Concentrating solutions for a biharmonic problem with supercritical growth

Abstract

In this paper we consider the following supercritical biharmonic problem:$$\begin{cases}\Delta^2 u= K(x)u^{p+\epsilon}&\text{in } \Omega,\\u> 0 &\text{in }\Omega,\\u=\Delta u=0&\text{on }\partial\Omega,\end{cases}$$where $K(x)\in C^3(\overline{\Omega})$ is a nonnegative function, $p=({N+4})/({N-4})$, $\epsilon> 0$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq6$. We show that, for $\epsilon$ small enough, there exists a family of concentrating solutions under certain assumptions on the critical points of the function $K(x)$.