Multiplicity of positive solutions for a class of nonlocal problem involving critical exponent

Abstract

In this paper, we study the following critical nonlocal problem \begin{equation*} \begin{cases} -\left(a-\lambda b\displaystyle\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\lambda |u|^{p-2}u+Q(x)|u|^{2}u, & x\in\Omega,\\ u=0, & x\in\partial\Omega, \end{cases} \end{equation*} where $a>0$, $b\ge0$, $20$ is a parameter, $\Omega$ is a smooth bounded domain in $\mathbb{R}^4$ and $Q(x)\in C({\overline{\Omega}})$ is a nonnegative function. By virtue of variational methods and delicate estimates, we prove that problem admits $k$ positive solutions for $\lambda>0$ sufficiently small, provided that the maximum of $Q(x)$ is achieved at $k$ interior points in $\Omega$.

Keywords

• nonlocal problem
• variational methods
• critical nonlinearity
• multiple positive solutions