On the existence of multiple, single-peaked solutions for a semilinear Neumann problem (Q1261970)

This paper is concerned with positive solutions of the following semilinear elliptic equation subject to homogeneous Neumann boundary conditions \[ -d\Delta u+u=| u |^{p-2}u \text{ in }\Omega,\quad \partial u/ \partial\nu=0 \text{ on } \partial \Omega \tag{I} \] where \(d\) is a positive constant, \(\Omega\) is a bounded domain in \(\mathbb{R}^ N\) \((N \geq 2)\) with a smooth boundary, and \(\nu\) is the unit outer normal to \(\partial \Omega\). \(p\) satisfies \(2<p<2N/(N-2)\) if \(N \geq 3\), and \(2<p<+\infty\) if \(N=2\). The goal of this paper is to establish a multiplicity result on the existence of nonconstant positive solutions of (I) and to show how the number of positive solutions is affected by the topology of \(\Omega\) or, more precisely, of \(\partial \Omega\). Moreover, it is proved that all solutions obtained have the property that each solution has at most one local maximum over \(\overline\Omega\), which is achieved at a point on the boundary of \(\Omega\). \textit{G. Mancini} and \textit{R. Musina} (Preprint) have obtained the same existence result. Meanwhile the author has generalized this result to exterior domain problems [see the following review] and to critical exponent problems in bounded domains (Preprint).