Sur le spectre de Fučik avec poids

Abstract

Let $\Omega$ be a bounded domain in $\mathbb R^N$ and $m_1$, $m_2$ two functions in $L^\infty(\Omega)$. In the present work, we study a new spectrum constitued by the set of pairs $(\alpha,\beta)$ of $\R ^2$ for which the problem $$ \left\{ \begin{array}{rcr} -\bigtriangleup u & = & \alpha m_1 u^+-\be m_2 u^- \quad\mbox{ in} \:\: \Omega,\\ u & = & 0 \quad \hspace{2,4cm}\quad\mbox{ on}\:\: \partial\Omega, \end{array} \right. $$ has a nontrivial solution, where $u^\pm=\di\max(0,\pm u)$. We study then the nonresonance with respect to this spectrum in a non autonomous problem.