Topological method for a class of the semilinear elliptic systems (Q2786414)

The article considers the semilinear elliptic system NEWLINE\[NEWLINE -\Delta U = \text{grad}_{U} H(x,U) \text{ in } \Omega, NEWLINE\]NEWLINE for \(\Omega \subset \mathbb{R}^n\) and \(U: \Omega \to \mathbb{R}^n\) with Dirichlet boundary conditions. The existence of two non-trivial weak solutions is proved using topological methods. In order to do this, the authors first assert that critical points of the associated functional indeed solve the given system in a weak sense. The authors then follow the cited article [\textit{A. M. Micheletti} and \textit{C. Saccon}, J. Differ. Equations 170, No. 1, 157--179 (2001; Zbl 1097.58010)] using the limit relative category of G. Fournier, D. Lupo, M. Ramos, and M. Willem in order to show existence of two critical points under certain conditions on \(H\) (most prominently, that the Hessian of \(H\) must be bounded between two adjacent eigenvalues of the Laplacian on the domain in the sense that \(\lambda_k \text{Id} < \text{Hess}_U H < \lambda_{k+1} \text{Id})\).