A prescribed scalar curvature-type equation: Almost critical manifolds and multiple solutions.
Let \(\Omega\) be a smooth bounded open set in \(\mathbb R^N\), \(N\geq 3\), \(a\in C^2(\overline{\Omega})\). The authors study the asymptotic behavior and existence of multiple solutions, as \(\delta\to 0\), for the perturbed boundary value problem: \(-\Delta u=(1+\delta a)u^{(N+2)/(N-2)}\) and \(u>0\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). Among other things it is established the non-existence in low dimensions of one-peak solutions (i. e. with energy close to \(S^{N/2}\), where \(S\) stands for the best Sobolev constant). The approach relies on an extension of Lyapunov-Schmidt-type finite-dimensional reduction.
nonlinear ellptic boundary value problem
scientific article; zbMATH DE number 2037131
scientific article