Ljusternik-Schnirelmann theorem for the generalized Laplacian
The author proves a Ljusternik-Schnirelman theorem for the generalized Laplacian \(\Delta_m (u)= \nabla ((m(|\nabla u|)/|\nabla u|) \nabla u)\). Consider the problem (1) \(-\Delta_m(u)= \lambda g(u)\) in a bounded subset \(\Omega \subset {\mathbb R}^N\) with the segment property and \(u=0\) on \(\partial \Omega\). The function \(m\colon [0,\infty[\to [0,\infty[\) is supposed to be increasing and continuous such that \(m(t)=0\) if and only if \(t=0\) and \(m(t)\to\infty\) as \(t\to\infty\); the function \(g\colon {\mathbb R}\to {\mathbb R}\) is odd and satisfies some growth condition. Using a Galerkin approximation, the pseudomonotonicity of \(-\Delta_m\) and a modified usual structure of a complementary system of Orlicz-Sobolev spaces, it is proved that (1) admits an infinite sequence of eigenvalues that tends to infinity.
scientific article; zbMATH DE number 1437350
scientific article