A note on the positive solutions for critical elliptic problems in exterior domains

We consider the existence of positive solutions for the elliptic Dirichlet problem $\left(\begin{array}{cc}-\Delta u+u=f\left(u\right)& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega ,\end{array}\right)$ where $\Omega \subset R^N$ ($N\ge 3$) is an exterior domain with smooth boundary $\partial \Omega \ne 0̸$ such that $R^N\setminus \Omega$ is bounded. If $f$ involves critical growth, the existing work only covers that $f\left(u\right)={\left|u\right|}^{p-2}u+\varepsilon {\left|u\right|}^{2^{\ast }-2}u$, where $2^{\ast }=\frac{2N}{N-2}$ is the Sobolev critical exponent and $\varepsilon >0$ is a sufficiently small parameter. In present paper, we remove this parameter, i.e., $f\left(u\right)={\left|u\right|}^{p-2}u+{\left|u\right|}^{2^{\ast }-2}u$, and establish the existence of positive solutions when $3\le N\le 6$ and $\frac{N+2}{N-2}\le p<2^{\ast }$ with $p>2$.