Existence of positive solutions for nonlinear elliptic equations with critical growth on exterior domains

In this paper, we prove the existence of solutions for nonlinear elliptic equation with critical growth $\left(\begin{array}{ccc}& {(-\Delta )}^{s}u+V\left(x\right)u={\left|u\right|}^{2_s^{\ast }-2}u& \text{in}\Omega ,\\ & u=0& R^N\setminus \Omega ,\end{array}\right)$ where $s\in (0,1],N>2s$, and $2_s^{\ast }=\frac{2N}{N-2s}$ is the critical Sobolev exponent. Here, $V\in L^{\frac{N}{2s}}\left(\Omega \right)$ is a given nonnegative potential, ${(-\Delta )}^s$ denotes the fractional Laplace operator, $\Omega \subset R^N$ is an exterior domain with smooth boundary $\partial \Omega \ne 0̸$ such that $R^N\setminus \Omega$ non-empty and bounded. Using barycentric functions and the minimax method, we establish the existence of a positive solution $u\in D_0^{s,2}\left(\Omega \right)$ under the assumption that $R^N\setminus \Omega$ is contained in a sufficiently small ball.