Existence for an elliptic system with nonlinear boundary conditions

Abstract

In this paper we prove the existence of a weak solution to the following system△_p u = △_q v = 0 in Ω,|∇u|^{p−2} ∂_ν u = f(x,u) − (α + 1)K(x)|u|^{α−1}u|v|^{β+1} + f1 on ∂Ω,|∇v|^{q−2} ∂_ν v = g(x,v) − (β + 1)K(x)|v|^{β−1}v|u|^{α+1} + g1 on ∂Ω,where Ω is a bounded domain in RN (N ≥ 2), f1, g1, f, g and K are functions that satisfy some conditions.

Keywords

• Steklov problems
• weights
• elliptic systems
• nonlinear boundary conditions.