Elliptic problem in an exterior domain driven by a singularity with a nonlocal Neumann condition

TL;DR

This paper proves the existence of ground state and infinitely many bounded solutions for a fractional elliptic problem with a singularity and a nonlocal Neumann boundary condition in an exterior domain.

Contribution

It introduces new existence results for solutions to a fractional elliptic problem with singular and nonlocal boundary conditions, extending previous work to exterior domains.

Findings

Existence of ground state solutions under specified conditions.

Existence of infinitely many bounded solutions.

Application of nonlocal Neumann boundary conditions in fractional problems.

Abstract

We prove the existence of ground state solution to the following problem. \begin{align*} (-\Delta)^{s}u+u&=\lambda|u|^{-\gamma-1}u+P(x)|u|^{p-1}u,~\text{in}~\mathbb{R}^N\setminus\Omega\\ N_su(x)&=0,~\text{in}~\Omega \end{align*} where $N\geq2$, $\lambda>0$, $0<s,\gamma<1$, $p\in(1,2_s^*-1)$ with $2_s^*=\frac{2N}{N-2s}$. % $0<s^-=\underset{{(x,y)\in\Omega\times\Omega}}{\inf}\{s(x,y)\}\leq s(x,y)\leq s^+=\underset{{(x,y)\in\Omega\times\Omega}}{\sup}\{s(x,y)\}<1$, $0<\gamma^-=\underset{{x\in\Omega}}{\inf}\{\gamma(x)\}\leq \gamma(x)\leq \gamma^+=\underset{{x\in\Omega}}{\sup}\{\gamma(x)\}<1$, $1-\gamma^-<1<p^-=\underset{x\in\Omega}{\inf}\{p(x)\}\leq p(x)\leq p^+=\underset{x\in\Omega}{\sup}\{p(x)\}<2_{s^-}^*=\underset{{x\in\Omega}}{\inf}\{2_s^*(x)\}$ with $2_s^*(x)=\frac{2N}{N-2\tilde{s}(s)}$ where $\tilde{s}(x)=s(x,x)$. Moreover, $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $(-\Delta)^s$ denotes the $s$-fractional Laplacian and finally $N_s$ denotes the nonlocal operator that describes the Neumann boundary condition which is given as follows. \begin{align*} N_{s}u(x)&=C_{N,s}\int_{\mathbb{R}^N\setminus\Omega}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy,~x\in\Omega. \end{align*} We further establish the existence of infinitely many bounded solutions to the problem.