Nontrivial solutions for some semilinear problems and applications to wave equations on balls and spheres
The abstract nonlinear equation \(Au=N(u)\) is studied in this paper. Here, \(A\) is a closed linear operator in a Hilbert space \(H\) satisfying some additional conditions and \(N\) is the (nonlinear) Nemitskii operator associated to a Carathéodory function \(g(x,s)\). The main assumption is that one simple eigenvalue of \(A\) crosses the function \(g(x,s)/s\) when \(s\) goes from \(-\infty\) to \(+\infty\). It is proved by using a continuation argument due to Berkovits and Mustonen that the above equation has at least two nontrivial solutions. This allows linear operators \(A\) which are not selfadjoint. Applications are given to both some semilinear wave equations on balls and spherical wave equations. Results depend on the space dimension \(n\), and the cases \(n=1,3\) are rather special.
scientific article; zbMATH DE number 807920