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

Abstract

We prove the existence of ground state solution to the following problem. \begin{align*} (-Δ)^{s}u+u&=λ|u|^{-γ-1}u+P(x)|u|^{p-1}u,~\text{in}~\mathbb{R}^N\setminusΩ\\ N_su(x)&=0,~\text{in}~Ω\end{align*} where $N\ge 2$, $\lambda >0$, $0<s,\gamma <1$, $p\in (1,2_s^{\ast }-1)$ with $2_s^{\ast }=\frac{2N}{N-2s}$. % $0<s^{-}=\underset{(x,y)\in \Omega \times \Omega }{\inf }\{s(x,y)\}\le s(x,y)\le s^{+}=\underset{(x,y)\in \Omega \times \Omega }{\sup }\{s(x,y)\}<1$, $0<{\gamma }^{-}=\underset{x\in \Omega }{\inf }\{\gamma (x)\}\le \gamma (x)\le {\gamma }^{+}=\underset{x\in \Omega }{\sup }\{\gamma (x)\}<1$, $1-{\gamma }^{-}<1<p^{-}=\underset{x\in \Omega }{\inf }\{p(x)\}\le p(x)\le p^{+}=\underset{x\in \Omega }{\sup }\{p(x)\}<2_{s^{-}}^{\ast }=\underset{x\in \Omega }{\inf }\{2_s^{\ast }(x)\}$ with $2_s^{\ast }(x)=\frac{2N}{N-2\tilde{s}(s)}$ 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\setminusΩ}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy,~x\inΩ. \end{align*} We further establish the existence of infinitely many bounded solutions to the problem.