Optimal decay of extremals for the fractional Sobolev inequality

Abstract

We obtain the sharp asymptotic behavior at infinity of extremal functions for the fractional critical Sobolev embedding.

1 Introduction and main result

Let $N>p>1$. In two seminal papers, Aubin [3] and Talenti [33] showed that the minimizers of the Sobolev quotient

$$\begin{aligned} \mathcal {S}_{p}=\inf _{u\in D^{1,p}(\mathbb {R}^N){\setminus }\{0\}} \frac{\Vert \nabla u\Vert ^p_{L^p(\mathbb {R}^N)}}{\displaystyle \left( \int _{\mathbb {R}^N} |u|^\frac{N\,p}{N-p}\, dx\right) ^\frac{N-p}{N}}, \end{aligned}$$

(1.1)

are given by the family of functions

$$\begin{aligned} U_{t}(x)={\mathcal C}\,t^\frac{p-N}{p}\,U\left( \frac{x-x_0}{t}\right) , \quad {\mathcal C}\in \mathbb {R}{\setminus }\{0\},\ t>0,\ x_0\in \mathbb {R}^N, \end{aligned}$$

(1.2)

where

$$\begin{aligned} U(x)=\mathcal {C}_{N,p}\,\big (1+|x|^{\frac{p}{p-1}}\big )^{\frac{p-N}{p}},\quad \mathcal {C}_{N,p}=\left( \int _{\mathbb {R}^N}\big (1+|x|^{\frac{p}{p-1}}\big )^{-N}\,dx\right) ^\frac{N\,p}{p-N}. \end{aligned}$$

For the limit case $p=1$, the problem was investigated by Federer and Fleming [14] and by Mazya [27].

On one side, these results establish an enlightening connection between the theory of Sobolev spaces and the theory of classical isoperimetric inequalities. On the other side, they provide a very powerful tool for the study of second order partial differential equations involving nonlinearities reaching the critical growth with respect to the Sobolev embedding. In the case $p=2$, these classification results were formally derived by Rosen [30].

The variational problem (1.1) is related to the following equation involving the p-Laplace operator $\Delta _p u=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$,

$$\begin{aligned} -\Delta _p u=|u|^{\frac{N\,p}{N-p}-2}\,u, \quad \hbox { in }\mathbb {R}^N. \end{aligned}$$

(1.3)

In fact, a nontrivial problem is that of proving that the only fixed sign solutions of this equation are precisely given by (1.2), for a suitable choice of the constant.

In the restricted class of radially symmetric fixed sign solutions to (1.3), this was shown by Guedda and Veron [17]. Recently, in [34, Corollary  1.3] for the case $1<p\le 2\,N/(N+2)$, in [10, Theorem  1.2] for the case $2\,N/(N+2)<p\le 2$ and in [32, Theorem 1.1] for the case $2<p<N$, it was proved that any positive weak solution to (1.3) is radially symmetric and radially decreasing about some point, thus answering positively to the classification of constant sign solutions to (1.3).

The result by Aubin and Talenti, as well as the previous results in the linear case $p=2$, strongly rely on the reduction of the problem to an ordinary differential equation which can be explicitly solved. We recall that more recently, the Aubin–Talenti result has been reproved in [8, Theorem 2] by means of very different techniques, based on Optimal Transport.

Let now $s\in (0,1)$, $p>1$ and $N>\textit{sp}$. The goal of this paper is to provide information about the asymptotic behavior at infinity of optimizers of the problem

$$\begin{aligned} \mathcal {S}_{p,s}:=\inf _{u\in D^{s,p}(\mathbb {R}^N){\setminus }\{0\}} \frac{\displaystyle \int _{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+\textit{sp}}}\,dx\,dy}{\displaystyle \left( \int _{\mathbb {R}^N} |u|^\frac{N\,p}{N-\textit{sp}}\,dx\right) ^\frac{N-\textit{sp}}{N}}, \end{aligned}$$

(1.4)

which is related to the fractional Sobolev embedding, see for example [28, Theorem 1]. Here

$$\begin{aligned} D^{s,p}(\mathbb {R}^N)=\left\{ u\in L^{{N\,p/(N-\textit{sp})}}(\mathbb {R}^N): \int _{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+\textit{sp}}}\,dx\,dy<\infty \right\} . \end{aligned}$$

In the limit case $p=1$, the sharp constant above has been determined in [15, Theorem 4.1] (see also [6, Theorem  4.10]). The relevant extremals are given by characteristic functions of balls, exactly as in the local case.

Problem (1.4) for $p>1$ is now related to the study of the nonlocal integro-differential equation

$$\begin{aligned} (-\Delta _p)^s u=|u|^{\frac{N\,p}{N-\textit{sp}}-2}\,u,\quad \hbox { in }\mathbb {R}^N, \end{aligned}$$

(1.5)

where, formally, the operator $(-\Delta _p)^s$ is defined on smooth functions as

$$\begin{aligned} (-\Delta _p)^s u(x) = 2\, \lim _{\varepsilon \searrow 0} \int _{\mathbb {R}^N {\setminus } B_\varepsilon (x)} \frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))}{|x - y|^{N+\textit{sp}}}\, dy, \quad x \in \mathbb {R}^N. \end{aligned}$$

This operator appears in some recent works like [2, 21]. See also [11, 18–20, 22] and the references therein for some existence and regularity results.

In the Hilbertian case $p=2$, it is known by [9, Theorem  1.1] that the family of functions

$$\begin{aligned} U_{t}(x)={\mathcal C}\,t^\frac{2\,s-N}{2}\,\left( 1+\left( \frac{|x-x_0|}{t}\right) ^2\right) ^{\frac{2\,s-N}{2}},\quad \mathcal {C}\in \mathbb {R}{\setminus }\{0\},\ t>0,\ x_0\in \mathbb {R}^N, \end{aligned}$$

(1.6)

is the only set of minimizers for the best Sobolev constant $\mathcal {S}_{2,s}$. More precisely, in [9, Theorem  1.1] it is proved that the family (1.6) provides all the minimizers of the following problem

$$\begin{aligned} \widetilde{\mathcal {S}}_{2,s}:=\inf _{u\in D^{s,2}(\mathbb {R}^N){\setminus }\{0\}} \frac{\big \Vert (-\Delta )^{s/2}\,u\big \Vert _{L^2(\mathbb {R}^N)}^2}{\displaystyle \left( \int _{\mathbb {R}^N} |u|^\frac{2\,N}{N-2\,s}\,dx\right) ^\frac{N-2\,s}{N}}, \end{aligned}$$

where the $L^2$ norm of $(-\Delta )^{s/2} u$ is defined in terms of the Fourier transform. By using [12, Proposition 3.6], one knows that

$$\begin{aligned} \int _{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2\,s}}\,dx\,dy=c\, \big \Vert (-\Delta )^{s/2}\,u\big \Vert _{L^2(\mathbb {R}^N)}^2, \end{aligned}$$

for some $c=c(N,s)>0$. This implies that (1.6) are the only solutions of (1.4) as well.

It is also known by [7, Theorem 1] that, for a suitable positive constant ${\mathcal C}={\mathcal C}(N,s)$, (1.6) are the only positive solutions of

$$\begin{aligned} (-\Delta )^s u=|u|^{\frac{2\,N}{N-2\,s}-2}\,u \quad \hbox { in }\mathbb {R}^N. \end{aligned}$$

(1.7)

The result in [7] is based upon the full equivalence between the weak solutions to (1.7) and the integral formulation

$$\begin{aligned} u(x)=\int _{\mathbb {R}^N}\frac{|u(y)|^{\frac{2\,N}{N-2\,s}-2}\,u}{|x-y|^{N-2\,s}}\,dy,\quad u\in L^{\frac{2\,N}{N-2\,s}}(\mathbb {R}^N), \end{aligned}$$

(1.8)

on the validity of some Kelvin transform and on moving plane arguments applied to (1.8), in the spirit of [23].

Unfortunately, in the nonlocal and nonlinear case $p\ne 2$ there is no Kelvin transform and no equivalent integral representation result. Furthermore, even restricting to the class of radially symmetric functions, establishing a classification result for the optimizers of (1.4) seems very hard. We conjecture that the optimizers are given by

$$\begin{aligned} U_{t}(x)={\mathcal C}\,t^\frac{\textit{sp}-N}{p}\,U\left( \frac{x-x_0}{t}\right) , \quad {\mathcal C}\in \mathbb {R}{\setminus }\{0\},\ t>0,\ x_0\in \mathbb {R}^N, \end{aligned}$$

(1.9)

where this time

$$\begin{aligned} U(x):=\mathcal {C}_{N,p,s}\,\big (1+|x|^{\frac{p}{p-1}}\big )^{\frac{\textit{sp}-N}{p}},\quad \mathcal {C}_{N,p,s}=\left( \int _{\mathbb {R}^N}\big (1+|x|^{\frac{p}{p-1}}\big )^{-N}\,dx\right) ^\frac{N\,p}{\textit{sp}-N}. \end{aligned}$$

(1.10)

Notice that (1.9) and (1.10) are consistent with the cases $p=2$ or $s=1$, in the last case we are back to the family of Aubin–Talenti functions (1.2) for the p-Laplacian operator.

In the main result of this paper, we prove that extremals for (1.4) have exactly the decay rate at infinity dictated by formula (1.10). Namely, we have the following.

Theorem 1.1

Let $U\in D^{s,p}(\mathbb {R}^N)$ be any minimizer for (1.4). Then $U\in L^\infty (\mathbb {R}^N)$ is a constant sign, radially symmetric and monotone function with

$$\begin{aligned} \lim _{|x|\rightarrow \infty }|x|^{\frac{N-\textit{sp}}{p-1}}U(x)=U_\infty , \end{aligned}$$

(1.11)

for some constant $U_\infty \in \mathbb {R}{\setminus }\{0\}$.

Remark 1.2

As it will be apparent from the proof of Theorem 1.1, the same conclusion (1.11) can be drawn for any constant sign, radially symmetric and monotone solution of the critical equation (1.5). In the local case this property is a plain consequence of the aforementioned classification result of [17].

The building blocks of Theorem 1.1 are a weak $L^q$ estimate for the minimizers (Proposition 3.3), a Radial Lemma for Lorentz spaces (Lemma 2.9) and the fact that the function

$$\begin{aligned} \Gamma (x):=|x|^{-\frac{N-\textit{sp}}{p-1}},\quad x\in \mathbb {R}^N{\setminus }\{0\}, \end{aligned}$$

is a weak solution of $(-\Delta _p)^su=0$ in $\mathbb {R}^N{\setminus } B_r$, for any $r>0$ (Theorem A.4). Then the crucial point will be constructing suitable barrier functions to be combined with a version of the comparison principle for $(-\Delta _p)^s$ recently obtained in [19]. Observe that for $s=1$, the function $\Gamma$ above is nothing but the fundamental solution of the p-Laplacian.

We wish to stress that Theorem 1.1 also provides a very useful tool for the investigation of existence of weak solutions for the nonlocal Brezis–Nirenberg problem in a smooth bounded domain $\Omega \subset \mathbb {R}^N$, i.e.

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta _p)^s\, u =\lambda \,|u|^{p-2}\,u+|u|^{\frac{N\,p}{N-\textit{sp}}-2}u &{} \hbox { in }\quad \Omega , \\ u = 0 &{} \hbox { in }\quad \mathbb {R}^N {\setminus } \Omega , \end{array}\right. } \end{aligned}$$

where $\lambda > 0$. This problem has been studied in [31] for $p=2$. For a general exponent $1<p<N/s$, by means of (1.11), one can estimate truncations of $U_t$ via a suitable cut-off function in terms of the sharp constant $\mathcal {S}_{p,s}$ without knowing the explicit form of the optimizers. Such a procedure is new even for the local case. These estimates allow to apply mountain pass or linking arguments by forcing the min-max levels to fall inside a compactness range for the energy functionals, see [29] for more details.

Plan of the paper

In Sect. 2 we set all the notations, definitions and basic facts that will be needed throughout the paper. Then in Sec. 3 we prove existence of solutions for (1.4), together with some basic integrability properties. We also prove that extremals have to be comparable to

$$\begin{aligned} x\mapsto |x|^{-\frac{N-\textit{sp}}{p-1}},\quad x\in \mathbb {R}^N{\setminus }\{0\}, \end{aligned}$$

at infinity (Corollary 3.7). Then the exact behavior (1.11) is proved in Sect. 4. The paper ends with “Appendix A”, containing a rigourous computation of the fractional p-Laplacian of a power function.

2 Preliminary results

2.1 Notation

In the following we will fix $s\in (0, 1)$, $p>1$ and N as the dimension, letting for brevity

$$\begin{aligned} p^*=\frac{\textit{Np}}{N-\textit{sp}}. \end{aligned}$$

We denote by $\omega _N$ the measure of the N-dimensional ball having unit radius. Moreover, $\mathbf {S}^{N-1}$ will denote $\{x\in \mathbb {R}^N:|x|=1\}$. For $E\subseteq \mathbb {R}^N$ measurable we denote by |E| its N-dimensional Lebesgue measure, by $E^c=\mathbb {R}^N{\setminus } E$ its complement and by $\chi _E$ its characteristic function. If $u:E\rightarrow \mathbb {R}$ is measurable we set

$$\begin{aligned}{}[u]_{W^{s, p}(E)}^p:=\int _{E\times E}\frac{|u(x)-u(y)|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy, \quad [u]_{s,p}:=[u]_{W^{s,p}(\mathbb {R}^N)}, \end{aligned}$$

and for any $q>0$

$$\begin{aligned} \Vert u\Vert _{L^q(E)}:=\left( \int _{E}|u|^q\, dx\right) ^{1/q},\quad \Vert u\Vert _q:=\Vert u\Vert _{L^q(\mathbb {R}^N)}. \end{aligned}$$

Finally, for $t\in \mathbb {R}$ we will use the notation

$$\begin{aligned} J_p(t)=|t|^{p-2}\,t. \end{aligned}$$

2.2 Elementary inequalities

We list here some useful inequalities on the function $J_p$. First, consider the case $p\ge 2$. We recall that

$$\begin{aligned} |J_p(a)-J_p(b)|\le (p-1)\,(|a|^{p-2}+|b|^{p-2})\,|a-b|,\quad a,b\in \mathbb {R},\quad p\ge 2, \end{aligned}$$

(2.1)

as a consequence of the mean value Theorem. In [19, eq. (2.7)] it is also proved the following inequality

$$\begin{aligned} J_p(a)-J_p(a+b)\le -2^{2-p}\,b^{p-1},\quad a\in \mathbb {R}, \ b\ge 0,\quad p\ge 2. \end{aligned}$$

(2.2)

Let us consider the case $p\in (1,2]$.

We recall the well-known monotonicity inequality

$$\begin{aligned} \big (J_p(a)-J_p(b)\big )\,(a-b)\ge c\,\frac{|a-b|^2}{(a^2+b^2)^\frac{2-p}{2}},\quad a,b\in \mathbb {R}{\setminus }\{0\},\quad p\in (1,2]. \end{aligned}$$

(2.3)

Next we prove the following inequality

$$\begin{aligned} J_p(a)-J_p(a-b)\ge \max \left\{ J_p(A)-J_p(A-b),\, \left( \frac{b}{2}\right) ^{p-1}\right\} ,\quad a\in [0,A], b\ge 0,\ p\in (1,2]. \end{aligned}$$

(2.4)

We distinguish two cases. First suppose that $a\ge b/2$. The function $t\mapsto J_p(t)-J_p(t-b)$ is readily seen to be decreasing on $[b/2, +\infty [$, so that

$$\begin{aligned} J_p(a)-J_p(a-b)\ge J_p(A)-J_p(A-b) \end{aligned}$$

in this case. On the other hand, if $a<b/2$, being $J_p$ odd and increasing we have

$$\begin{aligned} J_p(a)-J_p(a-b)\ge J_p(b-a)\ge J_p\left( \frac{b}{2}\right) , \end{aligned}$$

and thus (2.4) is proved.

2.3 Functional framework

We consider the space

$$\begin{aligned} D^{s,p}_0(\Omega ):=\Big \{u\in L^{p^*}(\Omega ): u\equiv 0 \,\text { in } \, \Omega ^c,\,\, [u]_{s,p}<+\infty \Big \},\quad D^{s,p}(\mathbb {R}^N):=D^{s,p}_0(\mathbb {R}^N), \end{aligned}$$

which is a Banach space with respect to the norm $[\,\cdot \, ]_{s,p}$. Our first aim is to prove, under suitable regularity assumptions on $\partial \Omega$, that $C^\infty _c(\Omega )$ is dense in $D^{s,p}_0(\Omega )$ with respect to the norm $[\,\cdot \, ]_{s,p}$. While this density result is well-known for $D^{s,p} (\Omega )\cap L^p (\Omega )$ (see for example [13]), we will need to remove the $L^p$ assumption in the following. Finally we will prove a comparison principle in a rather general space.

Theorem 2.1

Let $\Omega \subset \mathbb {R}^N$ be an open set such that $\partial \Omega$ is compact and locally the graph of a continuous function. Then $D^{s,p}_0(\Omega )$ is the completion of $C^\infty _c(\Omega )$ with respect to the norm $[\,\cdot \,]_{s,p}$.

Proof

Let $u\in D^{s,p}_0(\Omega )$. Reasoning on $u_+$ and $u_-$ separately (which still belong to $D^{s,p}_0(\Omega )$), we can suppose that u is nonnegative. Consider, for $\varepsilon >0$, the function $u_\varepsilon =(u-\varepsilon )_+$. Using the $1-$Lipschitzianity of $t\mapsto (t-\varepsilon )_+$ it is readily checked that

$$\begin{aligned} |u_\varepsilon (x)-u_\varepsilon (y)|^p \le |u(x)-u(y)|^p, \quad |u_\varepsilon (x)-u_\varepsilon (y)|^p \rightarrow |u(x)-u(y)|^p,\ \text { a.e. in }\mathbb {R}^{2N}. \end{aligned}$$

Therefore $u_\varepsilon \in D^{s,p}_0(\Omega )$ and by dominated convergence $[u_\varepsilon ]_{s,p}\rightarrow [u]_{s,p}$. This in turn implies that $u_\varepsilon \rightarrow u$ in $D^{s,p}_0(\Omega )$ by uniform convexity of the norm. Now Chebyshev’s inequality ensures that $\mathrm {supp}(u_\varepsilon )$ has finite measure, thus by Hölder’s inequality we get $u_\varepsilon \in L^p(\mathbb {R}^N)$. This yields

$$\begin{aligned} u_\varepsilon \in D^{s,p}_0(\Omega )\cap L^p(\mathbb {R}^N), \end{aligned}$$

and [13, Theorem 6] ensures that $u_\varepsilon$ can be approximated, in the norm $[\, \cdot \, ]_{s,p},$ by functions which belong to $C^\infty _c(\Omega )$.

We recall the following nonlocal Hardy inequality proved in [15, Theorem 2].

Proposition 2.2

(Hardy’s inequality) Let $N>\textit{sp}$. Then there exists $C=C(N, p, s)>0$ such that

$$\begin{aligned} \int _{\mathbb {R}^N}\frac{|u|^p}{|x|^{\textit{sp}}}\, dx\le C\,[u]_{s, p}^p,\quad \hbox { for every } u\in D^{s,p}(\mathbb {R}^N). \end{aligned}$$

(2.5)

We then define a suitable space where a comparison principle holds true. For any $\Omega \subset \mathbb {R}^N$ open set, we define

$$\begin{aligned} \begin{aligned} \widetilde{D}^{s,p}(\Omega ):=\Big \{u\in L^{p-1}_{\mathrm {\mathrm {loc}}}(\mathbb {R}^N)\cap L^{p^*}(\Omega ) \,:\,&\exists \, E\supset \Omega \text { with }E^c \text {compact},\, \mathrm {dist}(E^c, \Omega )>0 \\&\hbox { and }\,[u]_{W^{s,p}(E)}<+\infty \Big \}. \end{aligned} \end{aligned}$$

We wish to point out that the definition above is given having in mind the case of $\Omega$ being an exterior domain, i.e. the complement of a compact set. Essentially, we consider functions u which are regular in a slight enlargement of $\Omega$ and possibly rough far from $\Omega$.

The following expedient result will be used in the sequel.

Lemma 2.3

(Nash-type interpolation inequality) Let $1<p<\infty$ and $0<s<1$. For every $u\in L^{p-1}(B_R)$ such that $[u]_{W^{s,p}(B_R)}<+\infty$ we have

$$\begin{aligned} \Vert u\Vert ^p_{L^p(B_R)}\le C\,R^{\textit{sp}}\, [u]^p_{W^{s,p}(B_R)}+\frac{C}{R^\frac{N}{p-1}}\,\Vert u\Vert ^p_{L^{p-1}(B_R)}, \end{aligned}$$

(2.6)

for some $C=C(N,s,p)>0$.

Proof

We observe that it is enough to prove (2.6) for $R=1$, then the general case can be obtained with a simple scaling argument.

At first, we prove (2.6) for functions in $W^{s,p}(B_1)$. We can use a standard compactness argument: assume by contradiction that (2.6) is false on $W^{s,p}(B_1)$, then there exists a sequence $\{u_n\}_{n\in \mathbb {N}}\subset W^{s,p}(B_1)$ such that

$$\begin{aligned} \Vert u_n\Vert ^p_{L^p(B_1)}=1\quad \hbox { and }\quad [u_n]^p_{W^{s,p}(B_R)}+\Vert u_n\Vert ^p_{L^{p-1}(B_1)}\le \frac{1}{n}. \end{aligned}$$

(2.7)

In particular, the sequence is bounded in $W^{s,p}(B_1)$. Thus by compactness of the embedding $W^{s,p}(B_1)\hookrightarrow L^{p}(B_1)$ (see for example [12, Theorem 7.1]) we get that (up to a subsequence) it converges strongly in $L^p(B_1)$ to $u\in W^{s,p}(B_1)$. From (2.7) we now easily get a contradiction. This shows that (2.6) is true for functions in $W^{s,p}(B_1)$.

We now take $u\in L^{p-1}(B_1)$ with finite Gagliardo seminorm. Observe that

$$\begin{aligned} \big ||u(x)|-|u(y)|\big |\le |u(x)-u(y)|, \end{aligned}$$

(2.8)

so that

$$\begin{aligned} \big [|u|\big ]_{W^{s,p}(B_1)}\le [u]_{W^{s,p}(B_1)}. \end{aligned}$$

Thus we can assume u to be positive without loss of generality. We define the increasing sequence $u_n=\min \{u,n\}\in W^{s,p}(B_1)$. From the first part of the proof and 1-Lipschitzianity of the function $t\mapsto \min \{t,n\}$ we have

$$\begin{aligned} \begin{aligned} \Vert u_n\Vert ^p_{L^p(B_1)}&\le C\,[u_n]^p_{W^{s,p}(B_1)}+C\,\Vert u_n\Vert ^p_{L^{p-1}(B_1)}\\&\le C\,[u]^p_{W^{s,p}(B_1)}+C\,\Vert u_n\Vert ^p_{L^{p-1}(B_1)}. \end{aligned} \end{aligned}$$

Passing to the limit and using the Monotone Convergence we get the desired conclusion.

Lemma 2.4

Let $1<p<\infty$ and $0<s<1$. For every $u\in L^{p-1}_{\mathrm {\mathrm {loc}}}(\mathbb {R}^N)$, every $E\subset \mathbb {R}^N$ open set and every ball $B_R\subset E$, we have

$$\begin{aligned} \int _{E} \frac{|u(x)|^{p}}{(1+|x|)^{N+\textit{sp}}}\, dx\le C\,[u]^p_{W^{s,p}(E)}+C\,\Vert u\Vert ^p_{L^{p-1}(B_R)}, \end{aligned}$$

(2.9)

for some $C=C(N,p,s,R)>0$, blowing-up as $R\searrow 0$.

Proof

We assume that the right-hand side on (2.9) is finite, otherwise there is nothing to prove. For simplicity, we can suppose that $B_R$ is centered at the origin. From (2.6), we infer

$$\begin{aligned} \int _{B_R} \frac{|u|^p}{(1+|x|)^{N+\textit{sp}}}\,dx\le C\,R^{\textit{sp}}\, [u]^p_{W^{s,p}(B_R)}+\frac{C}{R^\frac{N}{p-1}}\,\Vert u\Vert ^p_{L^{p-1}(B_R)}. \end{aligned}$$

(2.10)

On the smaller ball $B_{R/2}$ (still centered at the origin), we have

$$\begin{aligned} \int _{E{\setminus } B_R} \int _{B_{R/2}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+\textit{sp}}}\,dy\,dx\le [u]^p_{W^{s,p}(E)}<+\infty . \end{aligned}$$

Since

$$\begin{aligned} |x-y|\le \frac{3}{2}\, |x|,\quad x\in E{\setminus } B_R,\ y\in B_{R/2}, \end{aligned}$$

we get

$$\begin{aligned} \begin{aligned} \int _{E{\setminus } B_R} \int _{B_{R/2}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+\textit{sp}}}\,dy\,dx&\ge c\,R^N\,\int _{E{\setminus } B_R} \frac{|u|^p}{|x|^{N+\textit{sp}}}\,dx\\&\quad -c\, \left( \int _{E{\setminus } B_R} \frac{1}{|x|^{N+\textit{sp}}}\,dx\right) \, \int _{B_{R/2}} |u|^p\,dy. \end{aligned} \end{aligned}$$

In conclusion, the previous estimate proves

$$\begin{aligned} \int _{E{\setminus } B_R} \frac{|u|^p}{(1+|x|)^{N+\textit{sp}}}\,dx\le \frac{C}{R^N}\,[u]_{W^{s,p}(E)}^p+\frac{C}{R^{N+\textit{sp}}}\, \int _{B_{R/2}} |u|^p\,dx. \end{aligned}$$

Using (2.6) to estimate the $L^p$ norm in the right-hand side gives

$$\begin{aligned} \int _{E{\setminus } B_R} \frac{|u|^p}{(1+|x|)^{N+\textit{sp}}}\,dx\le \frac{C}{R^N}\,\,[u]_{W^{s,p}(E)}^p+\frac{C}{R^{N\frac{p}{p-1}+\textit{sp}}}\, \left( \int _{B_{R/2}} |u|^{p-1}\,dy\right) ^\frac{p}{p-1}, \end{aligned}$$

(2.11)

possibly for a different constant $C=C(N,s,p)>0$. By summing up (2.10) and (2.11) we get the conclusion.$\square$

The next proposition shows that in the space $\widetilde{D}^{s, p}(\Omega )$, the operator $(-\Delta _p)^s$ is well defined.

Proposition 2.5

For any $u\in \widetilde{D}^{s,p}(\Omega )$, the operator

$$\begin{aligned} D^{s,p}_0(\Omega )\ni \varphi \mapsto \langle (-\Delta _p)^s u, \varphi \rangle :=\int _{\mathbb {R}^N\times \mathbb {R}^N}\frac{J_p(u(x)-u(y))\,(\varphi (x)-\varphi (y))}{|x-y|^{N+\textit{sp}}}\, dx\, dy \end{aligned}$$

is well defined and belongs to the dual space $(D^{s, p}_0(\Omega ))^*$.

Proof

We proceed as in [19, Lemma 2.3]. Let $E\supset \Omega$ be such that $E^c$ is compact, $\mathrm{dist}(E^c, \Omega )>0$ and $[u]_{W^{s,p}(E)}<+\infty$. Since $\varphi \equiv 0$ in $\Omega ^c$, we split the integral as

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^N\times \mathbb {R}^N}&\frac{J_p(u(x)-u(y))\,(\varphi (x)-\varphi (y))}{|x-y|^{N+\textit{sp}}}\, dx\, dy\\&=\int _{E\times E}\frac{J_p(u(x)-u(y))\,(\varphi (x)-\varphi (y))}{|x-y|^{N+\textit{sp}}}+2\int _{\Omega \times E^c}\frac{J_p(u(x)-u(y))\,\varphi (x)}{|x-y|^{N+\textit{sp}}}\, dx\, dy. \end{aligned} \end{aligned}$$

By Hölder’s inequality the first term is finite and defines a continuous linear functional on $D^{s,p}_0(\Omega )$. Let us focus on the second one. By using that $\varphi \equiv 0$ in $E^c$, we need to show that

$$\begin{aligned} \varphi \mapsto \int _{\Omega }\varphi (x)\left( \int _{E^c} \frac{J_p(u(x)-u(y))}{|x-y|^{N+\textit{sp}}}\, dy\right) dx, \end{aligned}$$

is a continuous linear functional on $D^{s,p}_0(\Omega )$. By means of Hardy’s inequality (2.5), we get that convergence of $\{\varphi _n\}_{n\in \mathbb {N}}$ in $D^{s,p}_0(\Omega )\subset D^{s,p}(\mathbb {R}^N)$ implies strong convergence in $L^p(\Omega )$ of $\{|x|^{-s}\varphi _n\}_{n\in \mathbb {N}}$. Thus to prove the claim it suffices to show that

$$\begin{aligned} x\mapsto |x|^s\int _{E^c} \frac{J_p(u(x)-u(y))}{|x-y|^{N+\textit{sp}}}\, dy\in L^{p'}(\Omega ). \end{aligned}$$

Being $E^c$ compact and $\mathrm{dist}(E^c, \Omega )\ge \delta >0$ it holds

$$\begin{aligned} |x-y|\ge C\,(1+|x|), \quad \hbox { for every } x\in \Omega ,\ y\in E^c, \end{aligned}$$

(2.12)

for some $C=C(E,\Omega )>0$. Thus, for almost every $x\in \Omega$, we can estimate

$$\begin{aligned}&\left| \int _{E^c} \frac{|x|^s\,J_p(u(x)-u(y))}{|x-y|^{N+\textit{sp}}}\, dy\right| \\&\quad \le C\left[ |E^c|\,\frac{|u(x)|^{p-1}}{(1+|x|)^{\frac{N+\textit{sp}}{p'}}}+\frac{1}{(1+|x|)^{N+s\,(p-1)}}\,\int _{E^c}|u|^{p-1}\, dy\right] . \end{aligned}$$

The first term belongs to $L^{p'}(\Omega )$ due to (2.9). For the second one this follows from a direct computation. This proves the claim and the proposition.$\square$

Definition 2.6

Let $u\in \widetilde{D}^{s,p}(\Omega )$ and $\Lambda \in (D^{s,p}_0(\Omega ))^*$. We say that $(-\Delta _p)^s u\le \Lambda$ weakly in $\Omega$ if for all $\varphi \in D^{s,p}_0(\Omega )$, $\varphi \ge 0$ in $\Omega$,

$$\begin{aligned} \int _{\mathbb {R}^{2N}}\frac{J_p(u(x)-u(y))\,(\varphi (x)-\varphi (y))}{|x-y|^{N+\textit{sp}}}\,dx\,dy \le \langle \Lambda , \varphi \rangle . \end{aligned}$$

Theorem 2.7

(Comparison principle in general domains) Let $\Omega \subset \mathbb {R}^N$ be an open set. Let $u, v\in \widetilde{D}^{s,p}(\Omega )$ satisfy

$$\begin{aligned} u\le v\ \hbox { in }\Omega ^c\quad \hbox { and }\quad (-\Delta _p)^s u\le (-\Delta _p)^s v\ \hbox { in } \Omega . \end{aligned}$$

Then $u\le v$ in $\Omega$.

Proof

It suffices to proceed as in [26, Lemma 9], we only need to prove that $w:=(u-v)_+$ is an admissible test function, i.e. it belongs to $D^{s,p}_0(\Omega )$. Clearly $w\equiv 0$ in $\Omega ^c$ and $w\in L^{p^*}(\mathbb {R}^N)$. To estimate the Gagliardo seminorm, let $E\supset \Omega$ be such that $E^c$ is compact, $\mathrm{dist}(E^c, \Omega )>0$ and

$$\begin{aligned} \int _{E\times E} \frac{|u(x)-u(y)|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy+\int _{E\times E} \frac{|v(x)-v(y)|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy<+\infty . \end{aligned}$$

(2.13)

Then

$$\begin{aligned} \int _{\mathbb {R}^{2N}} \frac{|w(x)-w(y)|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy=\int _{E\times E} \frac{|w(x)-w(y)|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy+2\int _{\Omega \times E^c} \frac{|w(x)|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy, \end{aligned}$$

and the first integral is finite due to

$$\begin{aligned} |w(x)-w(y)|^p\le C\,(|u(x)-u(y)|^p+|v(x)-v(y)|^p), \end{aligned}$$

and (2.13). For the second one we use (2.12), and since $|w(x)|^p\le C(|u(x)|^p+|v(x)|^p)$ we get

$$\begin{aligned} \int _{\Omega \times E^c} \frac{|w(x)|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy\le C\,|E^c| \int _{\Omega } \frac{|u(x)|^p}{(1+|x|)^{N+\textit{sp}}}\, dx+ C\,|E^c|\int _{\Omega } \frac{|v(x)|^p}{(1+|x|)^{N+\textit{sp}}}\, dx. \end{aligned}$$

The last two terms are finite, due the definition of $\widetilde{D}^{s,p}(\Omega )$ and (2.9).$\square$

Finally, for the reader’s convenience we recall the following result from [19]. The proof is identical to the one of [19, Lemma 2.8] and we omit it.

Proposition 2.8

(Non-local behavior of $(-\Delta _p)^s$) Let $N>\textit{sp}$ and let $\Omega \subset \mathbb {R}^N$ be an open set such that $\partial \Omega$ is compact and locally the graph of continuous functions. Suppose that $u\in \widetilde{D}^{s,p}(\Omega )$ weakly solves $(-\Delta _p)^su=f$ for some $f\in L^1_\mathrm{loc}(\Omega )\cap (D^{s,p}_0(\Omega ))^*$, in the sense that

$$\begin{aligned} \langle (-\Delta _p)^s u, \varphi \rangle =\int _{\Omega }f\,\varphi \, dx,\quad \hbox { for every } \varphi \in D^{s,p}_0(\Omega ). \end{aligned}$$

(2.14)

Let v be a measurable function with compact support $K:=\mathrm {supp}(v)$ such that

$$\begin{aligned} \mathrm{dist}(K,\Omega )>0,\quad \int _{\Omega ^c} |v|^{p-1}\, dx<+\infty , \end{aligned}$$

and define for a.e. Lebesgue point $x\in \Omega$ of u

$$\begin{aligned} h(x)=2\int _{K}\frac{J_p\big (\big (u(x)-u(y)\big )-v(y)\big )-J_p\big (u(x)-u(y)\big )}{|x-y|^{N+\textit{sp}}}\,dy. \end{aligned}$$

Then $u+v\in \widetilde{D}^{s,p}(\Omega )$ and $(-\Delta _p)^s(u+v)=f+h$ weakly.

2.4 Radial functions

For every measurable function $u:\mathbb {R}^N\rightarrow \mathbb {R}$ we define its distribution function

$$\begin{aligned} \mu _u(t)=\big |\{x\, :\, |u(x)|>t\}\big |,\quad t>0. \end{aligned}$$

Let $0< q<\infty$ and $0< \theta < \infty$, the Lorentz space $L^{q,\theta }(\mathbb {R}^N)$ is defined by

$$\begin{aligned} L^{q,\theta }(\mathbb {R}^N)=\left\{ u\, :\, \int _0^\infty t^{\theta -1}\,\mu _u(t)^\frac{\theta }{q}\,dt<+\infty \right\} . \end{aligned}$$

In the limit case $\theta =\infty$, this is defined by

$$\begin{aligned} L^{q,\infty }(\mathbb {R}^N)=\left\{ u\, :\, \sup _{t>0} t\,\mu _u(t)^\frac{1}{q}<+\infty \right\} , \end{aligned}$$

and we recall that this coincides with the weak $L^q$ space (see for example [24, page 106]).

Lemma 2.9

(Radial Lemma for Lorentz spaces) Let $0< \theta \le \infty$ and $0< q< \infty$. Let $u\in L^{q,\theta }(\mathbb {R}^N)$ be a non-negative and radially symmetric decreasing function. Then

$$\begin{aligned} \begin{aligned} 0\le u(x)&\le \left( \theta \,\omega _N^{-\frac{\theta }{q}}\,\int _0^\infty t^{\theta -1}\,\mu _u(t)^\frac{\theta }{q}\,dt\right) ^\frac{1}{\theta }\, |x|^{-\frac{N}{q}},\quad \hbox { if }\; \theta <\infty ,\\ 0\le u(x)&\le \left( \omega _N^{-\frac{1}{q}}\,\sup _{t>0} t\,\mu _u(t)^\frac{1}{q}\right) \,|x|^{-\frac{N}{q}},\quad \hbox { if }\; \theta =\infty . \end{aligned} \end{aligned}$$

Proof

For simplicity, we simply write $\mu$ in place of $\mu _u$ and suppose that $u=u(r)$ coincides with its right-continuous representative. We start with the case $\theta <\infty$. First of all, we prove that

$$\begin{aligned} \int _0^\infty t^{\theta -1}\,\mu (t)^\frac{\theta }{q}\,dt=\frac{N-\alpha }{N\,\theta \,\omega _N^{\alpha /N}}\,\int _{\mathbb {R}^N} \frac{u^\theta }{|x|^\alpha }\,dx, \end{aligned}$$

(2.15)

where the exponent $\alpha <N$ is given by the relation¹

$$\begin{aligned} \frac{\theta }{q}=\frac{N-\alpha }{N}. \end{aligned}$$

With a simple change of variable

$$\begin{aligned} \int _0^\infty t^{\theta -1}\,\mu (t)^\frac{\theta }{q}\, dt=\frac{1}{\theta }\, \int _0^\infty \mu (s^{1/\theta })^\frac{\theta }{q}\, ds. \end{aligned}$$

(2.16)

Then we observe that

$$\begin{aligned} \int _{\mathbb {R}^N} \frac{u^\theta }{|x|^\alpha }\, dx=\int _{\mathbb {R}^N} \frac{\displaystyle \int _0^\infty \chi _{\{t\,:\,u(x)^\theta >t\}}(s)\, ds}{|x|^\alpha }\, dx=\int _0^\infty \int _{\mathbb {R}^N} \frac{\chi _{\{t\,:\,u(x)>t^{1/\theta }\}}(s)}{|x|^\alpha }\, dx\, ds, \end{aligned}$$

and

$$\begin{aligned} \chi _{\{t\,:\,u(x)>t^{1/\theta }\}}(s)=\chi _{\{y\, :\, u(y)>s^{1/\theta }\}}(x). \end{aligned}$$

By assumption we have

$$\begin{aligned} \{y\, :\, u(y)>s^{1/\theta }\}=\left\{ y\, :\, |y|<\left( \frac{\mu (s^{1/\theta })}{\omega _N}\right) ^\frac{1}{N}\right\} =:B_{R(s)}, \end{aligned}$$

since the function u is radially decreasing. Thus we arrive at

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^N} \frac{u^\theta }{|x|^\alpha }\, dx&=\int _0^\infty \left( \int _{B_{R(s)}} \frac{1}{|x|^\alpha }\, dx\right) \, ds\\&=N\,\omega _N\, \int _0^\infty \int _0^{R(s)} \varrho ^{N-1-\alpha }\,d\varrho \, ds=\frac{N\,\omega _N}{N-\alpha }\, \int _0^\infty \frac{\mu (s^{1/\theta })^\frac{N-\alpha }{N}}{\omega _N^\frac{N-\alpha }{N}}\, ds\\&=\frac{N\,\omega _N^{\alpha /N}}{N-\alpha }\, \int _0^\infty \mu (s^{1/\theta })^\frac{\theta }{q}\, ds. \end{aligned} \end{aligned}$$

Using (2.16) we finally obtain

$$\begin{aligned} \int _0^\infty t^{\theta -1}\,\mu (t)^\frac{\theta }{q}\, dt=\frac{N-\alpha }{N\,\theta \,\omega _N^{\alpha /N}}\, \int _{\mathbb {R}^N} \frac{u^\theta }{|x|^\alpha }\, dx, \end{aligned}$$

which proves (2.15).

As for the decay estimate, thanks to (2.15) we have

$$\begin{aligned} \begin{aligned} +\infty >\int _{\mathbb {R}^N} \frac{u^\theta }{|x|^\alpha }\,dx&=N\,\omega _N\, \int _0^{+\infty } u(\varrho )^\theta \,\varrho ^{N-1-\alpha }\, d\varrho \\&\ge N\,\omega _N\, \int _0^{R} u(\varrho )^\theta \, \varrho ^{N-1-\alpha }\,d\varrho \ge N\,\omega _N\, u(R)^\theta \, \frac{R^{N-\alpha }}{N-\alpha }, \end{aligned} \end{aligned}$$

where $\alpha$ is as above. Recalling that $(N-\alpha )/\theta =N/q$, we get the desired conclusion.

For the case $\theta =\infty$, it is sufficient to observe that

$$\begin{aligned} \sup _{t>0} t\,\mu (t)^\frac{1}{q}=\omega _N^\frac{1}{q}\,\sup _{x\in \mathbb {R}^N} |x|^\frac{N}{q}\,u(x). \end{aligned}$$

Then the decay estimate easily follows.$\square$

3 Properties of extremals

3.1 Basic properties

We first observe that by homogeneity we can equivalently write

$$\begin{aligned} \mathcal {S}_{p,s}=\inf _{u\in D^{s,p}(\mathbb {R}^N)} \left\{ \int _{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+\textit{sp}}}\,dx\,dy\, :\, \int _{\mathbb {R}^N} |u|^{{Np/(N-\textit{sp})}}\,dx=1\right\} . \end{aligned}$$

(3.1)

Then we start with the following result.

Proposition 3.1

Let $1<p<\infty$ and $s\in (0,1)$ be such that $\textit{sp}<N$. Then:

• problem (3.1) admits a solution;

• for every $U\in D^{s,p}(\mathbb {R}^N)$ solving (3.1), there exist $x_0\in \mathbb {R}^N$ and $u:\mathbb {R}^+\rightarrow \mathbb {R}$ constant sign monotone function such that $U(x)=u(|x-x_0|)$;

• every minimizer $U\in D^{s,p}(\mathbb {R}^N)$ weakly solves

  $$\begin{aligned} (-\Delta _p)^s U=\mathcal {S}_{p,s}\,|U|^{p^*-2}\,U,\quad \hbox { in }\mathbb {R}^N, \end{aligned}$$

  that is

  $$\begin{aligned} \int _{\mathbb {R}^{2N}} \frac{J_p(U(x)-U(y))\, \big (\varphi (x)-\varphi (y)\big )}{|x-y|^{N+\textit{sp}}}\,dx\,dy=\mathcal {S}_{p,s}\, \int _{\mathbb {R}^N} |U|^{p^*-2}\,U\,\varphi \,dx, \end{aligned}$$

  (3.2)

  for every $\varphi \in D^{s,p}(\mathbb {R}^N)$.

Proof

The existence of a solution for (3.1) follows from the Concentration-Compactness Principle, see [25, Section I.4, Example  iii)].

It is not difficult to show that every solution of (3.1) must have costant sign. Indeed, for every admissible $u\in D^{s,p}(\mathbb {R}^N)$, still by (2.8) the function |u| is still admissible and does not increase the value of the functional. More important, the inequality sign in (2.8) is strict if $u(x)\,u(y)<0$, i.e. if u changes sign.

Radial symmetry of the solutions comes from the Pólya-Szegő principle for Gagliardo seminorms (see [1]), i.e. for every non-negative function $u\in D^{s,p}(\mathbb {R}^N)$ we have

$$\begin{aligned}{}[u^\#]^p_{s,p}\le [u]^p_{s,p}. \end{aligned}$$

(3.3)

Here $u^\#$ denotes the radially symmetric decreasing rearrangement of u. It is crucial to observe that inequality (3.3) is strict, unless u is (up to a translation) a radially symmetric decreasing function, see [15, Theorem A.1].

Finally, if U solves (3.1), then it minimizes as well the functional

$$\begin{aligned} u\mapsto [u]^p_{s,p}-\mathcal {S}_{p,s}\,\left( \int _{\mathbb {R}^N} |u|^{p^*}\,dx\right) ^\frac{p}{p^*}. \end{aligned}$$

Equation (3.2) is exactly the Euler–Lagrange equation associated with this functional, once it is observed that U has unitary $L^{p^*}$ norm.

Proposition 3.2

(Global boundedness) Let $U\in D^{s,p}(\mathbb {R}^N)$ be a non-negative solution of (3.1). Then we have $U\in L^\infty (\mathbb {R}^N)\cap C^0(\mathbb {R}^N)$.

Proof

Thanks to the properties of the minimizers contained in Proposition 3.1, it is enough to prove that $U\in L^\infty _{\mathrm {\mathrm {loc}}}(\mathbb {R}^N)$, since continuity then follows from [5, Theorem 3.13] (see also [19, Theorem  5.4] for a direct proof). With this aim, we just need to show that $U\in L^{q\,(p^*-1)}(\mathbb {R}^N)$ for some $q>N/(\textit{sp})$. This would imply that

$$\begin{aligned} U^{p^*-1}\in L^q(\mathbb {R}^N),\quad \hbox { for some }q>\frac{N}{\textit{sp}}, \end{aligned}$$

and thus $U\in L^\infty _{\mathrm {loc}}(\mathbb {R}^N)$ would automatically follow by [5, Theorem 3.8].

Let $M>0$ and $\alpha >1$, we set for simplicity $U_M=\min \{U,M\}$ and $g_{\alpha ,M}(t)=t\,\min \{t,\,M\}^{\alpha -1}$. Then we insert in (3.2) the test function $\varphi =g_{\alpha ,M}(U)\in D^{s,p}(\mathbb {R}^N)$. This yields

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2N}}&\frac{J_p(U(x)-U(y))\, \big (g_{\alpha ,M}(U(x))-g_{\alpha ,M}(U(y))\big )}{|x-y|^{N+\textit{sp}}}\,dx\,dy=\mathcal {S}_{p,s}\, \int _{\mathbb {R}^N} U^{p^*-p}\,U_{M}^{\alpha -1}\,U^p\,dx.\\ \end{aligned} \end{aligned}$$

We now observe that if we set

$$\begin{aligned} G_{\alpha ,M}(t)=\int _0^t g'_{\alpha ,M}(\tau )^\frac{1}{p}\,d\tau , \end{aligned}$$

by using [5, Lemma A.2] from the previous identity with simple manipulations we get

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2N}}&\frac{|G_{\alpha ,M}(U(x))-G_{\alpha ,M}(U(y))|^p}{|x-y|^{N+\textit{sp}}}\,dx\,dy\\&\le \mathcal {S}_{p,s}\, \left[ K_0^{\alpha -1}\,\int _{\mathbb {R}^N} U^{p^*}\,dx+\left( \int _{\{U\ge K_0\}} U^{p^*}\,dx\right) ^\frac{p^*-p}{p^*}\,\left( \int _{\mathbb {R}^N} \left( U_{M}^{(\alpha -1)}\,U^{p}\right) ^\frac{p^*}{p}\,dx\right) ^\frac{p}{p^*}\right] , \end{aligned} \end{aligned}$$

for some $K_0>0$ that will be chosen in a while. If we estimate from below the left-hand side by Sobolev inequality and use that U has unitary norm, we get²

$$\begin{aligned}&\left( \frac{p}{p+\alpha -1}\right) ^p\,\left( \int _{\mathbb {R}^N} \right. \left. \left( U^p\,U_M^{(\alpha -1)}\right) ^\frac{p^*}{p}\,dx\right) ^\frac{p}{p^*}\, \nonumber \\&\quad \le K_0^{\alpha -1} +\left( \int _{\{U\ge K_0\}} U^{p^*}\,dx\right) ^\frac{p^*-p}{p^*}\,\left( \int _{\mathbb {R}^N} \left( U_M^{(\alpha -1)}\,U^{p}\right) ^\frac{p^*}{p}\,dx\right) ^\frac{p}{p^*}. \end{aligned}$$

(3.4)

We now choose the parameters: we first take $\alpha >1$ such that

$$\begin{aligned} p^*+(\alpha -1)\,\frac{p^*}{p}=q\,(p^*-1),\quad \hbox { i. e. } \quad \alpha =p\,q\,\frac{(p^*-1)}{p^*}-(p-1), \end{aligned}$$

where $q>N/(\textit{sp})$, then we choose $K_0=K_0(\alpha ,U)>0$ such that

$$\begin{aligned} \left( \int _{\{U\ge K_0\}} U^{p^*}\,dx\right) ^\frac{p^*-p}{p^*}\le \frac{1}{2}\,\left( \frac{p}{p+\alpha -1}\right) ^p. \end{aligned}$$

With this choice we can absorb the last term on the right-hand side of (3.4) and thus obtain

$$\begin{aligned} \left( \frac{p}{p+\alpha -1}\right) ^p\, \left( \int _{\mathbb {R}^N} U^{p^*}\,U_M^{(\alpha -1)\,\frac{p^*}{p}}\,dx\right) ^\frac{p}{p^*}\le 2\,K_0^{\alpha -1}. \end{aligned}$$

If we now take the limit as M goes to $+\infty$, we finally get that $U\in L^{q\,(p^*-1)}(\mathbb {R}^N)$ for some $q>N/(\textit{sp})$, together with the estimate

$$\begin{aligned} \left\| U^{p^*-1}\right\| ^q_{q}\le \left( 2\,K_0^{\alpha -1}\,\left( \frac{p+\alpha -1}{p}\right) ^p\right) ^\frac{p^*}{p}, \end{aligned}$$

and thus the conclusion.$\square$

Proposition 3.3

(Borderline Lorentz estimate) Let $U\in D^{s,p}(\mathbb {R}^N)$ be a non-negative solution of (3.1). Then

$$\begin{aligned} U\in L^q(\mathbb {R}^N),\quad \hbox { for every } q>q_0:=\frac{(p-1)\,N}{N-\textit{sp}}. \end{aligned}$$

(3.5)

Moreover, we have $U\in L^{q_0,\infty }(\mathbb {R}^N)$ with the estimate

$$\begin{aligned} \sup _{t>0}t\,|\{U> t\}|^\frac{1}{q_0}\le \Vert U\Vert _{p^*-1}^{\frac{p^*-1}{p-1}}. \end{aligned}$$

(3.6)

Proof

We divide the proof in two parts: we first prove (3.5). Then we will use (3.5) to prove (3.6).

Part I: intermediate estimate. Given $0<\alpha <1$ and $\varepsilon >0$, we take the Lipschitz increasing function $\psi _\varepsilon :[0,+\infty )\rightarrow [0,+\infty )$ defined as

$$\begin{aligned} \psi _\varepsilon (t)=\int _0^t \left[ (\varepsilon +\tau )^\frac{\alpha -1}{p}+\frac{\alpha -1}{p}\,\tau \,(\varepsilon +\tau )^\frac{\alpha -1-p}{p}\right] ^p\,d\,\tau . \end{aligned}$$

We observe that

$$\begin{aligned} 0\le \psi _\varepsilon (t)\le \int _0^t (\varepsilon +t)^{\alpha -1}\,d\tau =\frac{1}{\alpha }\,[(\varepsilon +t)^\alpha -\varepsilon ^\alpha ]\le \frac{t^\alpha }{\alpha }, \end{aligned}$$

(3.7)

where in the second inequality we used that $0<\alpha <1$. We insert in (3.2) the test function $\varphi =\psi _\varepsilon (U)\in D^{s,p}(\mathbb {R}^N)$. This gives

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2N}}&\frac{J_p(U(x)-U(y))\, \big (\psi _{\varepsilon }(U(x))-\psi _\varepsilon (U(y)\big )}{|x-y|^{N+\textit{sp}}}\,dx\,dy=\mathcal {S}_{p,s}\, \int _{\mathbb {R}^N} U^{p^*-1}\,\psi _\varepsilon (U)\,dx.\\ \end{aligned} \end{aligned}$$

By defining

$$\begin{aligned} \Psi _\varepsilon (t):=\int _0^t \psi '_\varepsilon (\tau )^\frac{1}{p}\,d\tau =t\,(\varepsilon +t)^\frac{\alpha -1}{p}, \end{aligned}$$

if we proceed as in the previous proof and use (3.7), we get

$$\begin{aligned} \begin{aligned} \left( \int _{\mathbb {R}^N} \Psi _\varepsilon (U)^{p^*}\,dx\right) ^\frac{p}{p^*}&\le \frac{1}{\alpha }\,\Vert U\Vert ^{p^*+\alpha -1}_{\infty }\,|\{U>K_0\}|\\&\quad +\left( \int _{\{U\le K_0\}} U^{p^*}\,dx\right) ^\frac{p^*-p}{p^*}\left( \int _{\{U\le K_0\}} \Big (\psi _\varepsilon (U)\,U^{p-1}\Big )^\frac{p^*}{p}\,dx\right) ^\frac{p}{p^*}, \end{aligned} \end{aligned}$$

for $K_0>0$. Observe that we also used the previous Proposition to assure that $U\in L^\infty (\mathbb {R}^N)$. From (3.7) we get

$$\begin{aligned} 0\le \psi _\varepsilon (t)\,t^{p-1}\le \frac{1}{\alpha }\,[(\varepsilon +t)^\alpha -\varepsilon ^\alpha ]\,t^{p-1} \le \quad \frac{1}{\alpha }\,(\varepsilon +t)^{\alpha -1}\,t^{p}=\frac{1}{\alpha }\, \Psi _\varepsilon (t)^p. \end{aligned}$$

Thus we arrive at

$$\begin{aligned} \begin{aligned} \left( \int _{\mathbb {R}^N} \Psi _\varepsilon (U)^{p^*}\,dx\right) ^\frac{p}{p^*}&\le \frac{1}{\alpha }\,\Vert U\Vert ^{p^*+\alpha -1}_{\infty }\,|\{U>K_0\}|\\&\quad +\frac{1}{\alpha }\,\left( \int _{\{U\le K_0\}} U^{p^*}\,dx\right) ^\frac{p^*-p}{p^*}\,\left( \int _{\mathbb {R}^N} \Psi _\varepsilon (U)^{p^*}\,dx\right) ^\frac{p}{p^*}, \end{aligned} \end{aligned}$$

The level $K_0=K_0(\alpha ,U)>0$ is now chosen so that

$$\begin{aligned} \left( \int _{\{U\le K_0\}} U^{p^*}\,dx\right) ^\frac{p^*-p}{p^*}\le \frac{\alpha }{2}, \end{aligned}$$

which yields

$$\begin{aligned} \left( \int _{\mathbb {R}^N} \left( U\,(U+\varepsilon )^\frac{\alpha -1}{p}\right) ^{p^*}\,dx\right) ^\frac{p}{p^*}\le \frac{2}{\alpha }\,\Vert U\Vert ^{p^*+\alpha -1}_{\infty }\,|\{U>K_0\}|, \end{aligned}$$

for every $0<\alpha <1$. By taking the limit as $\varepsilon$ goes to 0, we get the desired integrability (3.5).

Part II: borderline Lorentz estimate. We now prove (3.6). For any $t>0$ we let $g_t(s)=\min \{t,\,s\}$, and define

$$\begin{aligned} G_t(s)=\int _0^sg'_t(\tau )^{\frac{1}{p}}d\tau =g_t(s). \end{aligned}$$

We test (3.2) with $g_t(U)$ and, thanks to [5, Lemma  A.2] and Sobolev inequality we get

$$\begin{aligned} \begin{aligned} \mathcal {S}_{p,s}\,\Vert g_t(U)\Vert _{p^*}^p&\le [g_t(U)]_{s, p}^p\le \int _{\mathbb {R}^{2N}} \frac{J_p(U(x)-U(y))\, \big (g_t(U(x))-g_t(U(y))\big )}{|x-y|^{N+\textit{sp}}}\,dx\,dy\\&\le \mathcal {S}_{p,s}\,\int _{\mathbb {R}^N} U^{p^*-1}\,g_t(U)\,dx. \end{aligned} \end{aligned}$$

We have $U\in L^{p^*-1}(\mathbb {R}^N)$, by choosing $q=p^*-1$ in (3.5). Thus we conclude that

$$\begin{aligned} t\,|\{U>t\}|^{\frac{1}{p^*}}\le \Vert g_t(U)\Vert _{p^*}\le \left( \int _{\mathbb {R}^N} U^{p^*-1}g_t(U)\,dx\right) ^\frac{1}{p}\le t^\frac{1}{p}\,\Vert U\Vert _{p^*-1}^\frac{p^*-1}{p}. \end{aligned}$$

This finally yields (3.6), after some elementary manipulations.$\square$

3.2 Decay estimates

As an intermediate step towards the proof of the asymptotic result (1.11), in this subsection we will prove that any (positive) solution of (3.1) verifies

$$\begin{aligned} \frac{1}{C}\,|x|^{-\frac{N-\textit{sp}}{p-1}}\le U(x)\le C\,|x|^{-\frac{N-\textit{sp}}{p-1}},\quad |x|> 1, \end{aligned}$$

for some $C=C(N,p,s,U)>1$, see Corollary 3.7 below.

In what follows, we will set for simplicity

$$\begin{aligned} \Gamma (x)=|x|^{-\frac{N-\textit{sp}}{p-1}},\quad x\in \mathbb {R}^N{\setminus }\{0\}, \end{aligned}$$

and

$$\begin{aligned} \widetilde{\Gamma }(x)=\min \{1,\,\Gamma (x)\}=\min \left\{ 1,\, |x|^{-\frac{N-\textit{sp}}{p-1}}\right\} ,\quad x\in \mathbb {R}^N. \end{aligned}$$

(3.8)

The following expedient result will be useful.

Lemma 3.4

With the notation above, we have

$$\begin{aligned} \frac{1}{C}\,\,|x|^{-N-\textit{sp}}\le (-\Delta _p)^s \widetilde{\Gamma }(x)\le C\,|x|^{-N-\textit{sp}},\quad \hbox { for }|x|>R>1, \end{aligned}$$

(3.9)

in weak sense, for some $C=C(N,p,s,R)>1$. The constant blows-up as R goes to 1.

Proof

From Theorem A.4, we know that $\Gamma$ belongs to $\widetilde{D}^{s,p}(B_R^c)$ and is a weak solution of $(-\Delta _p)^su=0$ in $B_R^c$ for any $R>1$. We then observe that the truncated function $\widetilde{\Gamma }$ can be written as

$$\begin{aligned} \widetilde{\Gamma }(x)=\Gamma (x)-\big (\Gamma (x)-1\big )_+. \end{aligned}$$

Thus we apply Proposition 2.8, with the choices

$$\begin{aligned} \Omega =B_R^c,\quad u=\Gamma ,\quad f\equiv 0,\quad v=-(\Gamma -1)_+, \end{aligned}$$

This yields for $|x|>R$

$$\begin{aligned} \begin{aligned} (-\Delta _p)^s \widetilde{\Gamma }(x)&=2\,\int _{B_1}\frac{J_p(\Gamma (x)-1)-J_p(\Gamma (x)-\Gamma (y))}{|x-y|^{N+\textit{sp}}}\, dy\\&=2\,\int _{B_1}\frac{J_p(\Gamma (y)-\Gamma (x))-J_p(1-\Gamma (x))}{|x-y|^{N+\textit{sp}}}\, dy. \end{aligned} \end{aligned}$$

(3.10)

We first prove the upper bound in (3.9). To this aim, by the monotonicity of $\Gamma$ we get

$$\begin{aligned} (\Gamma (y)-\Gamma (x))^{p-1}-(1-\Gamma (x))^{p-1}\le (\Gamma (y)-\Gamma (x))^{p-1}\le \Gamma (y)^{p-1},\quad |x|> R, \ |y|\le 1. \end{aligned}$$

Moreover

$$\begin{aligned} |x-y|\ge \frac{R-1}{R}\,|x|,\quad \hbox { for all } |x|>R \hbox { and } |y|<1. \end{aligned}$$

By spending these informations in (3.10), we obtain

$$\begin{aligned} (-\Delta _p)^s \widetilde{\Gamma }(x)\le \left( \frac{R}{R-1}\right) ^{N+\textit{sp}}\,\frac{2}{|x|^{N+\textit{sp}}}\int _{B_1} \Gamma (y)^{p-1}\, dy=\frac{C}{|x|^{N+\textit{sp}}}, \end{aligned}$$

as desired. Observe that we also used that $\Gamma \in L^{p-1}_{\mathrm {loc}}(\mathbb {R}^N)$.

In order to prove the lower bound, we need to distinguish between the case $1<p<2$ and the case $p\ge 2$. If $p\ge 2$, then $J_p(t)=|t|^{p-2}\,t$ is a convex superadditive function on $[0,\infty )$. Thus we get

$$\begin{aligned} J_p(\Gamma (y)-\Gamma (x))-J_p(1-\Gamma (x))\ge J_p(\Gamma (y)-1),\quad |x|>R>1>|y|. \end{aligned}$$

As for the kernel, we have

$$\begin{aligned} |x-y|<2\,|x|,\quad \hbox { if }|x|>|y|, \end{aligned}$$

(3.11)

thus in conclusion from (3.10) we get

$$\begin{aligned} (-\Delta _p)^s \widetilde{\Gamma }(x)\ge \frac{2^{1-N-\textit{sp}}}{|x|^{N+\textit{sp}}}\int _{B_1} \big (\Gamma (y)-1\big )^{p-1}\, dy=\frac{C}{|x|^{N+\textit{sp}}}. \end{aligned}$$

Using again that $\Gamma \in L^{p-1}_{\mathrm {loc}}(\mathbb {R}^N)$ and that $\Gamma >1$ in $B_1$ gives the lower bound in (3.9), in the case $p\ge 2$.

In the case $1<p<2$, we need to use (2.3), which gives

$$\begin{aligned} \begin{aligned} J_p(\Gamma (y)-\Gamma (x))-J_p(1-\Gamma (x))&\ge C\,\frac{(\Gamma (y)-1)}{\Big ((\Gamma (y)-\Gamma (x))^2+(1-\Gamma (x))^2\Big )^\frac{2-p}{2}}\\&\ge \frac{C}{2^\frac{2-p}{2}}\,\frac{(\Gamma (y)-1)}{(\Gamma (y)-\Gamma (x))^{2-p}}\ge C\,\Gamma (y)^{p-1}\, \left( 1-\frac{1}{\Gamma (y)}\right) . \end{aligned} \end{aligned}$$

By using this and (3.11) in (3.10), we get the desired lower bound for $1<p<2$ as well.   $\square$

In order to prove a lower bound for positive radially decreasing solutions of (3.1), we need to focus on the auxiliary problem

$$\begin{aligned} \mathcal {I}(R)=\inf _{u\in D^{s,p}(\mathbb {R}^N)}\Big \{[u]_{s, p}^p: u\ge \chi _{B_R}\Big \}. \end{aligned}$$

(3.12)

Proposition 3.5

Let $1<p<\infty$ and $s\in (0,1)$ be such that $\textit{sp}<N$. For any $R>0$, problem (3.12) has a unique solution $u_R>0$. Moreover, $u_R$ is radial, non-increasing and $u_R\in D^{s,p}(\mathbb {R}^N)$ solves in weak sense

$$\begin{aligned} \left\{ \begin{array}{rcll} (-\Delta _p)^s u_R&{}=&{}0&{}\quad \text { in } \overline{B_R}^{\,c},\\ u_R&{}\equiv &{} 1&{}\quad \hbox { in } \overline{B_R}. \end{array} \right. \end{aligned}$$

Proof

The existence of a solution follows easily by using the Direct Methods. Indeed, if $\{u_n\}_{n\in \mathbb {N}}\subset D^{s,p}(\mathbb {R}^N)$ is a minimizing sequence, then a uniform bound on their Gagliardo seminorms entails a uniform bound on the $L^{p^*}$ norms, by Sobolev inequality. Thus we have weak convergence (up to a subsequence) in $L^{p^*}(\mathbb {R}^N)$ to a function $u\in D^{s,p}(\mathbb {R}^N)$. Moreover, the constraint $u_n\ge \chi _{B_R}$ is stable with respect to weak convergence and thus it passes to the limit. Consequently, u is a minimizer. The uniqueness follows from strict convexity of the functional.

All the other required properties of $u_R$ follow as in the proof of Proposition 3.1, we just show that $u_R$ saturates the constraint $u_R\ge \chi _{B_R}$. For simplicity, we set $\mathcal {E}(u)=[u]_{s,p}^p$. Then from [16, Remark 3.3] we have

$$\begin{aligned} \mathcal {E}(\max \{u,\, t\})+\mathcal {E}(\min \{u,\, t\})\le \mathcal {E}(u),\quad \hbox { for every } u\in D^{s,p}(\mathbb {R}^N),\ t\in \mathbb {R}. \end{aligned}$$

(3.13)

In particular, $\min \{u_R,1\}$ is admissible and is still a minimizer. Thus by uniqueness it coincides with $u_R$.$\square$

Thanks to Lemma 3.4, we can prove a decay estimate for the solution of (3.12).

Proposition 3.6

The solution $u_1$ of problem (3.12) with $R=1$ satisfies

$$\begin{aligned} \frac{|x|^{-\frac{N-\textit{sp}}{p-1}}}{C}\le u_1(x)\le p^\frac{1}{p-1}\,|x|^{-\frac{N-\textit{sp}}{p-1}},\quad \hbox { for } |x|\ge 1, \end{aligned}$$

for some constant $C=C(N,p,s)>1$.

Proof

Observe that $u_1$ is continuous due to [19, Theorem 1.1]. We prove the two estimates separately.

Upper bound. We first observe that by using the scaling properties of the Gagliardo seminorm, we have

$$\begin{aligned} \mathcal {I}(R)=R^{N-\textit{sp}}\,\mathcal {I}(1). \end{aligned}$$

(3.14)

For every $R>1$, we set $u_1(R)=t \in (0,1)$. As in the previous proof, we set $\mathcal {E}(u)=[u]_{s,p}^p$. The function $\min \{u_1,\, t\}/t$ is admissible for problem (3.12) with $B_R$, then the minimality of $u_R$ gives

$$\begin{aligned} \mathcal {E}\left( \frac{\min \{u_1,\,t\}}{t}\right) \ge \mathcal {E}(u_R)=\mathcal {I}(R)=R^{N-\textit{sp}}\,\mathcal {I}(1), \end{aligned}$$

thanks to (3.14). Similarly, we get

$$\begin{aligned} \mathcal {E}\left( \frac{\max \{u_1-t,\, 0\}}{1-t}\right) \ge \mathcal {E}(u_1)=\mathcal {I}(1). \end{aligned}$$

then using the p-homogeneity of the energy and summing the previous two inequalities

$$\begin{aligned} \mathcal {E}(\min \{u_1,\, t\})+\mathcal {E}(\max \{u_1,\, t\})\ge \big (t^p\,R^{N-\textit{sp}}+(1-t)^p\big )\,\mathcal {I}(1). \end{aligned}$$

Using the submodularity of Gagliardo seminorms (3.13) in the left-hand side and simplifying we get

$$\begin{aligned} t^p\,R^{N-\textit{sp}}\le 1-(1-t)^p. \end{aligned}$$

By recalling the definition of t, we obtain

$$\begin{aligned} u_1(R)^p\, R^{N-s\, p}\le 1-(1-u_1(R))^p \end{aligned}$$

(3.15)

and since $1-(1-u_1(R))^p\le p\, u_1(R)$ we get

$$\begin{aligned} u_1(R)\le p^{\frac{1}{p-1}}\, R^{-\frac{N-\textit{sp}}{p-1}}. \end{aligned}$$

Lower bound. By using Proposition 2.8 with

$$\begin{aligned} \Omega =\overline{B_3}^{\,c},\quad u=u_1,\quad f\equiv 0,\quad v=-(u_1-u_1(2))_+, \end{aligned}$$

the truncated function

$$\begin{aligned} u=\min \{u_1,\,u_1(2)\}=u_1-(u_1-u_1(2))_+, \end{aligned}$$

satisfies weakly in $\overline{B_3}^{\,c}$

$$\begin{aligned} \begin{aligned} (-\Delta _p)^s u(x)&=2\int _{B_2}\frac{J_p(u_1(x)-u_1(2))-J_p(u_1(x)-u_1(y))}{|x-y|^{N+\textit{sp}}}\, dy\\&\ge 2\int _{B_1}\frac{J_p(u_1(y)-u_1(x))-J_p(u_1(2)-u_1(x))}{|x-y|^{N+\textit{sp}}}\, dy. \end{aligned} \end{aligned}$$

In the last passage we used that the integrand is nonnegative by the monotonicity of $u_1$. Recall that $u_1\equiv 1$ in $B_1$ and by (3.15) we have $u_1(2)<1=u_1(1)$. Then, it is readily checked that

$$\begin{aligned} (u_1(1)-u_1(x))^{p-1}-(u_1(2)-u_1(x))^{p-1}\ge c,\quad \hbox { for }|x|>3, \end{aligned}$$

for some constant $c=c(p,u_1(1)-u_1(2))>0$. Since also $|x-y|\le 2\,|x|$ for all $x\in B_2^c$ and $y\in B_1$, the previous discussion yields

$$\begin{aligned} (-\Delta _p)^s u(x)\ge \frac{2\,c\,|B_1|}{(2\,|x|)^{N+\textit{sp}}}=:\frac{c_1}{|x|^{N+\textit{sp}}},\quad \hbox { for }|x|>3. \end{aligned}$$

(3.16)

On the other hand, from Lemma 3.4, for every $\varepsilon >0$ we have

$$\begin{aligned} (-\Delta _p)^s (\varepsilon \,\widetilde{\Gamma }(x))\le \frac{c_2}{|x|^{N+\textit{sp}}}\,\varepsilon ^{p-1},\quad \hbox { for } |x|>3, \end{aligned}$$

(3.17)

where $\widetilde{\Gamma }$ is given in (3.8). Now choose $\varepsilon >0$ as follows

$$\begin{aligned} \varepsilon =\min \left\{ u_1(3)\,3^\frac{N-\textit{sp}}{p-1},\, \left( \frac{c_1}{c_2}\right) ^\frac{1}{p-1}\right\} , \end{aligned}$$

so that by (3.16) and (3.17) it holds

$$\begin{aligned} \left\{ \begin{array}{rcll} (-\Delta _p)^s (\varepsilon \,\widetilde{\Gamma }) &{}\le &{} (-\Delta _p)^s u&{}\quad \text { in } \overline{B_3}^{\,c},\\ \varepsilon \,\widetilde{\Gamma }&{}\le &{} u &{}\quad \text { in } \overline{B_3}. \end{array} \right. \end{aligned}$$

Therefore by Theorem 2.7 and the definitions of $\widetilde{\Gamma }$ and u we have

$$\begin{aligned} \varepsilon \,|x|^{-\frac{N-\textit{sp}}{p-1}}\le u(x)=u_1(x),\quad \hbox { for }|x|>3. \end{aligned}$$

In $\overline{B_3}{\setminus } B_1$ the estimate is simpler to obtain, indeed

$$\begin{aligned} u_1(3)\,|x|^{-\frac{N-\textit{sp}}{p-1}}\le u_1(3)\le u_1(x), \end{aligned}$$

thus we get the conclusion.$\square$

Finally, we can prove the aforementioned decay estimate for solutions of (3.1).

Corollary 3.7

(Sharp decay rate) Let $U\in D^{s,p}(\mathbb {R}^N)$ be a positive radially symmetric and decreasing solution of (3.1). Then

$$\begin{aligned} \left( \inf _{B_1} U\right) \,\frac{|x|^{-\frac{N-\textit{sp}}{p-1}}}{C}\le U(x)\le \left( \omega _N^{-\frac{1}{p^*}}\,\Vert U\Vert _{p^*-1}^{\frac{p^*-1}{p}}\right) ^\frac{p}{p-1}\,|x|^{-\frac{N-\textit{sp}}{p-1}},\quad |x|\ge 1, \end{aligned}$$

where the constant $C=C(N,p,s)>1$ is the same of Proposition 3.6.

Proof

The upper bound follows from the borderline $L^{q_0,\infty }$ estimate of (3.6), combined with the Radial Lemma 2.9.

As for the lower bound, by the weak Harnack inequality for positive supersolution of $(-\Delta _p)^s$ (see [19, Theorem 5.2]), we have

$$\begin{aligned} \lambda :=\inf _{B_1} U\ge C\,\left( \int _{B_2} U^{p-1}\,dx\right) ^\frac{1}{p-1}>0. \end{aligned}$$

Then the function $\lambda \, u_1$ is a lower barrier for U in $B_1^c$. Thus the lower bound follows from Theorem 2.7 and Proposition 3.6.$\square$

4 Proof of the main result

In this section we still denote by $\widetilde{\Gamma }$ the truncated function defined by (3.8), while U is a positive radially symmetric and decreasing solution of (3.1). As in the previous section we will systematically use the abuse of notation $U(x)=U(r)$ and $\widetilde{\Gamma }(x)=\widetilde{\Gamma }(r)$, for $r=|x|$.

Lemma 4.1

Suppose that

$$\begin{aligned} U(R)\ge A\,\widetilde{\Gamma }(R)\quad \text {for some }R>2. \end{aligned}$$

For any $\delta >0$ there exists $\theta =\theta (N, p, s, \delta ,U)<1$ such that

$$\begin{aligned} U(r)\ge (A-\delta )\,\widetilde{\Gamma }(r)\quad \hbox { for any }\theta \, R\le r\le R. \end{aligned}$$

Similarly, if

$$\begin{aligned} U(R)\le B\, \widetilde{\Gamma }(R)\quad \text {for some }R>2, \end{aligned}$$

then

$$\begin{aligned} U(r)\le (B+\delta )\, \widetilde{\Gamma }(r)\quad \hbox { for any }R\le r\le R/\theta . \end{aligned}$$

Proof

Consider the first statement and let $\theta <1$ to be determined. U is non increasing and

$$\begin{aligned} U(r)\le C\,r^{-\frac{N-\textit{sp}}{p-1}}, \quad r\ge 1, \end{aligned}$$

by Corollary 3.7. Then for any $\theta \, R\le r\le R$ it holds

$$\begin{aligned} \frac{U(R)}{\widetilde{\Gamma }(R)}-\frac{U(r)}{\widetilde{\Gamma }(r)}\le & {} U(R)\,\left( \frac{1}{\widetilde{\Gamma }(R)}-\frac{1}{\widetilde{\Gamma }(r)}\right) \le \frac{C}{R^{\frac{N-\textit{sp}}{p-1}}}\,\left( R^{\frac{N-\textit{sp}}{p-1}}-r^{\frac{N-\textit{sp}}{p-1}}\right) \\\le & {} C\,\left( 1-\theta ^{\frac{N-\textit{sp}}{p-1}}\right) . \end{aligned}$$

Therefore by hypothesis we get

$$\begin{aligned} \frac{U(r)}{\widetilde{\Gamma }(r)}\ge A-C\,\left( 1-\theta ^{\frac{N-\textit{sp}}{p-1}}\right) ,\quad \hbox { for } r\in [\theta \, R, R], \end{aligned}$$

which gives the first claim. The proof of the other statement is similar: for any $R\le r\le R/\theta$ it holds

$$\begin{aligned} \frac{U(r)}{\widetilde{\Gamma }(r)}-\frac{U(R)}{\widetilde{\Gamma }(R)}\le U(R)\,\left( \frac{1}{\widetilde{\Gamma }(r)}-\frac{1}{\widetilde{\Gamma }(R)}\right) \le \frac{C}{R^{\frac{N-\textit{sp}}{p-1}}}\,\left( r^{\frac{N-\textit{sp}}{p-1}}-R^{\frac{N-\textit{sp}}{p-1}}\right) \le C\,\left( \theta ^{-\frac{N-\textit{sp}}{p-1}}-1\right) , \end{aligned}$$

which gives

$$\begin{aligned} \frac{U(r)}{\widetilde{\Gamma }(r)}\le B+C\,\left( \theta ^{-\frac{N-\textit{sp}}{p-1}}-1\right) ,\quad \hbox { for } r\in [R, R/\theta ]. \end{aligned}$$

This completes the proof.$\square$

We are ready for the proof of the main result.

Theorem 4.2

There exists $U_\infty >0$ such that

$$\begin{aligned} \lim _{r\rightarrow +\infty } r^{\frac{N-\textit{sp}}{p-1}}U(r)=U_\infty . \end{aligned}$$

Proof

We can suppose that $p\ne 2$, since for $p=2$ the function U has an explicit expression. By virtue of Corollary 3.7 we readily have

$$\begin{aligned} \frac{1}{C}\le m:=\liminf _{r\rightarrow +\infty } \frac{U(r)}{\widetilde{\Gamma }(r)}\le \limsup _{r\rightarrow +\infty }\frac{U(r)}{\widetilde{\Gamma }(r)}=:M\le C, \end{aligned}$$

with C depending on U as well. Suppose by contradiction that $M-m>0$, and fix $0<\varepsilon _0<(M-m)/4$.

$\bullet$ ${Case~p>2}$. There exists $R_0=R_0(\varepsilon _0)>2$ such that

$$\begin{aligned} \frac{U(r)}{\widetilde{\Gamma }(r)}\ge m-\varepsilon _0,\quad \hbox { for }r\ge R_0, \end{aligned}$$

and we can choose an arbitrarily large $R>R_0$ such that

$$\begin{aligned} \frac{U(R)}{\widetilde{\Gamma }(R)}\ge M-\frac{M-m}{4}. \end{aligned}$$

Consider $\delta =(M-m)/4$. By Lemma 4.1, there exists $\theta <1$ so that for any such R it holds

$$\begin{aligned} \frac{U(r)}{\widetilde{\Gamma }(r)}\ge \frac{M+m}{2},\quad \hbox { for } r\in [\theta \, R, R]. \end{aligned}$$

(4.1)

Since R can be chosen arbitrarily large, we can suppose $\theta R>R_0$ as well. Consider, for any $0<\varepsilon <(M-m)/4$, the lower barrier $w(r)=g(r)\,\widetilde{\Gamma }(r)$ where g is the following step function

$$\begin{aligned} g(r)={\left\{ \begin{array}{ll} 0&{}\quad \text {if }\quad r<R_0,\\ m-\varepsilon _0&{}\quad \text {if }\quad R_0\le r<\theta R,\\ \frac{M+m}{2}&{}\quad \text {if }\quad \theta R\le r<\sqrt{\theta }R,\\ m+\varepsilon &{}\quad \text {if }\quad \sqrt{\theta }\,R<r. \end{array}\right. } \end{aligned}$$

It is easily seen that³ $w\in \widetilde{D}^{s,p}(\overline{B_R}^{\,c})$. Moreover, by using (4.1), it is readily verified that $w\le U$ in $\overline{B_R}$. We claim that, for sufficiently small $\varepsilon _0$ and $\varepsilon$ and sufficiently large R, it holds

$$\begin{aligned} (-\Delta _p)^s w\le (-\Delta _p)^s U,\quad \hbox { in }\quad \overline{B_R}^{\,c}. \end{aligned}$$

This would end the proof, since Theorem 2.7 would yield $U\ge w$ in $\mathbb {R}^N$ and then

$$\begin{aligned} m=\liminf _{r\rightarrow +\infty } r^{\frac{N-\textit{sp}}{p-1}}\,U(r)=\liminf _{r\rightarrow +\infty } \frac{U(r)}{\widetilde{\Gamma }(r)}\ge \liminf _{r\rightarrow +\infty } g(r)= m+\varepsilon , \end{aligned}$$

giving a contradiction. The function $w-(m+\varepsilon )\,\widetilde{\Gamma }$ is supported in $B_{\sqrt{\theta }\,R}\Subset B_{R}$ and thus using Proposition 2.8 with

$$\begin{aligned} \Omega =\overline{B_R}^{\,c},\quad u=(m+\varepsilon )\,\widetilde{\Gamma },\quad f= (-\Delta _p)^s\Big ((m+\varepsilon )\,\widetilde{\Gamma }\Big ) ,\quad v=w-(m+\varepsilon )\,\widetilde{\Gamma }, \end{aligned}$$

and (3.9), for any $|x|>R$ it holds

$$\begin{aligned} \begin{aligned} (-\Delta _p)^s w(x)&= (m+\varepsilon )^{p-1}\,(-\Delta _p)^s\widetilde{\Gamma }(x)\\&\quad +\int _{B_{\sqrt{\theta }\,R}}\frac{J_p\big ((m+\varepsilon )\,\widetilde{\Gamma }(x)-w(y)\big )-J_p\big ((m+\varepsilon )\,(\widetilde{\Gamma }(x)-\widetilde{\Gamma }(y))\big )}{|x-y|^{N+\textit{sp}}}\, dy\\&\le \frac{C}{|x|^{N+\textit{sp}}}+\int _{B_{\sqrt{\theta }\,R}}\frac{h(x, y)}{|x-y|^{N+\textit{sp}}}\, dy, \end{aligned} \end{aligned}$$

(4.2)

where

$$\begin{aligned} h(x, y)=J_p\big ((m+\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))\big )-J_p\big (w(y)-(m+\varepsilon )\,\widetilde{\Gamma }(x)\big ). \end{aligned}$$

We now decompose the last integral in (4.2) as follows

$$\begin{aligned} \int _{B_{\sqrt{\theta }\,R}}\, dy=\int _{B_{R_0}}\, dy+\int _{B_{\theta \,R}{\setminus } B_{R_0}}\, dy+\int _{B_{\sqrt{\theta }\,R}{\setminus } B_{\theta \,R}}\, dy, \end{aligned}$$

(4.3)

and proceed to estimate each term separately.

Being $R_0=R_0(\varepsilon _0)$ and h universally bounded, it holds

$$\begin{aligned} \int _{B_{R_0}}\frac{h(x, y)}{|x-y|^{N+\textit{sp}}}\, dy\le \Vert h\Vert _{L^\infty (\mathbb {R}^{2\,N})}\,\frac{\omega _N\,R_0^N}{\Big ||x|-R_0\Big |^{N+\textit{sp}}}\le \frac{C(\varepsilon _0)}{|x|^{N+\textit{sp}}}\,\left( 1-\theta \right) ^{-N-\textit{sp}}, \end{aligned}$$

(4.4)

where we used that (recall that we are assuming $\theta \,R> R_0$)

$$\begin{aligned} \Big ||x|-R_0\Big |\ge \left( 1-\frac{R_0}{R}\right) \,|x|\ge (1-\theta )\,|x|,\quad \hbox { for }|x|>R. \end{aligned}$$

For the second integral in (4.3), we notice that for $y\in B_{\theta \, R}{\setminus } B_{ R_0}$ and $x\in B_R^c$ we have

$$\begin{aligned} h(x, y)=J_p\big ((m+\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))\big )-J_p\big ((m+\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))-(\varepsilon +\varepsilon _0)\,\widetilde{\Gamma }(y)\big ). \end{aligned}$$

Observe that by (2.1), with simple manipulations we get

$$\begin{aligned} \begin{aligned} h(x, y)&\le c\, \Big [(m+\varepsilon )^{p-2}+(\varepsilon +\varepsilon _0)^{p-2}\Big ]\,(\varepsilon +\varepsilon _0)\,\widetilde{\Gamma }(y), \end{aligned} \end{aligned}$$

for $x\in B_R^c$, $y\in B_{\theta R}{\setminus } B_{R_0}$ and $c=c(p)>0$. Therefore, since

$$\begin{aligned} |x-y|\ge \Big ||x|-|y|\Big |\ge |x|-\theta \,R\ge \left( 1-\theta \right) \,|x|,\quad \hbox { for } x\in B_R^c,\ y\in B_{\theta \, R}, \end{aligned}$$

recalling the definition of $\widetilde{\Gamma }$ we get

$$\begin{aligned} \begin{aligned} \int _{B_{\theta R}{\setminus } B_{R_0}}\frac{h(x, y)}{|x-y|^{N+\textit{sp}}}\, dy&\le \frac{C\,(\varepsilon +\varepsilon _0)}{(1-\theta )^{N+\textit{sp}}\,|x|^{N+\textit{sp}}}\int _{B_{\theta R}{\setminus } B_{ R_0}}\frac{1}{|y|^{N-\textit{sp}}}\, dy\\&\le C\,\frac{(\varepsilon +\varepsilon _0)}{(1-\theta )^{N+\textit{sp}}}\,\frac{(\theta \,R)^{\textit{sp}}}{|x|^{N+\textit{sp}}}, \end{aligned} \end{aligned}$$

(4.5)

where $C=C(N,s,p,M+m)>0$. For the third integral in (4.3), for $y\in B_{\sqrt{\theta }R}{\setminus } B_{\theta R}$ we have

$$\begin{aligned} \begin{aligned} h(x, y)&=J_p\Big ((m+\varepsilon )(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))\Big )\\&\quad \, -J_p\left( (m+\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))+\left( \frac{M-m}{2}-\varepsilon \right) \,\widetilde{\Gamma }(y)\right) \\&\le J_p\Big (\underbrace{(m+\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))}_a\Big )\\ {}&\quad \, -J_p\Big (\underbrace{(m+\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))}_{a}+\underbrace{\left( \frac{M-m}{4}\right) \,\widetilde{\Gamma }(y)}_b\Big ), \end{aligned} \end{aligned}$$

since $\varepsilon <(M-m)/4$. The inequality (2.2) thus gives

$$\begin{aligned} h(x, y) \le -2^{2-p}\,\left( \frac{M-m}{4}\right) ^{p-1}\,\widetilde{\Gamma }(y)^{p-1}. \end{aligned}$$

Therefore, using

$$\begin{aligned} |x-y|\le 2\,|x|,\quad \hbox { for } |x|>R, \ |y|<\sqrt{\theta }\,R, \end{aligned}$$

we obtain

$$\begin{aligned} \int _{B_{\sqrt{\theta }\,R}{\setminus } B_{\theta \, R}}\frac{h(x, y)}{|x-y|^{N+\textit{sp}}}\, dy\le & {} -\frac{c\,(M-m)^{p-1}}{|x|^{N+\textit{sp}}}\int _{B_{\sqrt{\theta }\,R}{\setminus } B_{\theta \, R}} |y|^{\textit{sp}-N}\, dy\nonumber \\\le & {} -c\,\theta ^\frac{\textit{sp}}{2}\,\Big (1-\theta ^\frac{\textit{sp}}{2}\Big )\,(M-m)^{p-1}\,\frac{R^{\textit{sp}}}{|x|^{N+\textit{sp}}}, \end{aligned}$$

(4.6)

for a constant $c=c(N,s,p)>0$. Gathering toghether the estimates (4.2), (4.4), (4.5) and (4.6) we proved

$$\begin{aligned} \begin{aligned} (-\Delta _p)^s w(x)&\le \left( C+\frac{C(\varepsilon _0)}{(1-\theta )^{N+\textit{sp}}}\right) \,\frac{1}{|x|^{N+\textit{sp}}}\\&\quad -\left[ c\,\Big (1-\theta ^\frac{\textit{sp}}{2}\Big )\,(M-m)^{p-1}-\frac{C\,(\varepsilon +\varepsilon _0)}{(1-\theta )^{N+\textit{sp}}}\right] \,\frac{R^{\textit{sp}}\,\theta ^{\textit{sp}}\,}{|x|^{N+\textit{sp}}}. \end{aligned} \end{aligned}$$

Thus we can choose $\varepsilon +\varepsilon _0$ small enough (depending only on $N,p,s,M-m$ and the chosen minimizer U), so that the second term above is negative. For any such a choice we have, for any $|x|> R$,

$$\begin{aligned} (-\Delta _p)^s w(x)\le \frac{C(\varepsilon _0)}{|x|^{N+\textit{sp}}},\quad (-\Delta _p)^s U(x)= U(x)^{p^*-1}\ge \frac{1}{C\,|x|^{N+\frac{\textit{sp}}{p-1}}}, \end{aligned}$$

where in the last estimate we used Corollary 3.7. Since $p> 2$, for sufficiently large R it holds

$$\begin{aligned} \frac{1}{C\,|x|^{N+\frac{\textit{sp}}{p-1}}}\ge \frac{R^{\textit{sp}\,\frac{p-2}{p-1}}}{C\,|x|^{N+\textit{sp}}}\ge \frac{C\,(\varepsilon _0)}{|x|^{N+\textit{sp}}}, \end{aligned}$$

and thus the claim follows.

$\bullet$ ${Case~1<p<2}$. There exists $R_0=R_0(\varepsilon _0)>2$ such that

$$\begin{aligned} \frac{U(r)}{\widetilde{\Gamma }(r)}\le M+\varepsilon _0,\quad \hbox { for } r\ge R_0 \end{aligned}$$

and we can choose an arbitrarily large $R>R_0$ such that

$$\begin{aligned} \frac{U(R)}{\widetilde{\Gamma }(R)}\le m+\frac{M-m}{4}. \end{aligned}$$

As before, we consider $\delta =(M-m)/4$ in Lemma 4.1: there exists $\theta <1$ so that for any such R it holds

$$\begin{aligned} \frac{U(r)}{\widetilde{\Gamma }(r)}\le \frac{M+m}{2},\quad \hbox { for every } r\in [R, R/\theta ]. \end{aligned}$$

(4.7)

Since $U\in L^\infty (\mathbb {R}^N)$, there exists $\overline{C}>0$ such that $U\le \overline{C}\, \widetilde{\Gamma }$ in $\mathbb {R}^N$, then for any $0<\varepsilon <(M-m)/4$ we consider the upper barrier $w(r)=g(r)\,\widetilde{\Gamma }(r)$, where

$$\begin{aligned} g(r)= {\left\{ \begin{array}{ll} \overline{C}&{}\quad \text {if }\quad r<R_0,\\ M+\varepsilon _0&{}\quad \text {if }\quad R_0\le r<R,\\ \frac{M+m}{2}&{}\quad \text {if }\quad R\le r<R/\sqrt{\theta },\\ M-\varepsilon &{}\quad \text {if }\quad R/\sqrt{\theta }<r. \end{array}\right. } \end{aligned}$$

Again, it is easy to verify that $w\in \widetilde{D}^{s,p}(\overline{B_R}^{\,c})$. Using (4.7), we can verify that $w\ge U$ in $B_{R/\theta }$. We claim that, for sufficiently small $\varepsilon _0$ and $\varepsilon$ and sufficiently large R, it holds

$$\begin{aligned} (-\Delta _p)^s w\ge (-\Delta _p)^s U,\quad \hbox { in } B^c_{R/\theta }. \end{aligned}$$

This would end the proof, since the comparison principle of Theorem 2.7 would yield $U\le w$ in $\mathbb {R}^N$ and then

$$\begin{aligned} M=\limsup _{r\rightarrow +\infty } r^{\frac{N-\textit{sp}}{p-1}}\,U(r)=\limsup _{r\rightarrow +\infty } \frac{U(r)}{\widetilde{\Gamma }(r)}\le \limsup _{r\rightarrow +\infty } g(r)= M-\varepsilon , \end{aligned}$$

which gives again a contradiction. The function $w-(M-\varepsilon )\,\widetilde{\Gamma }$ is supported in $B_{R/\sqrt{\theta }}\Subset B_{R/\theta }$ and thus using again Proposition 2.8 with

$$\begin{aligned} \Omega =\overline{B_{R/\theta }}^{\,c},\quad u=(M-\varepsilon )\,\widetilde{\Gamma },\quad f= (-\Delta _p)^s\Big ((M-\varepsilon )\,\widetilde{\Gamma }\Big ) ,\quad v=w-(M-\varepsilon )\,\widetilde{\Gamma }, \end{aligned}$$

and (3.9), for any $|x|>R/\theta$ it holds

$$\begin{aligned} \begin{aligned} (-\Delta _p)^s w(x)&= (M-\varepsilon )^{p-1}\,(-\Delta _p)^s\,\widetilde{\Gamma }(x) \\&\quad +\int _{B_{R/\sqrt{\theta }}}\frac{J_p\big ((M-\varepsilon )\,\widetilde{\Gamma }(x)-w(y)\big )-J_p\big ((M-\varepsilon )\,(\widetilde{\Gamma }(x)-\widetilde{\Gamma }(y))\big )}{|x-y|^{N+\textit{sp}}}\, dy \\&\ge \frac{1}{C\,|x|^{N+\textit{sp}}}+\int _{B_{R/\sqrt{\theta }}}\frac{h(x, y)}{|x-y|^{N+\textit{sp}}}\, dy, \end{aligned} \end{aligned}$$

(4.8)

where

$$\begin{aligned} h(x, y)=J_p\big ((M-\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))\big )-J_p\big (w(y)-(M-\varepsilon )\,\widetilde{\Gamma }(x)\big ). \end{aligned}$$

As above, we now decompose the last integral in (4.8) as

$$\begin{aligned} \int _{B_{R/\sqrt{\theta }}}\, dy=\int _{B_{R_0}}\, dy+\int _{B_{R}{\setminus } B_{R_0}}\, dy+\int _{B_{R/\sqrt{\theta }}{\setminus } B_{R}}\, dy, \end{aligned}$$

and proceed to estimate each term separately.

Being $R_0=R_0(\varepsilon _0)$ and h universally bounded, as before we get

$$\begin{aligned} \int _{B_{R_0}}\frac{h(x, y)}{|x-y|^{N+\textit{sp}}}\, dy\ge -\frac{C(\varepsilon _0)}{|x|^{N+\textit{sp}}}, \end{aligned}$$

(4.9)

where this time we used that (recall that we are assuming $R>R_0$)

$$\begin{aligned} \Big ||x|-R_0\Big |\ge \left( 1-\frac{R_0}{R}\,\theta \right) \,|x|\ge (1-\theta )\,|x|,\quad \hbox { for }|x|>R/\theta . \end{aligned}$$

For $y\in B_{R}{\setminus } B_{ R_0}$ we have

$$\begin{aligned} h(x, y)=J_p\Big ((M-\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))\Big )-J_p\Big ((M-\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))+(\varepsilon +\varepsilon _0)\,\widetilde{\Gamma }(y)\Big ), \end{aligned}$$

and by subaddivity of $\tau \mapsto \tau ^{p-1}$, we get

$$\begin{aligned} h(x, y)\ge -(\varepsilon +\varepsilon _0)^{p-1}\,\widetilde{\Gamma }(y)^{p-1}. \end{aligned}$$

Therefore, the analogue of (4.5) is now

$$\begin{aligned} \int _{B_{R}{\setminus } B_{ R_0}}\frac{h(x, y)}{|x-y|^{N+\textit{sp}}}\, dy\ge -C\,(\varepsilon +\varepsilon _0)^{p-1}\frac{R^{\textit{sp}}}{|x|^{N+\textit{sp}}}, \end{aligned}$$

(4.10)

and again $C=C(N,s,p,M+m)>0$. For the previous estimate we also used that

$$\begin{aligned} |x-y|\ge |x|-R\ge \left( 1-\theta \right) \,|x|,\quad \hbox { for } |x|>R/\theta ,\ |y|<R. \end{aligned}$$

For $y\in B_{R/\sqrt{\theta }}{\setminus } B_{R }$ and $x\in B^c_{R/\theta }$ we have

$$\begin{aligned} \begin{aligned} h(x, y)&=J_p\Big (\underbrace{(M-\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))}_a\Big )\\&\quad \,-J_p\Big (\underbrace{(M-\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))}_a-\underbrace{\left( \frac{M-m}{2}-\varepsilon \right) \,\widetilde{\Gamma }(y)}_b\Big ). \end{aligned} \end{aligned}$$

Clearly

$$\begin{aligned} 0\le a=(M-\varepsilon )\,(\widetilde{\Gamma }(y)-\widetilde{\Gamma }(x))\le (M-\varepsilon )\,\widetilde{\Gamma }(y)=:A, \end{aligned}$$

so that (2.4) provides

$$\begin{aligned} h(x, y)\ge \max \left\{ (M-\varepsilon )^{p-1}-\left( \frac{M+m}{2}\right) ^{p-1},\,\left( \frac{M-m}{2}-\varepsilon \right) ^{p-1}2^{1-p}\right\} \,\widetilde{\Gamma }(y)^{p-1}. \end{aligned}$$

Proceeding as for (4.6) and using

$$\begin{aligned} |x-y|\le 2\,|x|,\quad \hbox { for } x\in B^c_{R/\theta }, \ y\in B_{R/\sqrt{\theta }}, \end{aligned}$$

we thus obtain

$$\begin{aligned} \int _{B_{R/\sqrt{\theta }}{\setminus } B_{R}}\frac{h(x, y)}{|x-y|^{N+sp}}\, dy \ge \frac{c}{|x|^{N+\textit{sp}}}\int _{B_{R/\sqrt{\theta }}{\setminus }B_{R}} |y|^{\textit{sp}-N}\, dy \ge c\,\frac{R^{\textit{sp}}}{|x|^{N+\textit{sp}}}, \end{aligned}$$

(4.11)

for a small constant c depending only on M and m. Gathering together the estimates (4.8), (4.9), (4.10) and (4.11), we proved

$$\begin{aligned} (-\Delta _p)^s w(x)\ge -\frac{C(\varepsilon _0)}{|x|^{N+\textit{sp}}}+\Big (c-C\,(\varepsilon +\varepsilon _0)^{p-1}\Big )\,\frac{R^{\textit{sp}}}{|x|^{N+\textit{sp}}}. \end{aligned}$$

in $B_{R/\theta }^c$. We can thus choose $\varepsilon _0$ and $\varepsilon$ small enough so that the second term above is positive. For any such choice we have, for any $|x|> R/\theta$,

$$\begin{aligned} (-\Delta _p)^s w(x)\ge -\frac{C(\varepsilon _0)}{|x|^{N+\textit{sp}}}+\frac{c}{2}\frac{R^{\textit{sp}}}{|x|^{N+\textit{sp}}}, \end{aligned}$$

and for sufficiently large R so that $c\,R^{\textit{sp}}>4\,C(\varepsilon _0)$ it holds

$$\begin{aligned} (-\Delta _p)^s w(x)\ge \frac{c}{4}\frac{R^{\textit{sp}}}{|x|^{N+\textit{sp}}}. \end{aligned}$$

By using Corollary 3.7 and the fact that $1<p<2$, for every $|x|\ge R/\theta$ we get

$$\begin{aligned} (-\Delta _p)^s U(x)= U^{p^*-1}(x)\le \frac{C}{|x|^{N+\frac{\textit{sp}}{p-1}}}\le \frac{C\,\theta ^{\textit{sp}\,\frac{2-p}{p-1}}}{R^{\textit{sp}\,\frac{2-p}{p-1}}|x|^{N+\textit{sp}}}. \end{aligned}$$

We thus conclude that $(-\Delta _p)^sU\le (-\Delta _p)^s w$ in $\overline{B_{R/\theta }}^{\,c}$ for R sufficiently large, as desired.$\square$

Appendix A: Power functions

We have the following result on power functions.

Lemma A.1

Let $0<(N-\textit{sp})/p<\beta <N/(p-1)$. For every $R>0$, the function $x\mapsto |x|^{-\beta }$ belongs to $\widetilde{D}^{s,p}(\overline{B_R}^{\,c})$.

Proof

A direct computation shows that $x\mapsto |x|^{-\beta }$ belongs to $L^{p-1}_{\mathrm {loc}}(\mathbb {R}^N)\cap L^{p^*}(B_R^c)$, when $\beta$ is as in the statement. We take $r<R$, then $E=\overline{B_r}^{\,c}\supset \overline{B_R}^{\,c}$ and we need to show

$$\begin{aligned} \Big [|x|^{-\beta }\Big ]_{W^{s,p}(B_r^c)}<+\infty ,\quad \text {for }\,\, \frac{N-\textit{sp}}{p}<\beta . \end{aligned}$$

(A.1)

We compute in polar coordinates

$$\begin{aligned}&\int _{B_r^c\times B_r^c}\frac{||x|^{-\beta }-|y|^{-\beta }|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy\\&\quad = \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\int _{r}^{+\infty }\int _r^{+\infty }\frac{|\varrho ^{-\beta }-t^{-\beta }|^p\,\varrho ^{N-1}\,t^{N-1}}{|\varrho \,\omega _1-t\,\omega _2|^{N+\textit{sp}}}\, d\varrho \, dt\, d\omega _1\, d\omega _2\\&\quad =2\int _{r}^{+\infty }\frac{\varrho ^{-\beta \, p}\,\varrho ^{2\,N-2}}{\varrho ^{N+\textit{sp}}}\int _r^{\varrho }\left| 1-\left( \frac{t}{\varrho }\right) ^{-\beta }\right| ^p\\&\qquad \times \, \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-(t\,\omega _2)/\varrho |^{N+\textit{sp}}}\left( \frac{t}{\varrho }\right) ^{N-1}\, dt\, d\varrho \\&\quad =2\int _r^{+\infty }\frac{\varrho ^{-\beta \, p}\,\varrho ^{2\,N-1}}{\varrho ^{N+\textit{sp}}}\int _{r/\varrho }^1|1-\xi ^{-\beta }|^p\,\xi ^{N-1}\int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-\xi \,\omega _2|^{N+\textit{sp}}}\, d\xi \, d\varrho . \end{aligned}$$

Let us now prove that for $0<\xi <1$ it holds

$$\begin{aligned} \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-\xi \,\omega _2|^{N+\textit{sp}}}\le \frac{C}{(1-\xi )^{1+\textit{sp}}}. \end{aligned}$$

Without loss of generality, we may assume that $\xi \ge 1/2$, since for $0<\xi <1/2$ the integral is uniformly bounded. By rotational invariance, we have

$$\begin{aligned} \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-\xi \,\omega _2|^{N+\textit{sp}}}= |\mathbf {S}^{N-1}|\int _{\mathbf {S}^{N-1}}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}, \end{aligned}$$

where $\mathbf{e}_1=(1,0,\dots ,0)$. By changing variable $\omega _2=(t,z)$ with

$$\begin{aligned} t=\pm \sqrt{1-|z|^2},\quad z\in B_1'\subset \mathbb {R}^{N-1}, \end{aligned}$$

we therefore get (the constant C may vary from a line to another)

$$\begin{aligned} \int _{\mathbf {S}^{N-1}}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}&= \int _{\mathbf {S}^{N-1}{\setminus }B_1(\mathbf{e_1})}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}+\int _{\mathbf {S}^{N-1}\cap B_1(\mathbf{e_1})}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}\\&\le C\left( 1+ \int _{B_1'}\frac{dz}{((1-\xi \, t)^2+\xi ^2\,|z|^2)^{\frac{N+\textit{sp}}{2}}}\right) \\&\le C\left( 1+\int _{B_1'}\frac{dz}{((1-\xi )^2+\xi ^2\,|z|^2)^{\frac{N+\textit{sp}}{2}}} \right) \\&\le C\left( 1+\frac{1}{(1-\xi )^{1+\textit{sp}}}\int _{B_{\frac{\xi }{1-\xi }}'}\frac{1}{ (1+|y|^2)^{\frac{N+\textit{sp}}{2}}}dy\right) \\&\le C\left( 1+\frac{1}{(1-\xi )^{1+\textit{sp}}}\int _{\mathbb {R}^{N-1}}\frac{1}{ (1+|y|^2)^{\frac{N+\textit{sp}}{2}}}dy\right) \end{aligned}$$

which proves the claim. Taking into account that for $0< \xi <1$ it also holds

$$\begin{aligned} \frac{|1-\xi ^{-\beta }|^p}{|1-\xi |^{1+\textit{sp}}}\le C\,(\xi ^{-\beta \, p}+|1-\xi |^{p\,(1-s)-1}) \end{aligned}$$

we therefore get

$$\begin{aligned} \Big [|x|^{-\beta }\Big ]_{W^{s,p}(B_r^c)}^p\le C\int _r^{+\infty }\varrho ^{N-1-p\,(s+\beta )}\,d\varrho \int _{r/\varrho }^1 \xi ^{N-1}\,\big (\xi ^{-\beta \, p}+|1-\xi |^{p\,(1-s)-1}\big )\, d\xi . \end{aligned}$$

All the integrals are now explicitly computable and one can readily get (A.1).$\square$

Lemma A.2

Let $0<(N-\textit{sp})/p<\beta <N/(p-1)$. For every $R>0$, it holds

$$\begin{aligned} (-\Delta _p)^s |x|^{-\beta }=C(\beta )\, |x|^{-\beta \,(p-1)-\textit{sp}}\quad \hbox { weakly in } \overline{B_R}^{\,c}, \end{aligned}$$

where the constant $C(\beta )$ is given by

$$\begin{aligned} C(\beta )=2\,\int _0^1 \varrho ^{\textit{sp}-1}\,\left[ 1-\varrho ^{N-\textit{sp}-\beta \,(p-1)}\right] \,\left| 1-\varrho ^{\beta }\right| ^{p-1}\, \Phi (\varrho )\,d\varrho , \end{aligned}$$

(A.2)

and

$$\begin{aligned} \Phi (\varrho )=\mathcal {H}^{N-2}(\mathbf {S}^{N-2})\,\int _{-1}^1 \frac{(1-t^2)^\frac{N-3}{2}}{\Big (1-2\,t\,\varrho +\varrho ^2\Big )^\frac{N+\textit{sp}}{2}}\, dt. \end{aligned}$$

(A.3)

Proof

Observe that

$$\begin{aligned} |x|^{-\beta \,(p-1)-\textit{sp}} \in L^{(p*)'}(B_R^c), \,\,\quad \text {for any }\beta >(N-\textit{sp})/p. \end{aligned}$$

Then, by Theorem 2.1 and Proposition 2.5 it suffices to show that

$$\begin{aligned} \int _{\mathbb {R}^{2N}}\frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dx\,dy=C(\beta )\,\int _\Omega |x|^{-\beta \,(p-1)-\textit{sp}}\,\varphi \,dx, \end{aligned}$$

for an arbitrary $\varphi \in C^{\infty }_c(\overline{B_R}^{\,c})$. For every such a $\varphi$ we consider the double integral

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2N}}\frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dx\,dy. \end{aligned} \end{aligned}$$

We observe that this is absolutely convergent, indeed

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2N}}&\frac{|J_p(|x|^{-\beta }-|y|^{-\beta })|}{|x-y|^{N+\textit{sp}}}\,|\varphi (x)-\varphi (y)|\,dx\,dy\\&=\int _{B_R^c\times B_R^c}\frac{|J_p(|x|^{-\beta }-|y|^{-\beta })|}{|x-y|^{N+\textit{sp}}}\,|\varphi (x)-\varphi (y)|\,dx\,dy\\&\quad +2\,\int _{B_R} \int _{\mathrm {supp}(\varphi )} \frac{|J_p(|x|^{-\beta }-|y|^{-\beta })|}{|x-y|^{N+\textit{sp}}}\,|\varphi (y)|\,dx\,dy\\&\le \Big [|x|^{-\beta }\Big ]_{W^{s,p}(B_R^c)}\,[\varphi ]_{W^{s,p}(B_R^c)}+C\,\Vert \varphi \Vert _{L^\infty }\,|\mathrm {supp}(\varphi )|\,\int _{B_R} |x|^{-\beta \,(p-1)}\,dx, \end{aligned} \end{aligned}$$

and both terms are finite, thanks to Lemma A.1. For $\delta >0$ we consider the conical set

$$\begin{aligned} \mathcal {O}_\delta =\{(x,y)\in \mathbb {R}^{2\,N}\, :\, (1-\delta )\,|x|\le |y|\le (1+\delta )\,|x|\}, \end{aligned}$$

then by the Dominated Convergence Theorem

$$\begin{aligned}&\lim _{\delta \searrow 0}\int _{\mathcal {O}_\delta ^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dy\,dx\nonumber \\&\quad =\int _{\mathbb {R}^{2N}}\frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x) -\varphi (y))\,dx\,dy. \end{aligned}$$

We now observe that

$$\begin{aligned} \begin{aligned}&\int _{\mathcal {O}_\delta ^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dy\,dx\\&\quad =2\,\int _{\mathbb {R}^N}\left( \int _{\mathcal {K}_\delta (x)^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,dy\right) \,\varphi (x)\,dx, \end{aligned} \end{aligned}$$

where for every $x\in \mathbb {R}^N$

$$\begin{aligned} \mathcal {K}_\delta (x)=\{y\in \mathbb {R}^N\, :\, (1-\delta )\,|x|\le |y|\le (1+\delta )\,|x|\}, \end{aligned}$$

and of course $\mathcal {K}_\delta (x)=\mathcal {K}_\delta (x')$ whenever $|x|=|x'|$. We set

$$\begin{aligned} f_\delta (x)=2\,\int _{\mathcal {K}_\delta (x)^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,dy,\quad x\in \mathbb {R}^N{\setminus }\{0\}, \end{aligned}$$

it is easily seen that $f_\delta$ is a radial function, homogeneous of degree $-\beta \,(p-1)-\textit{sp}$ (see [4, Lemma 6.2]). Thus for $x\not =0$ we have

$$\begin{aligned} f_\delta (x)=|x|^{-\beta \,(p-1)-\textit{sp}}\,f_{\delta }(\omega ),\quad \hbox { for }\quad \omega =\frac{x}{|x|}\in \mathbf {S}^{N-1}. \end{aligned}$$

(A.4)

We set

$$\begin{aligned} C(\beta ;\delta ):=f_{\delta }(\omega )=2\,\int _{\mathcal {K}_\delta (\omega )^c} \frac{J_p(1-|y|^{-\beta })}{|\omega -y|^{N+\textit{sp}}}\,dy,\quad \omega \in \mathbf {S}^{N-1}, \end{aligned}$$

which is independent of the direction $\omega$, by radiality of $f_\delta$. By taking the average over $\mathbf {S}^{N-1}$ and proceeding as in [4, Lemma B.2], we get

$$\begin{aligned} C(\beta ;\delta )=2\,\int _{|\varrho -1|\ge \delta } \varrho ^{N-1}\,|1-\varrho ^{-\beta }|^{p-2}\,(1-\varrho ^{-\beta })\,\Phi (\varrho )\,d\varrho , \end{aligned}$$

where $\Phi$ is defined in (A.3). We now decompose the integral defining $C(\beta ;\delta )$ and perform a change of variables, i.e.

$$\begin{aligned} \begin{aligned} C(\beta ;\delta )&=-2\,\int _0^{1-\delta } \varrho ^{N-1}\,|1-\varrho ^{-\beta }|^{p-1}\,\Phi (\varrho )\,d\varrho +2\,\int _{1+\delta }^\infty \varrho ^{N-1}\,|1-\varrho ^{-\beta }|^{p-1}\,\Phi (\varrho )\,d\varrho \\&=-2\,\int _0^{1-\delta } \varrho ^{N-1-\beta \,(p-1)}\,|\varrho ^{\beta }-1|^{p-1}\,\Phi (\varrho )\,d\varrho \\&\quad +2\,\int _0^{1/(1+\delta )} \varrho ^{-N-1}\,|1-\varrho ^{\beta }|^{p-1}\,\Phi (1/\varrho )\,d\varrho . \end{aligned} \end{aligned}$$

Finally, observe that

$$\begin{aligned} \Phi (1/\varrho )=\varrho ^{N+\textit{sp}}\,\Phi (\varrho ), \end{aligned}$$

thus the quantity $C(\beta ;\delta )$ can be written as

$$\begin{aligned} \begin{aligned} C(\beta ;\delta )&=2\,\int _0^{1-\delta } \left( 1-\varrho ^{N-\textit{sp}-\beta \,(p-1)}\right) \,\varrho ^{\textit{sp}-1}\,(1-\varrho ^{\beta })^{p-1}\,\Phi (\varrho )\,d\varrho \\&\quad +2\,\int _{1-\delta }^{1/(1+\delta )} \varrho ^{\textit{sp}-1}\,(1-\varrho ^{\beta })^{p-1}\,\Phi (\varrho )\,d\varrho . \end{aligned} \end{aligned}$$

(A.5)

Recall that $\varphi$ is compactly supported in $\overline{B_R}^{\,c}$, thus by using (A.4) we can estimate

$$\begin{aligned} \begin{aligned}&\left| \int _{\Omega } f_\delta \,\varphi \,dx-C(\beta )\,\int _\Omega |x|^{-\beta \,(p-1)-\textit{sp}}\,\varphi \,dx\right| \\&\quad \le \Vert \varphi \Vert _\infty \,R^{-\beta \,(p-1)-\textit{sp}}\,|\mathrm {supp}(\varphi )|\,\Big |C(\beta ;\delta )-C(\beta )\Big |. \end{aligned} \end{aligned}$$

In order to prove that $C(\beta ;\delta )$ converges to $C(\beta )$ as $\delta$ goes to 0, we decompose the function $\Phi$ defined in (A.3) as follows

$$\begin{aligned} \begin{aligned} \Phi (\varrho )&=\int _{-1}^{1/2} \frac{(1-t^2)^\frac{N-3}{2}}{(1-2\,t\,\varrho +\varrho ^2)^\frac{N+\textit{sp}}{2}}\, dt+\int _{1/2}^{1} \frac{(1-t^2)^\frac{N-3}{2}}{(1-2\,t\,\varrho +\varrho ^2)^\frac{N+\textit{sp}}{2}}\, dt=:\Phi _1(\varrho )+\Phi _2(\varrho ), \end{aligned} \end{aligned}$$

where we omitted the dimensional constant $\mathcal {H}^{N-2}(\mathbf {S}^{N-2})$ for simplicity. Let us start estimating $\Phi _1$. If we use that

$$\begin{aligned} 1-2\,t\,\varrho +\varrho ^2=(\varrho -t)^2+(1-t^2)\ge \frac{3}{4},\quad \hbox { if }-1\le t\le \frac{1}{2}, \end{aligned}$$

we get

$$\begin{aligned} 0\le \Phi _1(\varrho )\le C,\quad 0<\varrho <1. \end{aligned}$$

(A.6)

We now consider $\Phi _2(\varrho )$, discussing separately the cases $0<\varrho <1/2$ and $1/2\le \varrho <1$. We observe that for $0<\varrho <1/2$ we have

$$\begin{aligned} 1-2\,t\,\varrho +\varrho ^2=(1-\varrho )^2+2\,\varrho \,(1-t)\ge \frac{1}{4},\quad \hbox { if } \frac{1}{2}\le t\le 1. \end{aligned}$$

Then we get again

$$\begin{aligned} 0\le \Phi _2(\varrho )\le C,\quad \hbox { if } 0<\varrho <\frac{1}{2}. \end{aligned}$$

(A.7)

We are left with the term $\Phi _2(\varrho )$ for $1/2\le \varrho <1$. With simple manipulations⁴ we can write it as

$$\begin{aligned} \Phi _2(\varrho )=\frac{(2\,\varrho )^{-\frac{N-1}{2}}}{(1-\varrho )^{1+\textit{sp}}}\int _{0}^{\frac{\varrho }{(1-\varrho )^2}} \frac{\left( 2-\frac{(1-\varrho )^2}{2\,\varrho }\,\tau \right) ^\frac{N-3}{2}\,\tau ^\frac{N-3}{2}}{(1+\tau )^\frac{N+\textit{sp}}{2}}\, d\tau . \end{aligned}$$

In particular, we get

$$\begin{aligned} 0\le \Phi _2(\varrho )\le C\,(1-\varrho )^{-1-\textit{sp}},\quad \hbox { if }\frac{1}{2}\le \varrho <1. \end{aligned}$$

(A.8)

By using (A.6), (A.7) and (A.8), we thus obtain for the first integral in (A.5)

$$\begin{aligned} \lim _{\delta \searrow 0}2\,\int _0^{1-\delta } \left( 1-\varrho ^{N-\textit{sp}-\beta \,(p-1)}\right) \,\varrho ^{\textit{sp}-1}\,(1-\varrho ^{\beta })^{p-1}\,\Phi (\varrho )\,d\varrho =C(\beta ), \end{aligned}$$

and observe that the latter is finite, thanks to (A.8). It is only left to show that the other integral in (A.5) converges to 0. Still by (A.6) and (A.8), we obtain

where we assumed for simplicity that $p-1-\textit{sp}\not =0$. If $p-1-\textit{sp}>0$, the last term converges to 0. If $p-1-\textit{sp}<0$, we have

$$\begin{aligned} \left( \frac{\delta }{1+\delta }\right) ^{p-1-\textit{sp}}-\delta ^{p-1-\textit{sp}}= & {} \delta ^{p-1-\textit{sp}}\,\Big [(1+\delta )^{\textit{sp}+1-p}-1\Big ]\\\simeq & {} (\textit{sp}+1-p)\,\delta ^{p-\textit{sp}},\quad \hbox { as } \delta \searrow 0, \end{aligned}$$

and thus the integral converges to 0 again. Finally, the borderline case $p-1-\textit{sp}=0$ is treated similarly, we leave the details to the reader.

In conclusion, we get

$$\begin{aligned} \lim _{\delta \searrow 0}\int _{\Omega } f_\delta \,\varphi \,dx=C(\beta )\,\int _\Omega |x|^{-\beta \,(p-1)-\textit{sp}}\,\varphi \,dx, \end{aligned}$$

as desired.$\square$

Remark A.3

The previous result was proved in [15, Lemma 3.1] for the limit case $\beta =(N-\textit{sp})/p$. Our argument is different, since we rely on elementary estimates for the function $\Phi$, rather than on special properties of hypergeometric and beta functions like in [15].

Observe that the choice $\beta =(N-\textit{sp})/(p-1)$ is feasible in the previous results, since

$$\begin{aligned} \frac{N-\textit{sp}}{p}<\frac{N-\textit{sp}}{p-1}<\frac{N}{p-1}. \end{aligned}$$

Moreover, with such a choice we have $C(\beta )=0$ in (A.2). Then from Lemmas A.1 and A.2, we get the following.

Theorem A.4

For any $R>0$, $\Gamma (x)=|x|^{-\frac{N-\textit{sp}}{p-1}}$ belongs to $\widetilde{D}^{s,p}(\overline{B_R}^{\,c})$ and weakly solves $(-\Delta _p)^su=0$ in $\overline{B_R}^{\,c}$.