Positive solutions for semilinear elliptic systems with sign-changing potentials

Abstract

In this paper, we study the existence of positive solutions of the Dirichlet problem -Δu = λ p(x)f(u; v) ; -Δv = λ q(x)g(u; v); in D, and u = v = 0 on ∂∞D, where D ⊂ Rn (n ≥ 3) is an C1,1-domain with compact boundary and λ > 0. The potential functions p; q are not necessarily bounded, may change sign and the functions f; g : ℝ2 → ℝ are continuous with f(0; 0) > 0, g(0; 0) > 0. By applying the Leray- Schauder fixed point theorem, we establish the existence of positive solutions for λ sufficiently small.

Keywords

• elliptic systems
• positive solution
• green potential
• leray-shauder fixed point theorem
• primary 35j47,35j57
• secondary 35j08,35p30