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

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 $N\ge 2$, $\lambda >0$, $0<s,\gamma <1$, $p\in (1,2_s^{\ast }-1)$ with $2_s^{\ast }=\frac{2N}{N-2s}$. % $0<s^-=\underset{{(x,y)\in\Omega\times\Omega}}{\inf}{s(x,y)}\leq s(x,y)\leq s^+=\underset{{(x,y)\in\Omega\times\Omega}}{\sup}{s(x,y)}<1$,$0<\gamma^-=\underset{{x\in\Omega}}{\inf}{\gamma(x)}\leq \gamma(x)\leq \gamma^+=\underset{{x\in\Omega}}{\sup}{\gamma(x)}<1$,$1-\gamma^-<1<p^-=\underset{x\in\Omega}{\inf}{p(x)}\leq p(x)\leq p^+=\underset{x\in\Omega}{\sup}{p(x)}<2_{s^-}^=\underset{{x\in\Omega}}{\inf}{2_s^(x)}$with$2_s^(x)=\frac{2N}{N-2\tilde{s}(s)}$where$\tilde{s}(x)=s(x,x)$. Moreover, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $(-\mathrm{\Delta }{)}^s$ 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.