Mountain pass type solutions for quasilinear elliptic equations
Let \(\Omega\) be a bounded domain in \({\mathbb R}^N\) and let \(g:\overline{\mathbb R}\times {\mathbb R}\rightarrow {\mathbb R}\) be a continuous function. The paper deals with the problem \(-\text{div} (a(|\nabla u(x)|)\nabla u(x))=g(x,u)\) in \(\Omega\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The function \(a:{\mathbb R}_+\rightarrow {\mathbb R}\) is such that the mapping \(\phi (s)=sa(|s|)\) is an increasing homeomorphism from \({\mathbb R}\) onto \({\mathbb R}\). Under appropriate conditions on \(\phi\) and \(g\), the authors prove the existence of nontrivial weak solutions of mountain pass type in an Orlicz-Sobolev space. The proof is based on variational arguments and refined estimates. Several interesting examples are also given in the work. The paper is very well written and the techniques developed by the authors can be adapted for the treatment of wide classes of boundary value problems.
scientific article; zbMATH DE number 1525159