Hoang Quoc Toan, Bui Quoc Hung: On a generalization of the Landesman-Lazer condition and Neumann problem for nonuniformly semilinear elliptic equations in an unbounded domain with nonlinear boundary condition, p.301-317

Abstract:

This paper deals with the existence of weak solutions of Neumann problem for a nonuniformly semilinear elliptic equation :

where $\Omega \subset R^N$, $N\geq 3$ is an unbounded domain with smooth and bounded boundary $\partial \Omega$, $\bar{\Omega}=\Omega \cup \partial \Omega$, $h(x)\in L^{1}_{loc}(\bar{\Omega})$, $a(x)\in C(\bar{\Omega}) $, $a(x)\to +\infty$ as $\vert x\vert\to +\infty$, $f(x,s)$, $x\in \Omega$, $g(x,s)$, $x\in \partial \Omega$ are Carathéodory, $k(x)\in L^{2}(\Omega)$, $\theta(x)\in L^{\infty}(\bar{\Omega})$, $\theta(x)\ge 0$.
Our arguments is based on the minimum principle and rely essentially on a generalization of the Landesman-Lazer type condition.

Key Words: Semilinear elliptic equation, Non-uniform, Landesman-Lazer condition, Minimum principle.

2000 Mathematics Subject Classification: Primary: 35J20;
Secondary: 35J60, 58E05.

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