The uniqueness of symmetric single peak solutions for a Neumann problem involving critical exponent.
Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^N\), \(N\geq 5\). This paper is devoted to the semilinear Neumann boundary value problem \(-\Delta u+\lambda u=u^{2^*-1}+au^{q-1}\) in \(\Omega\), \(u>0\) in \(\Omega\), \(\partial u/\partial\nu =0\) on \(\partial\Omega\), where: \(\lambda>0\) is a large constant, \(a\) is a fixed positive constant, \(2^*=2N/(N-2)\) denotes the critical Sobolev exponent, and \(2<q<2^*\). The main result of the paper establishes the existence of a unique symmetric single peak solution. The proof is based on a constructive method and adequate elliptic estimates.
scientific article; zbMATH DE number 2037217