The effect of geometry of the domain boundary in an elliptic Neumann problem

Abstract

In this paper we consider the problem $$ \begin{cases} -\Delta u + \mu u= u^{2^* -1 -{\varepsilon}} \quad \hbox{in } \Omega \\ u>0 \quad \hbox{in } \Omega \ \quad \frac{\partial u}{\partial \nu} =0 \quad \hbox{on } \partial \Omega \end{cases} $$ where $\Omega$ is a bounded, smooth domain in ${\mathbb R}^N (N\geq 3),\ 2^* = \frac{2N}{N-2}, \ \mu>0, \ {\varepsilon} \in [0, \frac{4}{N-2})$, $\nu$ is the unit outward normal vector to $\partial \Omega$. We show the topological effect of superlevel and/or sublevel sets of the function of mean curvature of $\partial \Omega$ on the number of one- or multipeak positive solutions as ${\varepsilon} \to 0$ for fixed $\mu$ and $\mu \to +\infty$ for ${\varepsilon} =0$. The solutions obtained concentrate at some boundary points with negative mean curvature (for ${\varepsilon}>0 $ small) or positive mean curvature (for ${\varepsilon}=0, \ \mu$ large).