Multiple Ordered Positive Solutions of an Elliptic Problem Involving the p&q-Laplacian

F. J. S. A. Corrêa ; A. S. S. Corrêa ; J. R. Santos Junior

Journal of convex analysis, Tome 21 (2014) no. 4, pp. 1023-1042

Résumé

We are concerned with questions of existence and multiplicity of positive solutions of the elliptic problem $$\label{main problem} \left\{\begin{array}{rclcc} -div \;(\mathcal{K}(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u) = \lambda f(u) \mbox{in} \Omega , \\ u = 0 \mbox{on} \partial\Omega \end{array} \right. \leqno{(P_{\lambda})} $$ with $1$, where $\Omega \subset \mathbb{R}^{N}$ is a bounded smooth domain, $\lambda$ is a positive real parameter, $\mathcal{K}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is a $C^{1}$-function and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous functions which changes sign. We use variational methods.

Détail

Classification : 35J65, 34B15
Mots-clés : Laplacian, variational methods, multiplicity of positive solutions