On the behavior of least energy solutions of a fractional $(p,q(p))$-Laplacian problem as p goes to infinity

TL;DR

This paper investigates the asymptotic behavior of positive least energy solutions to a fractional $(p,q(p))$-Laplacian problem as p approaches infinity, revealing how solutions concentrate and depend on the domain's geometry.

Contribution

It provides a detailed analysis of the limit behavior of solutions to a fractional $(p,q(p))$-Laplacian problem as p tends to infinity, including the influence of parameters and domain geometry.

Findings

Solutions concentrate at a point in the domain as p increases.

The limit solutions are characterized by a variational problem involving the domain's inradius.

The asymptotic behavior depends on the ratio of q(p) to p and the parameters α, β.

Abstract

We study the behavior as $p\rightarrow\infty$ of $u_{p},$ a positive least energy solution of the problem \[ \left\{\begin{array} [c]{lll} \left[ \left( -\Delta_{p}\right) ^{\alpha}+\left( -\Delta_{q(p)}\right) ^{\beta}\right] u=\mu_{p}\left\Vert u\right\Vert _{\infty}^{p-2} u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{in} & \mathbb{R}^{N}\setminus\Omega\\ \left\vert u(x_{u})\right\vert =\left\Vert u\right\Vert _{\infty}, & & \end{array} \right. \] where $\Omega\subset\mathbb{R}^{N}$ is a bounded, smooth domain, $\delta_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & 0<\beta<\alpha<1\\ (1,\infty) & \mathrm{if} & 0<\alpha<\beta<1 \end{array} \right. \] and \[ \lim_{p\rightarrow\infty}\sqrt[p]{\mu_{p}}>R^{-\alpha}, \] with $R$ denoting the inradius of $\Omega.$

Equations

$$\left\{\begin{array}[c]{lll}\left[\left(-\Delta_{p}\right)^{\alpha}+\left(-\Delta_{q(p)}\right)^{\beta}\right]u=\mu_{p}\left\|u\right\|_{\infty}^{p-2}u(x_{u})\delta_{x_{u}}&\mathrm{in}&\Omega\\ u=0&\mathrm{in}&\mathbb{R}^{N}\setminus\Omega\\ \left|u(x_{u})\right|=\left\|u\right\|_{\infty},&&\end{array}\right.$$

$$\lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{\begin{array}[c]{lll}(0,1)&\mathrm{if}&0<\beta<\alpha<1\\ (1,\infty)&\mathrm{if}&0<\alpha<\beta<1\end{array}\right.$$

$$\underset{p\to \infty }{\lim }\sqrt[p]{{\mu }_p}>R^{-\alpha },$$

$$W_0^{s,m}(\Omega ):=\left\{u\in L^m({\mathbb{R}}^N):u=0\mathrm{in}{\mathbb{R}}^N\setminus \Omega \mathrm{and}{\left[u\right]}_{s,m}<\infty \right\},$$

$${\left[u\right]}_{s,m}:={\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{{\left|u(x)-u(y)\right|}^m}{{\left|x-y\right|}^{N+sm}}\mathrm{d}x\mathrm{d}y\right)}^{\frac{1}{m}}$$

$$1\leq r<m_{s}^{\ast}:=\left\{\begin{array}[c]{ll}\dfrac{Nm}{N-sm},&m<N/s,\\ \infty,&m\geq N/s.\end{array}\right.$$

$$W_0^{s,m}(\Omega )\hookrightarrow \hookrightarrow C_0(\overline{\Omega })\mathrm{if}m>N/s.$$

$${\lambda }_{s,m}:=\inf \left\{\frac{{\left[u\right]}_{s,m}^m}{{\Vert u\Vert }_{\infty }^m}:u\in W_0^{s,m}(\Omega )\setminus \left\{0\right\}\right\}$$

$$\underset{(x,y)\ne (0,0)}{\sup }\frac{\left|u(x)-u(y)\right|}{{\left|x-y\right|}^{s-\frac{N}{m}}}\le C{\left[u\right]}_{s,m},\forall u\in W_0^{s,m}(\Omega )$$

$$\big{\langle}\left(-\Delta_{m}\right)^{s}u,\varphi\big{\rangle}_{s,m}:=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\dfrac{\left|u(x)-u(y)\right|^{m-2}(u(x)-u(y))(\varphi(x)-\varphi(y))}{\left|x-y\right|^{N+sm}}\mathrm{d}x\mathrm{d}y,\quad\forall\,u,\varphi\in W_{0}^{s,m}(\Omega).$$

$$({\mathcal{L}}_{s,m}u)(x):=2{\int }_{{\mathbb{R}}^N}\frac{{\left|u(x)-u(y)\right|}^{m-2}(u(y)-u(x))}{{\left|x-y\right|}^{N+sm}}\mathrm{d}y.$$

$$\displaystyle\big{\langle}\left(-\Delta_{m}\right)^{s}u,\varphi\big{\rangle}_{s,m}$$

$$={\int }_{{\mathbb{R}}^N}\phi (x)\left({\int }_{{\mathbb{R}}^N}\frac{{\left|u(x)-u(y)\right|}^{m-2}(u(x)-u(y))}{{\left|x-y\right|}^{N+sm}}\mathrm{d}y\right)\mathrm{d}x$$

$$-{\int }_{{\mathbb{R}}^N}\phi (y)\left({\int }_{{\mathbb{R}}^N}\frac{{\left|u(x)-u(y)\right|}^{m-2}(u(x)-u(y))}{{\left|x-y\right|}^{N+sm}}\mathrm{d}x\right)\mathrm{d}y$$

$$={\int }_{{\mathbb{R}}^N}\phi (x)({\mathcal{L}}_{s,m}u)(x)\mathrm{d}x.$$

$$\left\{\begin{array}[c]{lll}\left[\left(-\Delta_{p}\right)^{\alpha}+\left(-\Delta_{q}\right)^{\beta}\right]u=\mu\left|u(x_{u})\right|^{p-2}u(x_{u})\delta_{x_{u}}&\mathrm{in}&\Omega\\ u=0&\mathrm{in}&\mathbb{R}^{N}\setminus\Omega,\\ \left|u(x_{u})\right|=\left\|u\right\|_{\infty}&&\end{array}\right.$$

$$\left\{\begin{array}[c]{lll}\left[\left(-\Delta_{p}\right)^{\alpha}+\left(-\Delta_{q}\right)^{\beta}\right]u=\mu\left\|u\right\|_{r}^{p-r}\left|u\right|^{r-2}u&\mathrm{in}&\Omega\\ u=0&\mathrm{in}&\mathbb{R}^{N}\setminus\Omega\end{array}\right.$$

$$\big{\langle}\left(-\Delta_{p}\right)^{\alpha}u,\varphi\big{\rangle}_{\alpha,p}+\big{\langle}\left(-\Delta_{q}\right)^{\beta}u,\varphi\big{\rangle}_{\beta,q}=\mu\left|u(x_{u})\right|^{p-2}u(x_{u})\varphi(x_{u})\quad\forall\,\varphi\in X(\Omega),$$

$$\underset{ϵ\to 0^{+}}{\lim }\frac{{\Vert u+ϵ\phi \Vert }_{\infty }^p-{\Vert u\Vert }_{\infty }^p}{pϵ}={\left|u(x_u)\right|}^{p-2}u(x_u)\phi (x_u),\forall \phi \in C(\overline{\Omega }).$$

$$E_{\mu }(u):=\frac{1}{p}{\left[u\right]}_{\alpha ,p}^p+\frac{1}{q}{\left[u\right]}_{\beta ,q}^q-\frac{\mu }{p}{\Vert u\Vert }_{\infty }^p,\mu >0,$$

$$0<\alpha <\beta <1\mathrm{and}N/\alpha <p<q$$

$$0<\beta <\alpha <1\mathrm{and}N/\beta <q<p.$$

$$X(\Omega):=\left\{\begin{array}[c]{lll}\left(W_{0}^{\beta,q}(\Omega),\left[\cdot\right]_{\beta,q}\right)&\mathrm{if}&0<\alpha<\beta<1\quad\mathrm{and}\quad N/\alpha<p<q\\ \left(W_{0}^{\alpha,p}(\Omega),\left[\cdot\right]_{\alpha,p}\right)&\mathrm{if}&0<\beta<\alpha<1\quad\mathrm{and}\quad N/\beta<q<p,\end{array}\right.$$

$$X(\Omega )\subset W_0^{\alpha ,p}(\Omega )\cap W_0^{\beta ,q}(\Omega )\mathrm{and}X(\Omega )\hookrightarrow \hookrightarrow C_0(\overline{\Omega }).$$

$$\mu >{\lambda }_{\alpha ,p}$$

$${\lambda }_{\alpha ,p}:=\inf \left\{\frac{{\left[u\right]}_{\alpha ,p}^p}{{\Vert u\Vert }_{\infty }^p}:u\in W_0^{\alpha ,p}(\Omega )\setminus \left\{0\right\}\right\}=\frac{{\left[e\right]}_{\alpha ,p}^p}{{\Vert e\Vert }_{\infty }^p}>0$$

$${\mathcal{N}}_{\mu }:=\left\{u\in W_0^{\alpha ,p}(\Omega )\setminus \left\{0\right\}:{\left[u\right]}_{\alpha ,p}^p+{\left[u\right]}_{\beta ,q}^q=\mu {\Vert u\Vert }_{\infty }^p\right\},$$

$${\mathcal{L}}_{\alpha ,p}u+{\mathcal{L}}_{\beta ,q}u=0\mathrm{in}D:=\Omega \setminus \left\{x_u\right\}$$

$$\left\{\begin{array}[c]{lll}\left[\left(-\Delta_{p}\right)^{\alpha}+\left(-\Delta_{q(p)}\right)^{\beta}\right]u=\mu_{p}\left|u(x_{p})\right|^{p-2}u(x_{p})\delta_{x_{p}}&\mathrm{in}&\Omega\\ u=0&\mathrm{in}&\mathbb{R}^{N}\setminus\Omega\\ \left|u(x_{p})\right|=\left\|u\right\|_{\infty}.&&\end{array}\right.$$

$$C_0^{0,s}(\overline{\Omega }):=\left\{u\in C_0(\overline{D}):{\left|u\right|}_s<\infty \right\},$$

$${\left|u\right|}_s:=\sup \left\{\frac{\left|u(x)-u(y)\right|}{{\left|x-y\right|}^s}:x,y\in \overline{\Omega }\mathrm{and}x\ne y\right\}.$$

$$\lim_{p\rightarrow\infty}\frac{q(p)}{p}=:Q\in\left\{\begin{array}[c]{lll}(0,1)&\mathrm{if}&0<\beta<\alpha<1\\ (1,\infty)&\mathrm{if}&0<\alpha<\beta<1\end{array}\right.$$

Full text

[TABLE]

where $\Omega\subset\mathbb{R}^{N}$ is a bounded, smooth domain, $\delta_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$

[TABLE]

and

[TABLE]

with $R$ denoting the inradius of $\Omega.$

Keywords: Asymptotic behavior, Dirac delta distribution, fractional Sobolev spaces, viscosity solutions.

2010 AMS Classification. 35D40, 35R11, 35J60.

1 Introduction

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N>1,$ and consider the Sobolev space of fractional order $s\in(0,1)$ and exponent $m>1,$

[TABLE]

where

[TABLE]

is the Gagliardo seminorm.

As it is well known, $\left(W_{0}^{s,m}(\Omega),\left[\cdot\right]_{s,m}\right)$ is a uniformly convex Banach space (also characterized as the closure of $C_{c}^{\infty}(\Omega)$ with respect to $\left[\cdot\right]_{s,m}$), compactly embedded into $L^{r}(\Omega)$ whenever

[TABLE]

Moreover,

[TABLE]

(The notation $A\hookrightarrow\hookrightarrow B$ means that the continuous embedding $A\hookrightarrow B$ is compact.) It follows that infimum

[TABLE]

is positive and, in fact, a minimum.

The compactness in (1) is consequence of the following Morrey’s type inequality (see [11])

[TABLE]

which holds whenever $m>N/s.$ If $m$ is sufficiently large, the positive constant $C$ in (2) can be chosen uniform with respect to $m$ (see [14, Remark 2.2]).

Let $\left(-\Delta_{m}\right)^{s}$ be the $s$-fractional $m$-Laplacian, the operator acting from $W_{0}^{s,m}(\Omega)$ into its topological dual, defined by

[TABLE]

We recall that $\left(-\Delta_{m}\right)^{s}u$ is the Gâteaux derivative at a function $u\in W_{0}^{s,m}(\Omega)$ of the Fréchet differentiable functional $v\mapsto m^{-1}\left[v\right]_{s,m}^{m}.$

An alternative pointwise expression for $\left(-\Delta_{m}\right)^{s}u$ is

[TABLE]

As argued in [16], this expression appears formally as follows

[TABLE]

In Section 2, we consider the nonhomogeneous problem

[TABLE]

where $\alpha,\beta,$ $p,q$ and $\mu>0$ satisfy suitable conditions, $x_{u}\in\Omega$ is a point where $u$ attains its sup norm ($\left|u(x_{u})\right|=\left\|u\right\|_{\infty}$), and $\delta_{x_{u}}$ is the Dirac delta distribution supported at $x_{u}.$

Proceeding as in [1] and [12], one can arrive at (4) as the limit case, as $r\rightarrow\infty,$ of the problem

[TABLE]

where $\left\|\cdot\right\|_{r}$ denotes the standard norm in the Lebesgue space $L^{r}(\Omega).$

As usual, we interpret (4) as an identity between functionals applied to the (weak) solution $u.$ Thus,

[TABLE]

where $X(\Omega)$ is an appropriate Sobolev space (that will be derived in the sequence). The functional at the left-hand side of (5) is the Gâteaux derivative of the Fréchet differentiable functional $v\mapsto p^{-1}\left[v\right]_{\alpha,p}^{p}+q^{-1}\left[v\right]_{\beta,q}^{q}$ at $u.$ However, the functional at the right-hand side is merely related to the right-sided Gâteaux derivative of the functional $\varphi\mapsto p^{-1}\left\|\varphi\right\|_{\infty}^{p}$ whenever $u$ assumes its sup norm at a unique point $x_{u}$. This has to do with the following fact (see Lemma 2.5 and Remark 2.6): if $u\in C(\overline{\Omega})$ assumes its sup norm only at $x_{u}\in\Omega,$ then

[TABLE]

Therefore, we define the formal energy functional associated with (4) by

[TABLE]

and formulate our hypotheses on $\alpha,\beta,$ $p$ and $q$ to guarantee the well-definiteness of this functional. For this, we take into account (1) and the following known facts:

• •

$W_{0}^{s,p}(\Omega)\not\hookrightarrow W_{0}^{s,q}(\Omega)$ for any $0<s<1\leq q<p\leq\infty$ (see [18, Theorem 1.1]),

• •

$W_{0}^{s_{2},m_{2}}(\Omega)\hookrightarrow W_{0}^{s_{1},m_{1}}(\Omega),$ whenever $0<s_{1}<s_{2}<1\leq m_{1}<m_{2}<\infty$ (see [3, Lemma 2.6]).

Thus, we assume that $\alpha,\beta,$ $p$ and $q$ satisfy one of the following conditions:

[TABLE]

or

[TABLE]

The assumption (6) provides the chain of embeddings $W_{0}^{\beta,q}(\Omega)\hookrightarrow W_{0}^{\alpha,p}(\Omega)\hookrightarrow\hookrightarrow C_{0}(\overline{\Omega})$ whereas (7) yields $W_{0}^{\alpha,p}(\Omega)\hookrightarrow W_{0}^{\beta,q}(\Omega)\hookrightarrow\hookrightarrow C_{0}(\overline{\Omega}).$ Therefore, the Sobolev space

[TABLE]

is the natural domain for the energy functional $E_{\mu}.$ Note that

[TABLE]

Once we have chosen $X(\Omega),$ a weak solution of (4) is defined (see Definition 2.2) by means of (5).

As for the parameter $\mu,$ we assume that

[TABLE]

where

[TABLE]

for some function $e\in W_{0}^{\alpha,p}(\Omega)\setminus\left\{0\right\}.$ The existence of $e$ is a consequence of the compact embedding of $W_{0}^{\alpha,p}(\Omega)$ into $C_{0}(\overline{\Omega})$ that holds in both cases (6) and (7).

It turns out that (8) is also a necessary condition for the existence of weak solutions (see Remark 2.3).

Assuming the above conditions on $\alpha,\beta,p,q$ and $\mu$ we show the existence of at least one positive weak solution that minimizes the energy functional either on $W_{0}^{\beta,q}(\Omega)\setminus\left\{0\right\},$ when (6) holds, or on the following Nehari-type set

[TABLE]

when (7) holds. Both type of minimizers are referred in this work as least energy solutions of (4). The reason behind the appearance of the Dirac delta is that the set where a minimizer of $E_{\mu}$ attains its sup norm is a singleton (as we will show).

We conclude Section 2 by observing that the weak solutions of (4) are also viscosity solutions of

[TABLE]

and use this fact to argue that nonnegative least energy solutions are strictly positive in $\Omega.$

In Section 3, we fix the fractional orders $\alpha$ and $\beta$ (with $\alpha\not=\beta$), allow $q$ and $\mu$ to depend suitably on $p$ ($q=q(p)$ and $\mu=\mu_{p}$) and denote by $u_{p}$ the positive least energy solution of the problem

[TABLE]

In the sequence we determine the asymptotic behavior of the pair $\left(u_{p},x_{p}\right)\in X(\Omega)\times\Omega,$ as $p$ goes to $\infty.$

Our main results are stated in Theorem 1.1 below, where, for each $s\in(0,1],$

[TABLE]

with $\left|\cdot\right|_{s}$ denoting the $s$-Hölder seminorm, defined by

[TABLE]

Theorem 1.1

Assume that

[TABLE]

and

[TABLE]

where $R$ is the inradius of $\Omega$ (i.e. the radius of the largest ball inscribed in $\Omega$).

Let $p_{n}\rightarrow\infty.$ There exist $x_{\infty}\in\Omega$ and $u_{\infty}\in C_{0}^{0,\beta}(\overline{\Omega})$ such that, up to a subsequence, $x_{u_{p_{n}}}\rightarrow x_{\infty}$ and $u_{p_{n}}\rightarrow u_{\infty}$ uniformly in $\overline{\Omega}.$ Moreover:

1. (i)

$0<u_{\infty}(x)\leq\left(\Lambda R^{\beta}\right)^{\frac{1}{Q-1}}\left(\operatorname{dist}(x,\partial\Omega)\right)^{\beta}\quad\forall\,x\in\Omega,$ 2. (ii)

$\operatorname{dist}(x_{\infty},\partial\Omega)=R,$ 3. (iii)

$u_{\infty}(x_{\infty})=\left\|u_{\infty}\right\|_{\infty}=R^{\beta}(\Lambda R^{\beta})^{\frac{1}{Q-1}},$ 4. (iv)

$\left|u_{\infty}\right|_{\beta}=(\Lambda R^{\beta})^{\frac{1}{Q-1}},$ 5. (v)

$\dfrac{\left|u_{\infty}\right|_{\beta}}{\left\|u_{\infty}\right\|_{\infty}}=R^{-\beta}=\min\left\{\dfrac{\left|v\right|_{\beta}}{\left\|v\right\|_{\infty}}:v\in C_{0}^{0,\beta}(\overline{\Omega})\setminus\left\{0\right\}\right\},$ 6. (vi)

$u_{\infty}$ is a viscosity solution of

[TABLE]

In the above equation the operators are defined according to the following notation, where $0<s<1$:

[TABLE]

There are a substantial amount of papers in the recent literature dealing with the asymptotic behavior of solutions as a parameter goes to infinity in problems that involve a combination of first order, local operators and nonlinearities of different homogeneity degrees (see [1], [4], [6], [7], [8], [10], [12], [17]). In [1], Alves, Ercole and Pereira determined the asymptotic behavior, as $p\rightarrow\infty,$ of the following problem of order $1$

[TABLE]

Their work motived us to formulate an adequate fractional version of (13) and study, in the present paper, the behavior of the corresponding least energy solutions as $p$ goes to infinity.

As for fractional operators, there are few works focusing such type of asymptotic behavior. Most of recent ones deal with the problem of determining the limit equation satisfied, in the viscosity sense, by the limit functions (as $m\rightarrow\infty$) of a family $\left\{u_{m}\right\}$ of minimizers. In general, such limit equation combines the operators $\mathcal{L}_{s}^{+},$ $\mathcal{L}_{s}^{-}$ and their sum

[TABLE]

We refer to this latter operator as $s$-Hölder infinity Laplacian, accordingly [5], where it was introduced. In that paper, Chambolle, Lindgren and Monneau studied the problem of minimizing the functional

[TABLE]

on the set

[TABLE]

where $g\in C^{0,s}(\partial\Omega)$ is given. After showing the existence of a unique minimizer $u_{m}\in X_{g}$ for this problem (assuming $m>N/s$), they proved that, up to a subsequence, $u_{m}\rightarrow u_{\infty}\in C^{0,s}(\overline{\Omega})$ uniformly and that this limit function is a viscosity solution of

[TABLE]

They also showed that $u_{\infty}$ is an optimal Hölder extension of $g$ in $\Omega.$

In [16], Lindqvist and Lindgren characterized the asymptotic behavior (as $m\rightarrow\infty$) of the only positive, normalized first eigenfunction $u_{m}$ of $(-\Delta_{m})^{s}$ in $W_{0}^{s,m}(\Omega).$ That is, $u_{m}>0$ in $\Omega,$ $\left\|u_{m}\right\|_{m}=1$ and $\left[u_{m}\right]_{s,m}^{m}=\Lambda_{s,m},$ where

[TABLE]

is the the first eigenvalue of $(-\Delta_{m})^{s}.$ Among several results, they proved that

[TABLE]

and that any limit function $u_{\infty}$ of the family $\left\{u_{m}\right\}$ is a positive viscosity solution of the problem

[TABLE]

In [14], Ferreira and Pérez-Llanos studied the asypmtotic behavior, as $m\rightarrow\infty,$ of the solutions of the problem

[TABLE]

for the cases $f=f(x)$ and $f=f(u)=\left|u\right|^{\theta(m)-2}u$ with $\Theta:=\lim_{m\rightarrow\infty}\theta(m)/m<1$ (that is, the exponent of the nonlinearity goes to infinity "sublinearly"). In the first case, they obtained different limit equations involving the operators $\mathcal{L}_{\infty},$ $\mathcal{L}_{\infty}^{+}$ and $\mathcal{L}_{\infty}^{-}$ according to the sign of the function $f(x).$ In the second case, they established the limit equation

[TABLE]

Such results in that paper are compatible with the ones obtained for the local operator in [2] for the first case and in [7] for the second case.

Recently, in [9], Rossi and Silva studied the problem of minimizing the Gagliardo seminorm $\left[\cdot\right]_{s,m}$ among the functions $v\in W^{s,m}(\mathbb{R}^{N})$ satisfying the constraints

[TABLE]

where the function $g$ in $\mathbb{R}^{N}\setminus\Omega$ and the constant $\alpha\in(0,\mathcal{L}^{N}(\Omega))$ are given, and $\mathcal{L}^{N}(D)$ denotes the $N$-dimensional Lebesgue volume of the subset $D\subset\mathbb{R}^{N}.$

They proved that, up to subsequences, the family $\left\{u_{m}\right\}$ of minimizers converges uniformly to a function $u_{\infty},$ as $m\rightarrow\infty,$ that solves the equation

[TABLE]

in the viscosity sense and also minimizes the $s$-Hölder seminorm $\left|\cdot\right|_{s}$ among the functions in $W^{s,\infty}(\mathbb{R}^{N})$ satisfying (15). Further, they showed the convergence of the respective extremal values, that is: $\left[u_{m}\right]_{s,m}\rightarrow\left|u_{\infty}\right|_{s}.$

More recently, in [13], Ercole, Pereira and Sanchis studied the asymptotic behavior of $u_{m},$ the positive solution of the minimizing problem

[TABLE]

where $\omega\in L^{1}(\Omega)$ is a positive weight satisfying $\left\|\omega\right\|_{1}=1.$ After showing that $u_{m}$ is the positive (weak) solution of the singular problem

[TABLE]

they proved that, up to subsequence, $\left\{u_{m}\right\}$ converges uniformly to a function $u_{\infty}\in C_{0}^{0,s}(\overline{\Omega})$ and $\sqrt[m]{\Lambda_{m}}\rightarrow\left|u_{\infty}\right|_{s}.$ Moreover, the limit function $u_{\infty}$ is a positive viscosity solution of

[TABLE]

satisfying

[TABLE]

where $Q_{s}(u):=\left|u\right|_{s}/\exp\left(\int_{\Omega}(\log\left|u\right|)\omega\mathrm{d}x\right).$

Our approach in this paper is inspired by the arguments and techniques developed in some of the works above mentioned and can be applied to the fractional version of [12] and also for studying a fractional version for the system considered in [17].

2 Existence of a positive least energy solution

In this section, we assume that $\mu$ satisfy (8) and that $\alpha,\beta,p$ and $q$ are related by one of the conditions (6) or (7). Our goal is to prove the existence of at least one positive least energy solution $u_{\mu}\in X(\Omega)\setminus\left\{0\right\}$ for the problem (4).

Remark 2.1

We recall that $\left[\left|u\right|\right]_{s,p}\leq\left[u\right]_{s,p}$ for all $u\in W_{0}^{s,m}(\Omega)$ since

[TABLE]

Definition 2.2

We say that a function $u\in X(\Omega)$ is a weak solution of (4) if $\left\|u\right\|_{\infty}=\left|u(x_{u})\right|$ and

[TABLE]

Remark 2.3

If $u\in X(\Omega)$ is a weak solution of (4), then (by taking $\varphi=u$)

[TABLE]

If, in addition, $u\not\equiv 0$ the definition of $\lambda_{\alpha,p}$ yields

[TABLE]

This shows that (8) is a necessary condition for the existence of a nontrivial weak solution.

Proposition 2.4

Suppose that $\alpha,\beta,p$ and $q$ satisfy (6). There exists at least one nonnegative function $u_{\mu}\in X(\Omega)\setminus\left\{0\right\}$ such that

[TABLE]

Proof. Let

[TABLE]

Since $X(\Omega)=W_{0}^{\beta,q}(\Omega)$ we have

[TABLE]

where $h:[0,\infty)\mapsto\mathbb{R}$ is given by

[TABLE]

Noting that $\lim_{t\rightarrow\infty}h(t)=\infty$ and

[TABLE]

we conclude that $E_{\mu}$ is coercive and bounded from below. Hence, by standards arguments of the Calculus of Variations (recall that $X(\Omega)=W_{0}^{\beta,q}(\Omega)\hookrightarrow W_{0}^{\alpha,p}(\Omega)\hookrightarrow\hookrightarrow C_{0}(\overline{\Omega})$) we can show that the functional $E_{\mu}$ assumes the global minimum value at a function $u_{\mu}\in X(\Omega).$

Now, in order to verify that $u_{\mu}\not\equiv 0$ we show that $E_{\mu}(v)<0=E_{\mu}(0)$ for some $v\in W_{0}^{\beta,q}(\Omega).$ Let $u\in W_{0}^{\alpha,p}(\Omega)\setminus\left\{0\right\}$ be such that

[TABLE]

By density and compactness, there exists a sequence $\left\{\varphi_{n}\right\}\subset C_{c}^{\infty}(\Omega)$ such that $\left[\varphi_{n}\right]_{\alpha,p}\rightarrow\left[u\right]_{\alpha,p}$ and $\left\|\varphi_{n}\right\|_{\infty}\rightarrow\left\|u\right\|_{\infty}.$ Therefore, there exists $n_{0}\in\mathbb{N}$ such that

[TABLE]

Since $\varphi_{n_{0}}\in X(\Omega)$ we have

[TABLE]

for some $t>0$ sufficiently small. Thus, $v:=t\varphi_{n_{0}}$ is such that $E_{\mu}(v)<0.$

According to Remark 2.1, $E_{\mu}(\left|u_{\mu}\right|)\leq E_{\mu}(u_{\mu}).$ Therefore, we can assume $u_{\mu}\geq 0$ in $\Omega.$

In the sequence we show that under (6) any minimizer of the energy functional $E_{\mu}$ is a weak solution of (4). For this we need the following result proved in [15].

Lemma 2.5

Let $u\in C(\overline{\Omega})$ and $\Gamma_{u}:=\left\{x\in\Omega:\left|u(x)\right|=\left\|u\right\|_{\infty}\right\}.$ Then,

[TABLE]

Remark 2.6

According to the notation of the Lemma 2.5, if

[TABLE]

for some $L\in\mathbb{R},$ then

[TABLE]

Of course this implies that $\Gamma_{u}$ is a singleton, say $\Gamma_{u}=\left\{x_{u}\right\},$ and therefore Lemma 2.5 yields

[TABLE]

Proposition 2.7

Suppose that $\alpha,\beta,p$ and $q$ satisfy (6). If $u\in X(\Omega)$ satisfies

[TABLE]

then $u$ is a weak solution of (4).

Proof. Let $\varphi\in X(\Omega)$ and $\epsilon>0.$ By hypothesis,

[TABLE]

where

[TABLE]

and

[TABLE]

As we already know (from the Introduction)

[TABLE]

According to Lemma 2.5

[TABLE]

Consequently,

[TABLE]

Now, repeating the above arguments with $\varphi$ replaced with $-\varphi$ we also conclude that

[TABLE]

It follows that (see Remark 2.6) $\Gamma_{u}=\left\{x_{u}\right\}$ and

[TABLE]

Now, let us analyze $E_{\mu}$ under the hypothesis (7). First we observe that $E_{\mu}$ is unbounded from below in $X(\Omega).$ In fact, this follows from the identity (where $e\in W_{0}^{\alpha,p}(\Omega)$ is given in (9))

[TABLE]

Thus, as usual, we look for a minimizer of $E_{\mu}$ restricted to Nehari-type set $\mathcal{N}_{\mu}$ given by (10).

Taking (7) into account, the following properties for a function $u\in X(\Omega)\setminus\left\{0\right\}$ can be easily verified

[TABLE]

and

[TABLE]

The latter property shows that $\mathcal{N}_{\mu}\not=\varnothing,$ since

[TABLE]

Moreover, combining (16) and (18) we obtain,

[TABLE]

for an arbitrary $u\in\mathcal{N}_{\mu}.$ Consequently,

[TABLE]

and

[TABLE]

Another property is that

[TABLE]

which also follows from (16), since

[TABLE]

Proposition 2.8

Suppose that $\alpha,\beta,p$ and $q$ satisfy (7). There exists at least one nonnegative function $u_{\mu}\in W_{0}^{\alpha,p}(\Omega)\setminus\left\{0\right\}$ such that

[TABLE]

Proof. Let $\left\{u_{n}\right\}\in\mathcal{N}_{\mu}$ be a minimizing sequence:

[TABLE]

Taking (20) into account and using compactness arguments, we can assume that $u_{n}$ converges to a function $u_{\mu}\in W_{0}^{\alpha,p}(\Omega)$ uniformly in $C(\overline{\Omega})$ and weakly in both Sobolev spaces $W_{0}^{\alpha,p}(\Omega)$ and $W_{0}^{\beta,q}(\Omega).$ Of course, $u_{\mu}\not\equiv 0$ since

[TABLE]

Hence,

[TABLE]

thus implying that $\theta u_{\mu}\in\mathcal{N}_{\mu},$ where

[TABLE]

Consequently,

[TABLE]

that is, $\theta=1,$ $u_{\mu}\in\mathcal{N}_{\mu}$ and $m_{\mu}=E_{\mu}(u_{\mu}).$

Remark 2.1 and (19) show that $\left|u_{\mu}\right|\in\mathcal{N}_{\mu}$ and $E_{\mu}(\left|u_{\mu}\right|)\leq E_{\mu}(u_{\mu}).$ Thus, we can assume that $u_{\mu}\geq 0$ in $\Omega.$

Proposition 2.9

Suppose that $\alpha,\beta,p$ and $q$ satisfy (7). If $u\in\mathcal{N}_{\mu}$ is such that

[TABLE]

then $u$ is a weak solution of (4).

Proof. Let $\varphi\in X(\Omega)$ be fixed. Since $u\in\mathcal{N}_{\mu}$ we have $\mu\left\|u\right\|_{\infty}^{p}-\left[u\right]_{\alpha,p}^{p}=\left[u\right]_{\beta,q}^{q}>0.$ Thus, by continuity there exists $\epsilon>0$ such that

[TABLE]

It follows that

[TABLE]

where

[TABLE]

Therefore, the function

[TABLE]

assumes a minimum value at $s=0.$ This implies that

[TABLE]

Using Lemma 2.5 and observing that $\tau(0^{+})=1$ and $u\in\mathcal{N}_{\mu}$ we compute

[TABLE]

Hence, (21) yields,

[TABLE]

Replacing $\varphi$ with $-\varphi$ we obtain

[TABLE]

Hence, according to Remark 2.6, $\Gamma_{u}=\left\{x_{u}\right\}$ and

[TABLE]

We gather the results above in the following theorem.

Theorem 2.10

Suppose that $\alpha,\beta,p$ and $q$ satisfy either (6) or (7), and that $\mu$ satisfies (8). Then (4) has at least one nonnegative least energy solution $u_{\mu}\in X(\Omega)\setminus\left\{0\right\}.$

We remark that $u_{\mu}\in X(\Omega)\setminus\left\{0\right\}$ given by Theorem 2.10 is a nonnegative weak solution of the fractional harmonic-type equation

[TABLE]

in the punctured domain $\Omega\setminus\left\{x_{u}\right\},$ since

[TABLE]

Consequently, if $p>\frac{1}{1-\alpha}$ and $q>\frac{1}{1-\beta}$ (see Remark 2.11) one can adapt the arguments developed in [14, Lemma 3.9] and [16, Proposition 11] to verify that $u_{\mu}$ is also a viscosity solution of

[TABLE]

(recall the definition of $\mathcal{L}_{s,m}$ in (3)). This means that $u_{\mu}$ is both a supersolution and a subsolution of (24), that is, $u_{\mu}$ meets the (respective) requirements:

• •

$(\mathcal{L}_{\alpha,p}\varphi)(x_{0})+(\mathcal{L}_{\beta,q}\varphi)(x_{0})\leq 0$ for every pair $\left(x_{0},\varphi\right)\in(\Omega\setminus\left\{x_{u_{\mu}}\right\})\times C_{c}^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

• •

$(\mathcal{L}_{\alpha,p}\varphi)(x_{0})+(\mathcal{L}_{\beta,q}\varphi)(x_{0})\geq 0$ for every pair $\left(x_{0},\varphi\right)\in D\times C_{c}^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

Remark 2.11

As observed in [16], if $D$ is a bounded domain of $\mathbb{R}^{N},$ $m>\frac{1}{s-1}$ and $\varphi\in C_{c}^{1}(\mathbb{R}^{N}),$ then the function $\mathcal{L}_{s,m}\varphi$ given by (3) is well defined and continuous at each point $x\in D.$ Obviously, the same holds for $\psi=\varphi+k,$ where $k$ is an arbitrary constant and $\varphi\in C_{c}^{1}(\mathbb{R}^{N}),$ since

[TABLE]

Moreover, it is simple to check that $u_{\mu}$ fulfills both requirements above even for test functions of the form $\psi=\varphi+k.$

It is interesting to notice that $u_{\mu}>0$ in $\Omega\setminus\left\{x_{u_{\mu}}\right\}$ as consequence of $u_{\mu}$ being a supersolution of (24). The argument comes from [16, Lemma 12]: by supposing that $u_{\mu}(x_{0})=0$ for some $x_{0}\in\Omega\setminus\left\{x_{u_{\mu}}\right\}$ and noting that $0\not\equiv u\geq 0,$ we can find a nonnegative and nontrivial test function $\varphi\in C_{c}^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

Hence,

[TABLE]

which leads to the contradiction $\varphi\equiv 0.$

3 Asymptotic behavior as p goes to infinity

Let $D$ be a bounded smooth (at least Lipschitz) domain of $\mathbb{R}^{N}.$ We recall that $\left(C_{0}^{0,s}(\overline{D}),\left|\cdot\right|_{s}\right)$ is a Banach space, but

[TABLE]

That is, $C_{c}^{\infty}(D)$ is not $\left|\cdot\right|_{s}$-dense in $C_{0}^{0,s}(\overline{D}).$

However, we have the following lemma that follows from [13, Lemma 9].

Lemma 3.1

Let $v\in C_{0}^{0,s}(\overline{D}).$ There exists a sequence $\left\{v_{k}\right\}\subset C_{c}^{\infty}(D)$ such that

[TABLE]

Now, returning to our bounded domain $\Omega,$ let

[TABLE]

It is the inradius of $\Omega$: the radius of the largest ball inscribed in $\Omega.$

Let $B_{R}(x_{0})$ be a ball centered at $x_{0}\in\Omega$ with radius $R$ and let $\phi_{R}:\overline{B_{R}(x_{0})}\rightarrow[0,R]$ be the distance function to the boundary $\partial B_{R}(x_{0}),$ that is,

[TABLE]

It is simple to verify that $\phi_{R}\in C_{0}^{0,s}(\overline{B_{R}(x_{0})}),$ for every $s\in(0,1],$ with

[TABLE]

Moreover, it is clear that $\phi_{R}$ extended by zero outside $B_{R}(x_{0})$ belongs to $C_{0}^{0,s}(\overline{\Omega})$ and its $s$-Hölder seminorm is preserved. In particular, such an extension is a Lipschitz function vanishing outside $\Omega.$ Hence,

[TABLE]

(Note that we are considering $\Omega$ at least a Lipschitz domain.) Consequently, we can apply [13, Lemma 7] to conclude that

[TABLE]

The proof of the following proposition is adapted from [16] where (14) is proved.

Proposition 3.2

For each $s\in(0,1]$ one has

[TABLE]

Proof. The second equality in (27) follows from (25). Since $\phi_{R}\in C_{0}^{0,s}(\overline{\Omega})\setminus\left\{0\right\}$ to prove the third equality in (27) it suffices to verify that

[TABLE]

Let $v\in C_{0}^{0,s}(\overline{\Omega})\setminus\left\{0\right\}.$ According to Lemma 3.1, there exists a sequence $\left\{v_{k}\right\}\subset C_{c}^{\infty}(\Omega)$ such that

[TABLE]

Hence, (14) yields

[TABLE]

concluding the proof of the third equality in (27)

Now, let us prove that

[TABLE]

First, observing that

[TABLE]

we obtain from (25) and (26) that

[TABLE]

To prove that

[TABLE]

we fix $m_{0}>\frac{N}{s}$ and take, for each $m$ sufficiently large, $u_{m}\in W_{0}^{s,m}(\Omega)$ such that $\left\|u_{m}\right\|_{\infty}=1$ and

[TABLE]

According to (2), we have

[TABLE]

The estimate (29) implies that $\left\{u_{m}\right\}$ is uniformly bounded in the Hölder space $C_{0}^{0,\beta-\frac{N}{m_{0}}}(\overline{\Omega}),$ which is compactly embedded in $C_{0}(\overline{\Omega}).$ It follows that, up to a subsequence, $\left\{u_{m}\right\}$ converges uniformly in $\overline{\Omega}$ to a function $u\in C_{0}(\overline{\Omega})$ such that $\left\|u\right\|_{\infty}=1.$

For each $1<k<m,$ we have, by Hölder’s inequality,

[TABLE]

Making $m\rightarrow\infty$, using the uniform convergence, Fatou’s Lemma and the above estimate we obtain

[TABLE]

Therefore,

[TABLE]

Since $R^{-s}\leq\left|u\right|_{s}$ (according to (28)) we obtain (30).

In the remaining of this section we fix $\alpha,\beta\in(0,1),$ with $\alpha\not=\beta,$ and consider $q$ a continuous function of $p$ satisfying

[TABLE]

We maintain the notation $q$ instead of $q(p)$ to simplify the presentation. Note that (31) implies that

[TABLE]

Moreover, $q<p$ if $Q\in(0,1)$ and $p<q$ if $Q\in(1,\infty).$

Our goal is to study the asymptotic behavior, as $p\rightarrow\infty,$ of the least energy solution $u_{p}$ of the problem

[TABLE]

where $\mu_{p}$ satisfies

[TABLE]

with $R$ denoting the inradius of $\Omega.$

This condition guarantees that

[TABLE]

for all $p$ sufficiently large, say $p>p_{0}$. Moreover, by taking a larger $p_{0}$ one of the conditions (6) or (7) is fulfilled. So, according to Theorem 2.10, for each $p>p_{0}$ the problem (32) has at least one positive least energy solution

[TABLE]

Remark 3.3

Combining (26) and (33) we have

[TABLE]

Consequently, $\mu_{p}\left\|\phi_{R}\right\|_{\infty}^{p}>\left[\phi_{R}\right]_{\alpha,p}^{p}$ for all $p$ large enough.

Proposition 3.4

Suppose (31) and (33) hold. Then,

[TABLE]

Proof. We assume that $p$ is large enough so that $u_{p}$ exists according to Theorem 2.10.

Since $u_{p}$ is a weak solution of (32) and $W_{0}^{\beta,q}(\Omega)$ is continuously embedded into $C(\overline{\Omega})$ we have

[TABLE]

so that

[TABLE]

Hence, taking into account the first equality in (27) and (33) we easily check that the second limit in (35) is consequence of the first one.

Let us then prove the first limit (35).

We start with the case $Q\in(1,\infty),$ where necessarily $p<q$ (and $0<\alpha<\beta$). After isolating $\left[u_{p}\right]_{\beta,q}$ in (36) we obtain

[TABLE]

Let

[TABLE]

(Note from Remark 3.3 that $t$ is well-defined). It is simple to verify that

[TABLE]

Noticing that

[TABLE]

we obtain

[TABLE]

where

[TABLE]

Since

[TABLE]

we can verify that

[TABLE]

Hence,

[TABLE]

Combining this with (37) we obtain the first limit in (35).

Now, let us analyze the case $Q\in(0,1),$ where necessarily $q<p$ (and $0<\beta<\alpha$). In this case,

[TABLE]

where

[TABLE]

(which is also well-defined according to Remark 3.3). It follows that

[TABLE]

where $a_{p}$ is also given by (38). Consequently,

[TABLE]

After isolating $\left[u_{p}\right]_{\beta,q}$ in (36) we obtain

[TABLE]

which combined with (39) provides the first limit in (35).

Corollary 3.5

Suppose (31) and (33) hold. Then,

[TABLE]

Proof. It follows from the second limit in (35) combined with the estimates

[TABLE]

In the next proposition we prove that the limit functions of the family $\left\{u_{p}\right\}_{p>p_{0}},$ as $p\rightarrow\infty,$ belongs to $C_{0}^{0,\beta}(\overline{\Omega})$ and minimize the quotient $\left|u\right|_{\beta}/\left\|u\right\|_{\infty}$ in $C_{0}^{0,\beta}(\overline{\Omega})\setminus\left\{0\right\}.$

Proposition 3.6

Let $\left\{p_{n}\right\}$ and $\left\{q_{n}\right\}$ satisfying (31), with $p_{n}\rightarrow\infty,$ and let $\mu_{n}:=\mu_{p_{n}}$ satisfying (33). Then, there exist $u_{\infty}\in C_{0}^{0,\beta}(\overline{\Omega})$ and $x_{\infty}\in\Omega$ such that, up to subsequences, $u_{p_{n}}\rightarrow u_{\infty}$ uniformly in $\overline{\Omega}$ and $x_{p_{n}}\rightarrow x_{\infty}\in\Omega,$ with

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

Proof. Since $\Omega$ is bounded, we can assume that (passing to a subsequence) $x_{p_{n}}$ converges to a point $x_{\infty}\in\overline{\Omega}.$ Fix m_{0}>N/\beta\and assume that $n$ is large enough so that $m_{0}<\left\{p_{n},q_{n}\right\}.$

Taking into account the inequality (2), we have (as in Proposition 3.2)

[TABLE]

The first limit in (35) implies that $\left\{u_{p_{n}}\right\}$ is uniformly bounded in the Hölder space $C_{0}^{0,\beta-\frac{N}{m_{0}}}(\overline{\Omega}),$ which is compactly embedded in $C_{0}(\overline{\Omega}).$ It follows that, up to a subsequence, $\left\{u_{p_{n}}\right\}$ converges uniformly in $\overline{\Omega}$ to a function $u_{\infty}\in C_{0}(\overline{\Omega}).$ Of course, $\left\|u_{\infty}\right\|=u_{\infty}(x_{\infty})$ and, by virtue of the second limit in (35),

[TABLE]

so that $x_{\infty}\not\in\partial\Omega.$

Now, if $m>m_{0}$ and $n$ is sufficiently large such that $q_{n}>m$, Hölder’s inequality yields

[TABLE]

Hence, combining the first limit in (35) and Fatou’s Lemma,

[TABLE]

Therefore,

[TABLE]

It follows that $u_{\infty}\in C_{0}^{0,\beta}(\overline{\Omega}).$ Hence, observing that

[TABLE]

we obtain

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

Remark 3.7

Considering Corollary 3.5 we can reproduce the proof of Proposition 3.6 to conclude that, in the case $Q\in(0,1),$ the limit function is more regular: $u_{\infty}\in C_{0}^{0,\alpha}(\overline{\Omega})$ and, moreover,

[TABLE]

These estimates are also valid in the complementary case $Q\in(1,\infty),$ where obviously the $\beta$-regularity is better that $\alpha$-regularity since $0<\alpha<\beta.$

Corollary 3.8

One has

[TABLE]

and, therefore, the maximum point $x_{\infty}$ of $u_{\infty}$ is also a maximum point of the distance function to the boundary $\partial\Omega.$

Proof. For each $x\in\Omega$ let $y_{x}\in\partial\Omega$ be such

[TABLE]

Then, since $u_{\infty}(y_{x})=0$ and $\left|u_{\infty}\right|_{\beta}=\left(\Lambda R^{\beta}\right)^{\frac{1}{Q-1}},$ we get

[TABLE]

Hence, observing that $\operatorname{dist}(x,\partial\Omega)=\left|x-y_{x}\right|\leq R$ and $u_{\infty}(x_{\infty})=R^{\beta}(\Lambda R^{\beta})^{\frac{1}{Q-1}},$ we obtain

[TABLE]

so that

[TABLE]

In the sequel, we argue that the function $u_{\infty}$ is a viscosity solution of the equation

[TABLE]

in $\Omega\setminus\left\{x_{\infty}\right\}$ (the operators $\mathcal{L}_{\alpha}^{+}$ and $\mathcal{L}_{\alpha}^{-}$ are defined in (12)). This means that $u_{\infty}$ is both a supersolution and a subsolution of (41) or, equivalently, $u_{\infty}$ meets the (respective) requirements:

• •

$\max\left\{\left(\mathcal{L}_{\alpha}^{+}\varphi\right)(x_{0}),\left(\left(\mathcal{L}_{\beta}^{+}\varphi\right)(x_{0})\right)^{Q}\right\}\leq\max\left\{\left(-\mathcal{L}_{\alpha}^{-}\varphi\right)(x_{0}),\left(\left(-\mathcal{L}_{\beta}^{-}\varphi\right)(x_{0})\right)^{Q}\right\}$ for every the pair $\left(x_{0},\varphi\right)\in(\Omega\setminus\left\{x_{\infty}\right\})\times C_{c}^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

• •

$\max\left\{\left(\mathcal{L}_{\alpha}^{+}\varphi\right)(x_{0}),\left(\left(\mathcal{L}_{\beta}^{+}\varphi\right)(x_{0})\right)^{Q}\right\}\geq\max\left\{\left(-\mathcal{L}_{\alpha}^{-}\varphi\right)(x_{0}),\left(\left(-\mathcal{L}_{\beta}^{-}\varphi\right)(x_{0})\right)^{Q}\right\}$ for every the pair $\left(x_{0},\varphi\right)\in(\Omega\setminus\left\{x_{\infty}\right\})\times C_{c}^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

A proof of the following result (where $t^{\pm}=\max\left\{\pm t,0\right\}$), adapted from [5, Lemma 6.5], can be found in [14, Lemma 6.1].

Lemma 3.9

If $s\in(0,1),$ $\varphi\in C_{c}^{1}(\mathbb{R}^{N}),$ $\lim\limits_{m\rightarrow\infty}z_{m}\rightarrow x,$ then,

[TABLE]

where

[TABLE]

and

[TABLE]

Proposition 3.10

The function $u_{\infty}$ is a viscosity solution of (41) in the punctured domain $\Omega\setminus\left\{x_{\infty}\right\}.$ Moreover, $u_{\infty}>0$ in $\Omega.$

Proof. We give a sketch of the proof based on [14] and [16].

In order to verify that $u_{\infty}$ is a supersolution of (41) in $\Omega\setminus\left\{x_{\infty}\right\}$ we fix a pair $\left(x_{0},\varphi\right)\in(\Omega\setminus\left\{x_{\infty}\right\})\times C_{c}^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

Since $x_{0}\not=x_{\infty}=\lim x_{n},$ we can assume that there exist $n_{0}\in\mathbb{N}$ and a ball $B_{\rho}(x_{0}),$ centered at $x_{0}$ and with radius $\rho,$ such that

[TABLE]

Hence,

[TABLE]

in the viscosity sense.

By standard arguments, we can construct a sequence $\left\{z_{n}\right\}\subset B_{\rho}(x_{0})$ such that $z_{n}\rightarrow x_{0}$ and

[TABLE]

It follows that the function $\psi_{n}:=\varphi+k_{n}$ satisfies

[TABLE]

Consequently, (see Remark 2.11)

[TABLE]

The inequality can be write as

[TABLE]

where $A_{p_{n}}:=A_{p_{n}}(\varphi(z_{n})),$ $A_{q_{n}}:=A_{q_{n}}(\varphi(z_{n})),$ $B_{p_{n}}:=B_{p_{n}}(\varphi(z_{n}))$ and $B_{q_{n}}:=B_{q_{n}}(\varphi(z_{n})).$

We have

[TABLE]

where the latter equality follows from Lemma 3.9. Analogously, we compute

[TABLE]

Therefore, (43) yields

[TABLE]

which shows that $u_{\infty}$ is a viscosity supersolution of (41) in $\Omega\setminus\left\{x_{\infty}\right\}.$

Similarly, by symmetric arguments, we can prove that $u_{\infty}$ is a viscosity subsolution of (41) in $\Omega\setminus\left\{x_{\infty}\right\}.$

The positivity of $u_{\infty}$ in $\Omega$ comes from the fact that u_{\infty}\is a supersolution of (41). Indeed, adapting the argument of [16, Lemma 22], if $u_{\infty}(x_{0})=0,$ then either

[TABLE]

for a nonnegative, nontrivial $\varphi\in C_{c}^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

In the first case, this yields

[TABLE]

and leads to the contradiction $\varphi\equiv 0.$ Obviously, in the second case we arrive at the same contradiction.

Proof of Theorem 1.1. It follows by gathering Proposition 3.6, Corollary 3.8 and Proposition 3.10.

4 Acknowledgements

G. Ercole was partially supported by CNPq/Brazil (306815/2017-6 and 422806/2018-8) and Fapemig/Brazil (CEX-PPM-00137-18).