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:

35J10 Schrodinger operator, See also {35Pxx}

35J60 Nonlinear PDE of elliptic type

35J30 General theory of higher-order, elliptic equations , See also {31A30, 31B30}

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
$$ \ds\liminf_{|x|\imp_\infty}V(x)\geq \min_{x\in
\Omega} V(x) > 0.$$

Keywords: Nonlinear eigenvalue problem, Schr\"{o}dinger equation, P-Laplacian, Unbounded domain