Nasreddine Megrez
A Nonlinear Elliptic Eigenvalue Problem in Unbounded Domain
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 56, 39-48(2002)
MSC:
Abstract: We find a localization of $\lambda$ such that the problem
$$(P_{\lambda}):\;\;\left\{ \begin{array}{l} -\Lap_p u+V(x)|u|^{p-2}u=\lambda f(x,u)\\ u_{|\partial\Omega}=0\\ \ds\lim_{|x|\imp \infty }u(x)=0 \end{array} \right .$$ has a solution, where $\Omega \subset \R^N$ is an
unbounded domain, $N>p\geq 2$, $f:\Omega \times \R\to \R$ is a
continuous function, and $V\in L^{p}_{loc}(\Omega)$ is a
continuous potential on $\Omega$ satisfying
$$\ds\liminf_{|x|\imp_\infty}V(x)\geq \min_{x\in \Omega} V(x) > 0.$$