A variational viewpoint on the existence of three solutions for some nonlinear elliptic problems
The problem studied is \[ \Delta u+ g(x, u)= h\quad \text{in} \quad \Omega,\quad u= 0\quad\text{on} \quad \partial\Omega,\tag{P} \] where \(\Omega\) is an open bounded subset of \(\mathbb{R}^n\), \(h: \Omega\mapsto \mathbb{R}\) is a given function and \(g: \Omega\times \mathbb{R}\mapsto \mathbb{R}\) has asymptotically nonsymmetric behaviour, i.e., is such that \[ \alpha= \lim_{s\to + \infty} g(x, s)/s\quad \text{and} \quad \beta= \lim_{s\to - \infty} g(x, s)/s \] and some eigenvalues \(\lambda_k\) of the equation \(\Delta u+ \lambda u= 0\) with \(u\) in \(H^1_0(\Omega)\) fall between \(\alpha\) and \(\beta\), and \(e_k\) the associated eigenfunction, with \(e_1> 0\) in \(\Omega\). Under the weaker assumption that \(g: \Omega\times \mathbb{R}\mapsto \mathbb{R}\) is only a Carathéodory function, the existence of at least three solutions of (P) is proved. The prove is based on a very simple variational argument with \(h= h_0+ te_1\). Some regions in the \((\alpha, \beta)\) plane, where the number of solutions is at least three for all large positive \(t\), are described.
scientific article; zbMATH DE number 896472