On a generalization of the Landesman-Lazer condition and Neumann problem for nonuniformly semilinear elliptic equations in an unbounded domain with nonlinear boundary condition

• Published in: Bulletin mathématiques de la Société des sciences mathématiques de Roumanie 2014-01, Vol.57 (105) (3), p.301-317
• Main Authors: Toan, Hoang Quoc, Hung, Bui Quoc
• Format: Article
• Language: English
• ISSN: 1220-3874; 2065-0264

This paper deals with the existence of weak solutions of Neumann problem for a nonuniformly semilinear elliptic equation : $\left\{ {_{\frac{{\partial u}}{{\partial n}} = g\left( {x,u} \right)on\partial \Omega }^{ - div\left( {h\left( x \right)\nabla u} \right) + a\left( x\right)u = \lambda \theta \left( x \right)u + f\left( {x,u} \right) - k\left( x \right)in\Omega }} \right.$ where Ω ⊂ RN, N ≥ 3 is an unbounded domain with smooth and bounded boundary ∂Ω, Ω̅ = Ω ∪ ∂Ω, h(x) ∊ $L_{loc}^1\left( \Omega \right)$, a(x) ∊ C(Ω̅), a(x) → +∞ as |x| → +∞, f(x, s), x ∊ Ω, g(x, s), x ∊ ∂Ω, are Carathéodory, k(x) ∊ L²(Ω), θ(x) ∊ L∞ (Ω̅), θ(x) ≥ 0. Our arguments is based on the minimum principle and rely essentially on a generalization of the Landesman-Lazer type condition.