Multiple solutions for a semilinear nonhomogeneous 1 elliptic system

Xiaodong Zhao and Lin Chen

Abstract

In this paper, by the Mountain Pass Theory and Ekeland’s variational principle, we consider the existence and multiplicity of nontrivial solutions for the nonhomogeneous semilinear elliptic system    −∆u + u = α α+β f(x)|u| α−2u|v| β + l1(x), x ∈ Ω, −∆v + v = β α+β f(x)|u| α|v| β−2v + l2(x), x ∈ Ω, ∂u ∂n = λg(x)|u| q−2u, ∂v ∂n = µh(x)|v| q−2 v, x ∈ ∂Ω, where Ω is a bounded domain in R N with smooth boundary, α > 1, β > 1 satisfying 2 < α+β < 2 ∗ (2∗ = 2N N−2 if N ≥ 3, 2 ∗ = ∞ if N = 2), 1 < q < 2, the pair of parameters (λ, µ) ∈ R 2 \ {(0, 0)}