Arbitrary Number of Positive Solutions For an Elliptic Problem with Critical Nonlinearity

Abstract

We show that the critical nonlinear elliptic Neumann problem

\[ \Delta u -\mu u + u^{7/3} = 0 \ \ \mbox{in} \ \Om, \ \ u >0 \ \mbox{in} \ \Om \ \mbox{and} \ \frac{ \partial u}{\partial \nu} = 0 \ \ \mbox{on} \ \partial \Om \]

where \( \Om \) is a bounded and smooth domain in ${\mathbb{R}}^5$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists ${\mu }_K>0$ such that for $0<\mu <{\mu }_K$ , the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega$, as $\mu \to 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.