Concentration estimates and multiple solutions to elliptic problems at critical growth

Abstract

In this paper, we consider the problem $-\Delta u =|u| ^{2^*-2}u+\lambda u$ in $\Omega$, $u= 0$ on $\partial \Omega$, where $\Omega$ is an open regular bounded subset of $\mathbb R^N$ $(N\geq 3)$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\lambda>0$. Our main result asserts that, if $N\geq 7$, the problem has infinitely many solutions and, from the point of view of the compactness arguments employed here, the restriction on the dimension $N$ cannot be weakened.