An application of a mountain pass theorem
The present paper is devoted to study the following Dirichlet problem: \[ -\Delta u=f(x,u), \quad x\in\Omega,\;u\in H^1_0(\Omega),\tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), with \(f(x,t)\) asymptotically linear in \(t\) at infinity. Under suitable assumption on \(f\), the author is using a version of the mountain pass theorem to prove existence of at least one positive solution of (1). The author indicates that his method also works for the case, when \(f(x,t)\) is superlinear in \(t\) at infinity.
scientific article; zbMATH DE number 1757443