline{u}}_{\lambda }(x),\\ \lambda [t^{-\eta } + f_k(x, t)] &{} \textrm{if} \ \ {\underline{u}}_{\lambda }(x) \le t \le {\overline{u}}_{\lambda }(x),\\ \lambda [{\overline{u}}_{\lambda }^{-\eta } + f_k(x,{\overline{u}}_{\lambda })] &{} \textrm{if} \ \ {\overline{u}}_{\lambda }(x)< t, \end{array} \right. \end{aligned}$$

and the functional $\Phi _\lambda : W_0^{s_1,p}(\Omega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \Phi _{\lambda }(u) = \frac{1}{p} \left[ u \right] _{s_1,p}^{p} + \frac{1}{q} \left[ u \right] _{s_2,p}^{q} - \int _{\Omega } G_{\lambda }(x,u) \textrm{d}x, \ \ \text {for all} \ \ u \in W_{0}^{s_1,p}(\Omega ) \end{aligned}$$

where $G(x,t) = \displaystyle \int _{0}^{t} g_{\lambda }(x,s) \textrm{d}s$.

By Proposition 3 of [18] we have $\Phi _\lambda \in C^{1}(W_0^{s_1,p}(\Omega ), {\mathbb {R}})$. Moreover, using the hypotheses $({{\textbf {H}}})$ we have, $\Phi _{\lambda }$ is coercive and sequently weakly lower semicontinuous. Thus, there exists $u_{\lambda }:= u_{k,\lambda } \in W_0^{s_1,p}(\Omega )$ such that,

$$\begin{aligned} \Phi _{\lambda }(u_{\lambda }) = \text {min}\bigg [\Phi _{\lambda }(u); \ u \in W_0^{s_1,p}(\Omega )\bigg ]. \end{aligned}$$

Thus, $\Phi _\lambda '(u_\lambda ) = 0$, that is,

$$\begin{aligned} \bigg \langle A_{s_1,p}(u_{\lambda }) + A_{s_2,q}(u_{\lambda }),\varphi \bigg \rangle = \int _{\Omega }g_{\lambda }(x,u_{\lambda }) \varphi \textrm{d}x, \ \ \text {for all} \ \ \varphi \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

(24)

Testing the Eq. (24) with $\varphi = (u_\lambda -{\overline{u}}_{\lambda })^+ \in W_0^{s_1,p}(\Omega )$ and using the inequality (23), we find

$$\begin{aligned}&\bigg \langle A_{s_1,p}(u_{\lambda }) + A_{s_2,q}(u_{\lambda }),(u_{\lambda } - {\overline{u}}_{\lambda })^+ \bigg \rangle \\&\quad = \int _{\Omega } g_{\lambda }(x,u_{\lambda })(u_{\lambda } - {\overline{u}}_{\lambda })^+ \textrm{d}x\\&\quad = \int _{\Omega } \lambda [{\overline{u}}_{\lambda }^{-\eta } + f_k(x,{\overline{u}}_{\lambda })](u_{\lambda } - {\overline{u}}_{\lambda })^+ \textrm{d}x\\&\quad \le \int _{\Omega } [\lambda {\underline{u}}_{\lambda }^{-\eta } + 1](u_{\lambda } - {\overline{u}}_{\lambda })^+ \textrm{d}x \ \ \ (\text {for all} \ \ 0< \lambda \le \lambda _0)\\&\quad = \bigg \langle A_{s_1,p}({\overline{u}}_{\lambda }) + A_{s_2,q}({\overline{u}}_{\lambda }),(u_{\lambda } - {\overline{u}}_{\lambda })^+ \bigg \rangle \end{aligned}$$

and so, by Proposition 2$u_{\lambda } \le {\overline{u}}_{\lambda }$.

Analogously, testing (24) with the function $\varphi = ({\underline{u}}_{\lambda } - u_\lambda )^+ \in W_0^{s_1,p}(\Omega )$ and using (7), we have,

$$\begin{aligned}&\bigg \langle A_{s_1,p}(u_{\lambda }) + A_{s_2,q}(u_{\lambda }),({\underline{u}}_{\lambda } - u_{\lambda })^+ \bigg \rangle = \int _{\Omega } g_{\lambda }(x,u_{\lambda })({\underline{u}}_{\lambda } - u_{\lambda })^+ \textrm{d}x\\&= \int _{\Omega } \lambda [{\underline{u}}_{\lambda }^{-\eta } + f_k(x,{\underline{u}}_{\lambda })]({\underline{u}}_{\lambda } - u_{\lambda })^+ \textrm{d}x\\&\ge \int _{\Omega } \lambda [c_0{\underline{u}}_\lambda ^{\tau -1} - c_2{\underline{u}}^{\theta -1}]({\underline{u}}_{\lambda } - u_{\lambda })^+ \textrm{d}x \ \ \ (\text {for all} \ \ 0< \lambda \le \lambda _0)\\&= \bigg \langle A_{s_1,p}({\underline{u}}_{\lambda }) + A_{s_2,q}({\underline{u}}_{\lambda }),({\underline{u}}_{\lambda } - u_{\lambda })^+ \bigg \rangle \end{aligned}$$

and so, by Proposition 2 we have $u_{\lambda } \le {\overline{u}}_{\lambda }$.

Therefore,

$$\begin{aligned} u_\lambda \in [{\underline{u}}_\lambda ,{\overline{u}}_\lambda ] \Rightarrow u_\lambda \in S_\lambda \Rightarrow (0,\lambda _0] \subseteq {\mathcal {L}}. \end{aligned}$$

For item (ii), it is sufficient to argue as in the Proposition 4, replacing ${\overline{u}}_\lambda$ with $u \in S_\lambda$, we show that ${\underline{u}}_\lambda \le u$ for all $u \in S_\lambda$. For show that $S_\lambda \subseteq \text {int}[(C_{s_1}^0(\Omega ))_+]$ we use the maximum principle, see [9]. $\square$

Proposition 7

If hypotheses $({{\textbf {H}}}_k)$ hold, $\lambda \in {\mathcal {L}}$ and $\mu \in (0,\lambda )$, then $\mu \in {\mathcal {L}}$.

Proof

Let $\lambda \in {\mathcal {L}}$, so we can find $u_{\lambda } \in S_{\lambda } \subseteq \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+]$. Consider the Dirichlet problem,

$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta _p)^{s_1}u + (-\Delta _q)^{s_2}u = \vartheta c_0u(x)^{\tau -1} - \lambda c_2u^{\theta -1} &{} \textrm{in} \ \ \Omega ,\\ u= 0 &{} \textrm{in} \ \mathrm {I\!R\!}^N\setminus \Omega ,\\ u> 0 &{} \textrm{in} \ \Omega \end{array} \right. \end{aligned}$$

(25)

with $0< \vartheta <\lambda$ and $1< \tau< q<p<\theta$. As we did in the proposition, we can find a unique solution ${\tilde{u}}_{\vartheta } \in \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+]$ to the problem (25) and, in addition, we can show that ${\tilde{u}}_{\vartheta }^{-\eta } \in L^{1}(\Omega )$. Since, for all $0<\vartheta _1 < \vartheta _2 \le \lambda$, we have $\vartheta _1 c_0u(x)^{\tau -1} - \lambda c_2u^{\theta -1} \le \vartheta _2 c_0u(x)^{\tau -1} - \lambda c_2u^{\theta -1}$, by comparison principle results that ${\tilde{u}}_{\vartheta _1} \le {\tilde{u}}_{\vartheta _2}$. Note that, by Proposition 5${\tilde{u}}_{\lambda } = {\underline{u}}_{\lambda }$, so

$$\begin{aligned} {\tilde{u}}_{\mu } \le {\underline{u}}_{\lambda } \le u_{\lambda }. \end{aligned}$$

Define the Caracthéodory function,

$$\begin{aligned} \gamma _{\lambda }(x,t) = \left\{ \begin{array}{llll} \mu [ {\tilde{u}}_{\mu }^{-\eta } + f_k(x, {\tilde{u}}_{\mu })] &{} \textrm{if} \ \ t< {\tilde{u}}_{\mu }(x),\\ \mu [t^{-\eta } + f_k(x, t)] &{} \textrm{if} \ \ {\tilde{u}}_{\mu }(x) \le t \le {\tilde{u}}_{\mu }(x),\\ \mu [{\tilde{u}}_{\mu }^{-\eta } + f_k(x,{\tilde{u}}_{\mu })] &{} \textrm{if} \ \ {\tilde{u}}_{\mu }(x)< t, \end{array} \right. \end{aligned}$$

Let $\Upsilon _{\lambda }: W_{0}^{s_1,p}(\Omega ) \rightarrow {\mathbb {R}}$ the $C^1$-functional defined by

$$\begin{aligned} \Upsilon _{\lambda }(u) = \frac{1}{p} \left[ u \right] _{s_1,p}^{p} + \frac{1}{q} \left[ u \right] _{s_2,p}^{q} - \int _{\Omega } \Gamma _{\mu }(x,u) \textrm{d}x, \ \ \text {for all} \ \ u \in W_{0}^{s_1,p}(\Omega ) \end{aligned}$$

where $\Gamma _{\lambda }(x,t) = \displaystyle \int _{0}^{t} \gamma (x,s) \textrm{d}s$.

Note that, $\Upsilon _{\lambda }$ is coercive and sequentially wekly lower semicontinuous. So,

$$\begin{aligned} \Upsilon _{\mu }(u_{\mu }) = \text {min}\left[ \Upsilon _{\mu }(u); \ u \in W_0^{s_1,p}(\Omega )\right] . \end{aligned}$$

is attained by a function $u_{\mu }:= u_{k,\mu } \in W_0^{s_1,p}(\Omega )$.

Thus, $\Upsilon _{\mu }'(u_{\mu }) = 0$, that is,

$$\begin{aligned} \bigg \langle A_{s_1,p}(u_{\mu }) + A_{s_2,q}(u_{\mu }),\varphi \bigg \rangle = \int _{\Omega }\gamma _{\mu }(x,u_{\mu }) \varphi \textrm{d}x, \ \ \text {for all} \ \ \varphi \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

(26)

Testing the Eq. (26) with $\varphi = (u_\mu - u_{\lambda })^{+} \in W_{0}^{s_1,p}(\Omega )$, using the Proposition 2 and $0<\mu <\lambda$ we show that $u_{\mu } \le u_{\lambda }$. In addition, testing the Eq. (26) with the function $\varphi = ({\tilde{u}}_{\mu } - u_{\mu })^+ \in W_0^{s_1,p}(\Omega )$, using the Proposition 2 and the fact ${\tilde{u}}_{\mu }$ is unique solution of the problem (25), we show ${\tilde{u}}_{\mu } \le u_{\mu }$.

So we have proved that,

$$\begin{aligned} u_{\mu } \in [{\tilde{u}}_{\mu }, u_{\lambda }] \Rightarrow u_{\mu } \in S_{\mu } \subseteq \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \ \ \text {and so} \ \ \mu \in {\mathcal {L}}. \end{aligned}$$

$\square$

Proposition 8

If hypotheses $({{\textbf {H}}}_k)$ hold, $\lambda \in {\mathcal {L}}$, $u_{\lambda } \in S_{\lambda } \subseteq \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }}))_+\right]$ and $\mu < \lambda$, then $\mu \in {\mathcal {L}}$ and there exists $u_\mu \in S_\mu$ such that

$$\begin{aligned} u_\lambda - u_\mu \in \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }}))_+\right] . \end{aligned}$$

Proof

By Proposition 6 we know that $\mu \in {\mathcal {L}}$ and we can find $u_\mu := u_{k,\mu } \in S_{\mu } \subseteq \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }})_+)\right]$ such that $u_\mu \le u_\lambda$. Let $\rho = \Vert u_{\lambda } \Vert _{\infty }$ and ${\widehat{E}}_{k,\rho } > 0$ be as postulated by hypothesis $({{\textbf {H}}}_k) \ (v)$. We have

$$\begin{aligned} (-\Delta _p)^{s_1} u_{\mu }(x)&+ (-\Delta _q)^{s_2} u_{\mu }(x) + \lambda {\widehat{E}}_{k,\rho } u_{\mu }(x)^{p-1} - \lambda u_{\mu }(x)^{-\eta } \\&\le \mu f_k(x,u_{\mu }(x)) + \lambda {\widehat{E}}_{k,\rho } u_{\mu }(x)^{p-1}\\&= \lambda \left[ f_k(x,u_{\mu }(x))+{\widehat{E}}_{k,\rho } u_{\mu }(x)^{p-1}\right] - \left( \lambda - \mu \right) f_k(x,u_{\mu }(x)) \\&\le \lambda \left[ f_k(x,u_{\mu }(x)) + {\widehat{E}}_{k,\rho } u_{\mu }(x)^{p-1}\right] \\&= (-\Delta _p)^{s_1} u_{\lambda }(x) + (-\Delta _q)^{s_2} u_{\lambda }(x) + \lambda {\widehat{E}}_{k,\rho } u_{\lambda }(x)^{p-1} - \lambda u_{\lambda }(x)^{-\eta }. \end{aligned}$$

Note that, the function $g(t) = \lambda {\widehat{E}}_{k,\rho }t^{p-1} - \lambda t^{-\eta }$ is nondecreasing in ${\mathbb {R}}_{0}^{+}$, thus, by Proposition 3 we have $u_\lambda - u_\mu \in \text {int}\left[ (C_{s_1}^{0}({\overline{\Omega }}))_+\right] .$ $\square$

Proposition 9

Assume that the hypotheses $({{\textbf {H}}}_k)$ hold. Then $\lambda ^{*} = \sup {\mathcal {L}} < +\infty ,$ for each $k \in {\mathbb {N}}$.

Proof

By hypotheses H(i), (ii) and (iii) we can find ${\widehat{\lambda }} > 0$ such that

$$\begin{aligned} t^{p-1} \le {\widehat{\lambda }} f_k(x,t) \ \ \text {for all} \ \ x \in \Omega , \ \ \text {all} \ \ t \ge 0. \end{aligned}$$

(27)

Let $\lambda >\lambda ^{*}$ and suppose that $\lambda \in {\mathcal {L}}$. Then, there exists $u_\lambda := u_{k,\lambda } \in S_{\lambda } \subseteq \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+]$, that is, $u_\lambda$ is a solution of the problem ($P_{k,\lambda }$). Consider $\Omega _0 \subset \subset \Omega$ and $m_0 = \displaystyle \min _{{\overline{\Omega }}}u_{\lambda } > 0$. For $\delta \in (0,1)$ small we set $m_0^\delta = m_0 + \delta$. Let $\rho = \Vert u_\lambda \Vert _{\infty }$ and ${\widehat{E}}_{k,\rho } > 0$ be as postulated by H(v). We have,

$$\begin{aligned} (-\Delta _p)^{s_1} m_0^{\delta }&+ (-\Delta _q)^{s_2} m_0^{\delta } + \lambda {\widehat{E}}_{k,\rho } (m_0^\delta )^{p-1} - \lambda (m_0^\delta )^{-\eta } \\&\le \lambda {\widehat{E}}_{k,\rho } (m_0^\delta )^{p-1} + \chi (\delta ) \ \ (\text {with} \ \ \chi (\delta ) \rightarrow 0^+ \ \ \text {as} \ \ \delta \rightarrow 0^+)\\&= \left[ \lambda {\widehat{E}}_{k,\rho } + 1\right] m_0^{p-1} + \chi (\delta ) \\&\le {\widehat{\lambda }} f_k(x, m_0) + \lambda {\widehat{E}}_{k,\rho } (m_0^\delta )^{p-1} + \chi (\delta ) \ \ (\text {see} \ (27))\\&= \lambda \left[ f_k(x,m_0) + {\widehat{E}}_{k,\rho } (m_0^\delta )^{p-1}\right] - (\lambda - {\widehat{\lambda }}) f_k(x, m_0)+ \chi (\delta )\\&\le \lambda \left[ f_k(x,u_\lambda (x)) + {\widehat{E}}_{k,\rho } u_{\lambda }^{p-1}\right] \ \ \text {for} \ \delta (0,1) \ \ \text {small enough}.\\&= (-\Delta _p)^{s_1} u_{\lambda }(x) + (-\Delta _q)^{s_2} u_{\lambda }(x) + \lambda {\widehat{E}}_{k,\rho } u_{\lambda }(x)^{p-1} - \lambda u_{\lambda }(x)^{-\eta }. \end{aligned}$$

where we have used the hypotheses H(iv), (v) and the fact $\chi (\delta ) \rightarrow 0^{+}$ as $\delta \rightarrow 0^{+}$. By strong comparison principle we have

$$\begin{aligned} u_\lambda - m_0^{\delta } \in \text {int}[(C_{s_1}^{0}(\Omega ))_+] \ \ \text {for} \ \ \delta \in (0,1) \ \ \text {small enough} \end{aligned}$$

which contradicts with the definition of $m_0$. Consequently, it holds $0<\lambda ^* \le {\widehat{\lambda }} < \infty$. $\quad \square$

Proposition 10

If hypotheses $({{\textbf {H}}}_k)$ hold and $\lambda \in (0,\lambda ^*)$, then problem ($P_{k,\lambda }$) has least two positive solutions

$$\begin{aligned} u_0, {\hat{u}} \in \text {int}[(C_{s_1}^{0}(\Omega ))_+] \ \ \text {with} \ \ u_0 \le {\hat{u}} \ \ \text {and} \ \ u_0 \ne {\hat{u}}. \end{aligned}$$

Proof

Let $0<\lambda< \vartheta < \lambda ^*$. By Proposition 9$\lambda ,\vartheta \in {\mathcal {L}}$. Thus, by Proposition 8 we can find $u_0 \in S_{\lambda } \subseteq \text {int}[(C_{s_1}^{0}(\Omega ))_+]$ and $u_{\vartheta } \in S_{\vartheta } \subseteq \text {int}[(C_{s_1}^{0}(\Omega ))_+]$ such that

$$\begin{aligned} u_{\vartheta } - u_0 \in S_{\lambda } \subseteq \text {int}[(C_{s_1}^{0}(\Omega ))_+]. \end{aligned}$$

From Proposition 8, we know that $u_\lambda \le u_0$, hence $u_0^{-\eta } \in L^{1}(\Omega )$. Consider the Carathéodory function

$$\begin{aligned} {\widehat{\omega }}_{\lambda }(x,t) = \left\{ \begin{array}{llll} \lambda [{u}_{0}^{-\eta } + f_k(x, {u}_{0})] &{} \textrm{if} \ \ t< {u}_{0}(x),\\ \lambda [t^{-\eta } + f_k(x, t)] &{} \textrm{if} \ \ {u}_{0}(x) \le t \le {u}_{\vartheta }(x),\\ \lambda [{u}_{\vartheta }^{-\eta } + f_k(x,{u}_{\vartheta })] &{} \textrm{if} \ \ {u}_{\vartheta }(x)< t \end{array} \right. \end{aligned}$$

and define the $C^1$-functional ${\widehat{\mu }}_{\lambda }: W_{0}^{s_1,p}(\Omega ) \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} {\widehat{\mu }}_{\lambda }(u) = \frac{1}{p} \left[ u \right] _{s_1,p}^{p} + \left[ u \right] _{s_2,p}^{q} - \int _{\Omega } {\widehat{W}}_{\lambda }(x,u) dx \ \ \text {for all} \ \ u \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

where ${\widehat{W}}_{\lambda }(t,x) = \displaystyle \int _{0}^{t} {\widehat{\omega }}_\lambda (x,s) \textrm{d}s$.

Consider also another Carathéodory function

$$\begin{aligned} \omega _{\lambda }(x,t) = \left\{ \begin{array}{llll} \lambda [{u}_{0}^{-\eta }(x) + f_k(x, {u}_{0})] &{} \textrm{if} \ \ t \le {u}_{0}(x),\\ \lambda [t^{-\eta } + f_k(x, t)] &{} \textrm{if} \ \ {u}_{0}(x) < t \end{array} \right. \end{aligned}$$

and define the $C^1$-functional $\mu _{\lambda }: W_{0}^{s_1,p}(\Omega ) \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \mu _{\lambda }(u) = \frac{1}{p} \left[ u \right] _{s_1,p}^{p} + \left[ u \right] _{s_2,p}^{q} - \int _{\Omega } W_{\lambda }(x,u) dx \ \ \text {for all} \ \ u \in W_{0}^{s_1,p}(\Omega ) \end{aligned}$$

where $W_{\lambda }(t,x) = \displaystyle \int _{0}^{t}\omega _\lambda (x,s) \textrm{d}s$.

It is clear that,

$$\begin{aligned} {\widehat{\mu }}_{\lambda }(u)\bigg \vert _{[0,u_{\theta }]} = \mu _{\lambda }(u)\bigg \vert _{[0,u_{\theta }]} \ \ \text {and} {\widehat{\mu }}'_{\lambda }(u)\bigg \vert _{[0,u_{\theta }]} = \mu '_{\lambda }(u)\bigg \vert _{[0,u_{\theta }]} \end{aligned}$$

(28)

Let $K_\mu = \big \{u \in W_{0}^{s_1,p}(\Omega ); \mu '(u) = 0\big \}$. Using the same arguments used in ([18], Proposition 8) we can show that

$$\begin{aligned} K_{{\widehat{\mu }}_{\lambda }}&\subseteq [u_0,u_\theta ] \cap \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \end{aligned}$$

(29)

$$\begin{aligned} K_{\mu _{\lambda }}&\subseteq [u_0) \cap \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \end{aligned}$$

(30)

From (30), we can assume that $K_{\mu _{\lambda }}$ is finite. Otherwise, we already have an infinity of positive smooth solutions of ($P_{k,\lambda }$) bigger than $u_0$ and so we are done. In addition, we can assume that

$$\begin{aligned} K_{\mu _{\lambda }} \cap [u_0,u_{\theta }] = \{u_0\}. \end{aligned}$$

(31)

Moreover, it is clear that ${\widehat{\mu }}_{\lambda }$ is coercive and sequentially weakly lower semicontinuous. So there exists ${\tilde{u}}_0 \in W_{0}^{s_1,p}(\Omega )$ such that,

$$\begin{aligned} {\widehat{\mu }}_{{\tilde{u}}_{0}} = \min \bigg [{\widehat{\mu }}_{\lambda }(u); \ \ u \in W_{0}^{s_1,p}(\Omega )\bigg ] \end{aligned}$$

from (29) we have

$$\begin{aligned} {\tilde{u}}_{0} \in K_{{\widehat{\mu }}_{\lambda }} \subseteq [u_0,u_{\theta }] \cap \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \end{aligned}$$

and so, from (28) and (31) results ${\tilde{u}}_{0} = u_0$. Therefore,

$$\begin{aligned} u_0 \in \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \ \text {is a local} \ \ W_{0}^{s_1,p}(\Omega )-\text {minimizer of} \ \ \mu _{\lambda }. \end{aligned}$$

Consequently, there exists $\rho \in (0,1)$ such that,

$$\begin{aligned} \mu _{\lambda }(u_0) < \inf \bigg [\mu _{\lambda }(u); \ \ \left[ u-u_0 \right] _{s_1,p} = \rho \bigg ] = m_{\lambda }. \end{aligned}$$

Note that, if $u \in \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+]$, then on account of hypothesis $({{\textbf {H}}}_k \ (ii))$ we have,

$$\begin{aligned} \mu _{\lambda }(tu) \rightarrow -\infty \ \ \text {as} \ \ t\rightarrow \infty \end{aligned}$$

and moreover, classical arguments, which can be found in ([2, 18]), along with conditions $({{\textbf {H}}}_k)$ show that the function $\mu _{\lambda }$ satisfies the Cerami condition. By mountain pass theorem, there exists ${\widehat{u}} \in W_{0}^{s_1,p}(\Omega )$ such that,

$$\begin{aligned} {\widehat{u}} \in K_{\mu _{\lambda }} \subseteq [u_0) \cap \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \end{aligned}$$

and $m_\lambda \le \mu _{\lambda }({\widehat{u}})$. So, we have ${\widehat{u}} \in S_{\lambda }, \ u_0 \le {\widehat{u}}$ and ${\widehat{u}} \ne u_0$. $\square$

Proposition 11

If hypotheses $({{\textbf {H}}}_k)$ hold, then $\lambda ^* \in {\mathcal {L}}$.

Proof

Let $\{\lambda _n\} \subset (0,\lambda ^*)$ be such that $\lambda _n \rightarrow \lambda ^*$. We have $\{\lambda _n\}_{n\ge 1} \subseteq {\mathcal {L}}$ and of the proof of Proposition 10 we find $u_n \in S_{\lambda _n} \subseteq \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+]$ such that,

$$\begin{aligned} \mu _{\lambda _n}(u_n)&= \frac{1}{p} \left[ u_n \right] _{s_1,p}^{p} + \left[ u_n \right] _{s_2,q}^{q} - \lambda _n\int _{\Omega } [u_n^{1-\eta } + f_k(x,u_n).u_n] \textrm{d}x \\&= \frac{1}{p} \left[ u_n \right] _{s_1,p}^{p} + \frac{1}{q} \left[ u_n \right] _{s_2,q}^{q} - \left[ u_n \right] _{s_1,p}^{p} - \left[ u_n \right] _{s_2,p}^{p} \ \ ( \text {Since} \ \ u_n \in S_{\lambda _n})\\&= \left( \frac{1}{p} - 1\right) \left[ u_n \right] _{s_1,p}^{p} + \left( \frac{1}{q} - 1\right) \left[ u_n \right] _{s_2,q}^{q} < 0 \ \ \text {for all} \ \ n \in {\mathbb {N}}. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \bigg \langle A_{s_1,p}(u_n) + A_{s_2,q}(u_n),\varphi \bigg \rangle = \int _{\Omega }[\lambda _n u_n^{-\eta } + f_k(x,u_n)]\varphi \textrm{d}x, \ \ \text {for all} \ \ \varphi \in W_{0}^{s_1,p}(\Omega ). \nonumber \\ \end{aligned}$$

(32)

Arguing as in the proof of Proposition 13 in [2], we obtain that at least for a subsequence,

$$\begin{aligned} u_n \rightarrow u_* \ \ \text {in} \ \ W_{0}^{s_1,p}(\Omega ) \ \ \text {as} \ n \rightarrow \infty . \end{aligned}$$

By Proposition 8, ${\tilde{u}}_{\lambda _1} \le u_n$ for all $n \in {\mathbb {N}}$. Therefore, we see $u_* \ne 0$ and $u_*^{-\eta }\varphi \le {\tilde{u}}_{\lambda _1}^{-\eta }\varphi \in L^1(\Omega )$ for all $\varphi \in W_{0}^{s_1,p}(\Omega )$. In (32), we pass to the limit as $n \rightarrow \infty$ and we obtain

$$\begin{aligned} \bigg \langle A_{s_1,p}(u_*) + A_{s_2,q}(u_*),\varphi \bigg \rangle = \int _{\Omega }[\lambda ^* u_*^{-\eta } + f_k(x,u_*)]\varphi \textrm{d}x, \ \ \text {for all} \ \ \varphi \in W_{0}^{s_1,p}(\Omega ). \end{aligned}$$

that is,

$$\begin{aligned} u_* \in S_{\lambda ^*} \subseteq \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \ \ \text {and so} \ \ \lambda ^* \in {\mathcal {L}}. \end{aligned}$$

$\square$

So, summarizing the situation for problem ($P_{k,\lambda }$), we can state the following bifurcation-type theorem.

Theorem 1

If hypotheses $({{\textbf {H}}}_k)$ hold, then we can find $\lambda ^*>0$ such that

• 1. For every $\lambda \in (0,\lambda ^*)$ problem ($P_{k,\lambda }$) has at least two nontrivial positive solutions

  $$\begin{aligned}u_0, {\hat{u}} \in \text {int}[(C_{s_1}^{0}(\Omega ))_+] \ \ \text {with} \ \ u_0 \le {\hat{u}} \ \ \text {and} \ \ u_0 \ne {\hat{u}}. \end{aligned}$$

• 2. For $\lambda = \lambda ^*$ problem ($P_{k,\lambda }$) has one nontrivial positive solution

  $$\begin{aligned} u_* \in \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \ \ \text {and so} \ \ \lambda ^* \in {\mathcal {L}}. \end{aligned}$$

• 3. For $\lambda > \lambda ^*$ problem ($P_{k,\lambda }$) has no nontrivial positive solution.

6 Existence of positive solution for $P_\lambda$

We denote by $u:= u_{k,\lambda }$ the solution of the problem ($P_{k,\lambda }$) given by Theorem 1. Thus, we obtain

Proposition 12

Let $u:= u_{k,\lambda } \in W_{0}^{s_1,p}(\Omega )$ be a positive weak solution to the problem in ($P_{k,\lambda }$), then $u \in L^{\infty }({\bar{\Omega }})$. Moreover, there exists $k >1$ sufficiently large such that,

$$\begin{aligned} \Vert u \Vert _\infty \le M_k. \end{aligned}$$

Proof

The arguments of the proof is taken from the celebrated article of [22] with appropriate modifications. We will proceed with the smooth, convex and Lipschitz function $g_\epsilon (t)=\left( \epsilon ^2+t^2\right) ^{\frac{1}{2}}$ for every $\epsilon >0$. Moreover, $g_\epsilon (t) \rightarrow |t|$ as $t \rightarrow 0$ and $\left| g_\epsilon ^{\prime }(t)\right| \le 1$. Let $0<\psi \in C_c^{\infty }(\Omega )$ and choose $\varphi =\psi \left| g_\epsilon ^{\prime }(u)\right| ^{p-2} g_\epsilon ^{\prime }(u)$ as the test function.

By Lemma 5.3 of [22] for all $\psi \in C_c^{\infty }(\Omega ) \cap {\mathbb {R}}^{+}$, we obtain

$$\begin{aligned} \begin{aligned} \langle A_{s_1,p}(g_\epsilon (u)), \psi \rangle + \langle A_{s_2,q}(g_\epsilon (u)), \psi \rangle \le \lambda \int _{\Omega }\left( \frac{1}{|u|^{\eta } }+ | f_k(x,u)| \right) \left| g'_\epsilon (u)\right| ^{p-1} \psi \textrm{d} x \end{aligned} \end{aligned}$$

By Fatou’s Lemma as $\varepsilon \rightarrow 0$ we have

$$\begin{aligned} \begin{aligned} \langle A_{s_1,p}(u), \psi \rangle + \langle A_{s_2,q}(u), \psi \rangle \le \lambda \int _{\Omega }\left( \frac{1}{|u|^{\eta } }+ | f_k(x,u)| \right) \psi \textrm{d} x \end{aligned} \end{aligned}$$

(33)

Define $u_n = \min \{(u-M_k^{\gamma })^{+}, n\}$ for each $n \in {\mathbb {N}}$ and $\gamma >0$. Let $\beta > 1$, $\delta > 0$ and consider $\psi _\delta = (u_n + \delta )^{\beta } - \delta ^{\beta }$. Thus, $\psi _\delta = 0$ in $\{u\le M_k^{\gamma }\}$ and using $\psi _\delta$ in (33) we obtain

$$\begin{aligned} \begin{aligned} \langle A_{s_1,p}(u), \psi _\delta \rangle + \langle A_{s_2,q}(u), \psi _\delta \rangle \le \lambda \int _{\Omega }\left( \frac{1}{|u|^{\eta } }+ | f_k(x,u)| \right) ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \end{aligned} \end{aligned}$$

By Lemma 5.4 in [22] to follow the estimates,

$$\begin{aligned} \begin{aligned}&\langle A_{s_1,p}(u), \psi _\delta \rangle + \langle A_{s_2,q}(u), \psi _\delta \rangle \\&\ge \beta \left( \displaystyle \frac{p}{\beta + p -1} \right) ^{p} \left[ (u_n + \delta )^{\frac{\beta + p -1}{p}}\right] _{s_1,p}^{p} + \beta \left( \displaystyle \frac{q}{\beta + q -1} \right) ^{q} \left[ (u_n + \delta )^{\frac{\beta + q -1}{q}}\right] _{s_2,q}^{q}\\&\ge \beta \left( \displaystyle \frac{p}{\beta + p -1} \right) ^{p} \left[ (u_n + \delta )^{\frac{\beta + p -1}{p}}\right] _{s_1,p}^{p} \end{aligned} \end{aligned}$$

consequently,

$$\begin{aligned} \begin{aligned} \beta \left( \displaystyle \frac{p}{\beta + p -1} \right) ^{p} \left[ (u_n + \delta )^{\frac{\beta + p -1}{p}}\right] _{s_1,p}^{p} \le \lambda \int _{\Omega }\left( \frac{1}{|u|^{\eta } }+ | f_k(x,u)| \right) ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \end{aligned} \end{aligned}$$

and thus,

$$\begin{aligned} \begin{aligned}&\left[ (u_n + \delta )^{\frac{\beta + p -1}{p}}\right] _{s_1,p}^{p} \le \lambda \displaystyle \frac{1}{\beta } \left( \displaystyle \frac{\beta + p -1}{p} \right) ^{p} \int _{\Omega }\left( \frac{1}{|u|^{\eta } }+ | f_k(x,u)| \right) \\&\quad ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \end{aligned} \end{aligned}$$

(34)

Using the estimates (5), for $M_k > 1$ we have,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left( \frac{1}{|u|^{\eta } }+ | f_k(x,u)| \right) ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \\&\quad \le \int _{\Omega }\left( \frac{1}{|u|^{\eta } } + C.M_k^{2\theta } \vert u \vert ^{r-1}\right) ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \\&\quad = \int _{\{u \ge M_k^\gamma \}}\left( \frac{1}{|u|^{\eta } } + C.M_k^{2\theta } \vert u \vert ^{r-1}\right) ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \\&\quad = \int _{\{u \ge M_k^\gamma \}} ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x + \int _{\{u \ge M_n \}} C.M_k^{2\theta } \vert u \vert ^{r-1}((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \\&\quad \le \int _{\{u \ge M_k^\gamma \}} M_k^{2\theta } ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x + \int _{\{u \ge M_k^\gamma \}} C.M_k^{2\theta } \vert u \vert ^{r-1}((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \\&\quad \le \int _{\Omega } M_k^{2\theta } ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x + \int _{\Omega } C.M_k^{2\theta } \vert u \vert ^{r-1}((u_k + \delta )^\beta - \delta ^{\beta }) \textrm{d} x\\&\quad \le C.M_k^{2\theta } \left( \vert \Omega \vert ^{\frac{\sigma -1}{\sigma }} + \Vert u \Vert ^{r-1}_{L^{p_{s_1}^*}(\Omega )} \right) \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )} \end{aligned} \end{aligned}$$

where C is a constant independent of k and $\sigma = \displaystyle \frac{p_{s_1}^{*}}{p_{s_1}^{*}-r+1}$. Moreover, observe that the function $u:= u_k$ satisfies $u \le {\overline{u}}$ where ${\overline{u}}$ is a supersolution of the problem (20) does not depend on k, we have $\Vert u \Vert ^{r-1}_{L^{p_{s_1}^*}(\Omega )} \le C_0 \Vert {\overline{u}} \Vert ^{r-1}_{\infty }$ independent of k. Thus,

$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left( \frac{1}{|u|^{\eta } }+ | f_k(x,u)| \right) ((u_n + \delta )^\beta - \delta ^{\beta }) \textrm{d} x \\&\quad \le K M_k^{2\theta } \left( \vert \Omega \vert ^{\frac{\sigma -1}{\sigma }} + \Vert {\overline{u}} \Vert ^{r}_{\infty } \right) \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )}\\&\quad = K_0 M_k^{2 \theta } \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )} \end{aligned} \end{aligned}$$

(35)

with $K_0$ independent of k.

By Sobolev inequality, triangle inequality and $(u_n + \delta )^{\beta + p -1} \ge \delta ^{p-1} (u_n + \delta )^{\beta }$

$$\begin{aligned} \begin{aligned} \left[ (u_n + \delta )^{\frac{\beta + p -1}{p}}\right] _{s_1,p}^{p}&\ge S \Vert (u_n+ \delta )^\beta - \delta ^{\beta } \Vert _{L^{p_{s_1}^{*}}(\Omega )}^p \\&\ge \left( \displaystyle \frac{\delta }{2}\right) ^{p-1} \left[ \int _{\Omega } \vert (u_n+\delta )^{\frac{p_{s_1}^{*}\beta }{p}} \textrm{d}x \right] ^{\frac{p}{p_{s_1}^{*}}} - \delta ^{\beta + p-1} \vert \Omega \vert ^{\frac{p}{p_{s_1}^{*}}}\\&\ge \left( \displaystyle \frac{\delta }{2}\right) ^{p-1} \Vert (u_n + \delta )^{\frac{\beta }{p}} \Vert _{L^{p_{s_1}^{*}}(\Omega )}^{p} - M_k^{2\theta } \delta ^{\beta +p-1} \vert \Omega \vert ^{\frac{p}{p_{s_1}^{*}}}, \end{aligned} \end{aligned}$$

(36)

in the estimate above we using that $M_k > 1$.

Using the estimates (36) and (35) in (34), we obtain

$$\begin{aligned} \begin{aligned}&\left\| \left( u_n+\delta \right) ^{\frac{\beta }{p}}\right\| _{L^{p_{s_1}^{*}}(\Omega )}^p \le \left( \frac{2}{\delta }\right) ^{p-1}\Bigg [\left( \frac{ (\beta +p-1)^p}{\beta p^p}\right) \\&\qquad K_0 M_k^{2\theta } \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )} +\delta ^{\beta +p-1}|\Omega |^ \frac{p}{p_s^*}\Bigg ] \\&= \left( \frac{2}{\delta }\right) ^{p-1} \left( \frac{ (\beta +p-1)^p}{\beta p^p}\right) K_0 M_k^{2\theta } \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )} +\delta ^{\beta }|\Omega |^ \frac{p}{p_s^*}\\&\le \left( \frac{2}{\delta }\right) ^{p-1} \left( \frac{ (\beta +p-1)^p}{\beta p^p}\right) K_0 M_k^{2\theta } \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )} + |\Omega |^ {\frac{p}{p_s^*} - 1} \int _{\Omega } (u_n + \delta )^{\beta } \textrm{d}x \end{aligned} \end{aligned}$$

By Holder’s inequality, we have

$$\begin{aligned} \delta ^{\beta } = \vert \Omega \vert ^{-1} \int _{\Omega } \delta ^{\beta } \textrm{d}x \le \vert \Omega \vert ^{-1} \int _{\Omega } (u_n + \delta )^{\beta } \textrm{d}x \le \vert \Omega \vert ^{-\frac{1}{\sigma }} \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )}. \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{aligned}&\left\| \left( u_n+\delta \right) ^{\frac{\beta }{p}}\right\| _{L^{p_{s_1}^{*}}(\Omega )}^p \le \left( \frac{2}{\delta }\right) ^{p-1} \left( \frac{ (\beta +p-1)^p}{\beta p^p}\right) K_0 M_k^{2\theta } \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )} \\&+ |\Omega |^ {\frac{p}{p_s^*} - \frac{1}{\sigma }} \Vert (u_n + \delta )^{\beta } \Vert _{L^{\sigma }(\Omega )}.\\ \end{aligned} \end{aligned}$$

Since, $\displaystyle \frac{1}{\beta }\left( \frac{\beta +p-1}{p}\right) ^p \ge 1$ we can deduce that

$$\begin{aligned} \begin{aligned} \left\| \left( u_n+\delta \right) ^{\frac{\beta }{p}}\right\| _{L^{p_{s_1}^{*}}(\Omega )}^p&\le \frac{1}{\beta }\left( \frac{\beta +p-1}{p}\right) ^p M_k^{2\theta } \left\| \left( u_n+\delta \right) ^\beta \right\| _q\left( \frac{K_0}{\delta ^{p-1}}+|\Omega |^{\frac{p}{p_s^*}-\frac{1}{\sigma }}\right) \\ \end{aligned} \end{aligned}$$

Now choose, $\delta > 0$ such that $\delta ^{p-1} = K_0 \vert \Omega \vert ^{\frac{1}{\sigma } - \frac{p}{p_{s_1}^{*}}}$ and $\beta >1$ such that, $\left( \displaystyle \frac{\beta +p-1}{p}\right) ^p \ge \beta ^{p}$. Thus,

$$\begin{aligned} \begin{aligned} \left\| \left( u_n+\delta \right) ^{\frac{\beta }{p}}\right\| _{L^{p_{s_1}^{*}}(\Omega )}^p \le C M_k^{2\theta } \beta ^{p-1} \left\| \left( u_n+\delta \right) ^\beta \right\| _{L^{\sigma }(\Omega )}\\ \end{aligned} \end{aligned}$$

For $\tau = \sigma \beta$ and $\alpha =\displaystyle \frac{p_{s_1}^{*}}{\sigma p}$ we obtain,

$$\begin{aligned} \begin{aligned} \left\| u_n+\delta \right\| _{L^{\alpha \tau }(\Omega )}^\beta \le C M_k^{2\theta } \beta ^{p-1} \left\| u_n+\delta \right\| _{L^{\tau }(\Omega )}^\beta \\ \end{aligned} \end{aligned}$$

and therefore,

$$\begin{aligned} \begin{aligned} \left\| u_n+\delta \right\| _{L^{\alpha \tau }(\Omega )} \le \left( C M_k^{2\theta }\right) ^{\frac{\sigma }{\tau }} \left( \displaystyle \frac{\tau }{\sigma }\right) ^{(p-1)\frac{\sigma }{\tau }} \left\| u_n+\delta \right\| _{L^{\tau }(\Omega )}. \end{aligned} \end{aligned}$$

Taking, $\tau _0 = \sigma$, $\tau _{m+1} = \alpha \tau _m = \alpha ^{m+1} \sigma$, then after performing m iterations we obtain the inequality

$$\begin{aligned} \begin{aligned} \left\| u_n+\delta \right\| _{L^{\tau _{m+1}}(\Omega )}&\le \left( C M_k^{2\theta }\right) ^{\displaystyle \sum _{i=0}^{m}\frac{\sigma }{\tau _i}} \left( \displaystyle \prod _{i=1}^{m}\left( \displaystyle \frac{\tau _i}{\sigma }\right) ^{\frac{\sigma }{\tau _i}}\right) ^{(p-1)} \left\| u_n+\delta \right\| _{L^{\tau }(\Omega )}\\&= \left( C M_k^{2\theta }\right) ^{\displaystyle \sum _{i=1}^{m}\frac{1}{\alpha ^{i}}} \left( \displaystyle \prod _{i=1}^{m}\alpha ^{\frac{i}{\alpha ^i}}\right) ^{(p-1)} \left\| u_n+\delta \right\| _{L^{\tau }(\Omega )} \end{aligned} \end{aligned}$$

Therefore, on passing the limit as $m \rightarrow \infty$, we get

$$\begin{aligned} \begin{aligned} \Vert u_n \Vert _{L^{\infty }(\Omega )} \le \left\| u_n+\delta \right\| _{L^{\infty }(\Omega )} \le C^{\frac{\alpha }{\alpha -1}} M_k^{\frac{2\theta \alpha }{\alpha -1}} \alpha ^{\frac{(p-1)\alpha }{(\alpha -1)^2}} \left\| u_n+\delta \right\| _{L^{\sigma }(\Omega )} \le C_1 M_k^{\frac{2\theta \alpha }{\alpha -1}}. \end{aligned}\nonumber \\ \end{aligned}$$

(37)

In the last inequality we use the fact, $u \le {\overline{u}}$, where ${\overline{u}} \in L^{\infty }(\Omega )$ is a supersolution of the problem (20) and thus, $u_n = \min \{(u - M_k^{\gamma })^{+}, n\} \le (u-M_k^{\gamma })^+ \le u^{+} \le {\overline{u}}$, for each $n \in {\mathbb {N}}$ and k large enough (such that $\Vert {\overline{u}} \Vert \le M_k^{\gamma })$.

Therefore, as $n \rightarrow \infty$ we obtain

$$\begin{aligned} \Vert (u - M_k^{\gamma })^+ \Vert _{\infty } \le M_k \end{aligned}$$

for $M_k$ sufficiently large and $\displaystyle \frac{2\theta \alpha }{\alpha -1} < 1$. Consequently, since $M_k \rightarrow \infty$ as $k \rightarrow \infty$ we have, for $\gamma < 1$, there exists $k>1$ large enough such that,

$$\begin{aligned} \Vert u \Vert _{\infty } \le M_k. \end{aligned}$$

Also, by (37), the embedding $W^{s_1,p}_0(\Omega ) \hookrightarrow L^{\sigma }(\Omega )$ and since $u_n = \min \{(u - M_k^{\gamma })^{+}, n\} \le (u-M_k^{\gamma })^+ \le u^{+}\le |u|$ we can establish

$$\begin{aligned} \Vert u_n\Vert _{L^{\infty }\left( \Omega \right) } \le CM_k^{\frac{2\theta \alpha }{\alpha -1}}[u]_{s_1,p}. \end{aligned}$$

Therefore, as $n \rightarrow \infty$ we obtain

$$\begin{aligned} \Vert u\Vert _{L^{\infty }\left( \Omega \right) } \le CM_k^{\frac{2\theta \alpha }{\alpha -1}}[u]_{s_1,p}, \end{aligned}$$

for $k>1$ large enough fixed. $\square$

Theorem 2

If hypotheses $({{\textbf {H}}})$ hold, then we can find $\lambda ^*=\lambda ^*(k)>0$ (k as in Proposition 12) such that

• 1. For every $\lambda \in (0,\lambda ^*)$ problem ($P_\lambda$) has at least two nontrivial positive solutions

  $$\begin{aligned}u_0, {\hat{u}} \in \text {int}[(C_{s_1}^{0}(\Omega ))_+] \ \ \text {with} \ \ u_0 \le {\hat{u}} \ \ \text {and} \ \ u_0 \ne {\hat{u}}. \end{aligned}$$

• 2. For $\lambda = \lambda ^*$ problem ($P_\lambda$) has one nontrivial positive solution

  $$\begin{aligned} u_* \in \text {int}[(C_{s_1}^{0}({\overline{\Omega }}))_+] \ \ \text {and so} \ \ \lambda ^* \in {\mathcal {L}}. \end{aligned}$$

• 3. For $\lambda > \lambda ^*$ problem ($P_\lambda$) has no nontrivial positive solution.

Proof

By Theorem 1, for each $\lambda \in (0,\lambda ^{*}]$ and $k \in {\mathbb {N}}$ there exists $u_{k,\lambda }$ such that,

Moreover, 1, 2 and 3 holds to the problem ($P_{k,\lambda }$), by Theorem 1.

Using the Proposition 12, we have $\Vert u_{k,\lambda } \Vert _{\infty } < M_k$ for some $k > 1$ large enough. Thus, $u_\lambda := u_{k,\lambda }(x) \le M_k$ and therefore $f_k(x,u_\lambda ) = f(x,u_\lambda )$, in other words $u_\lambda$ satisfies the problem ($P_\lambda$). $\square$