Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three

Abstract

In this paper, we are interested in the following nonlocal problem with critical exponent \begin{align*} \begin{cases} -\left(a-b\displaystyle\int_{\Omega}|\nabla u|^2{\rm d}x\right)\Delta u=\lambda |u|^{p-2}u+|u|^{4}u, &\quad x\in\Omega,\\ u=0,  &\quad x\in\partial\Omega, \end{cases} \end{align*} where $a,b$ are positive constants, $20$ is a parameter. By variational methods, we prove that problem has a positive ground state solution $u_b$ for $\lambda>0$ sufficiently large. Moreover, we take $b$ as a parameter and study the asymptotic behavior of $u_b$ when $b\searrow0$.