Existence of two nontrivial solutions for a class of elliptic eigenvalue problems (Q1580900)

Given a bounded domain \(\Omega\) in \({\mathbb R}^n\), \(n \geq 3\) with a regular boundary \(\partial \Omega\), the following family \((P_{\lambda, \mu})\) of elliptic eigenvalue problems is considered: \[ -\Delta u = \lambda (f(u) + \mu g(u)) \quad \text{in }\Omega, \qquad u = 0 \quad \text{on }\partial \Omega \] where \(\lambda\), \(\mu \in \mathbb R\) and \(f, g: {\mathbb R} \to {\mathbb R}\) are continuous. The main result gives sufficient conditions on \(f\) and \(g\) under which \((P_{\lambda, \mu})\) has at least two weak solutions in \(H^1_0(\Omega)\). The proof is based on the mountain pass theorem and suitable inequalities involving \(\lambda\) and \(\mu\).