Existence Of Solutions Of Weakly Coupled Semilinear Elliptic Systems With Indefinite Weights

In this paper, we study the existence of solutions of semilinear elliptic systems with sign-changing weights. This paper is constituted with three chapters.In chapter 1, wo introduce the background and main results in weakly coupled semilinear elliptic systems. We illustrate difficulties about the weakly coupled semilinear elliptic systems, and ideas to solve them.In chapter 2, we consider the existence of nontrivial solutions of weakly coupled semilinear elliptic systems:whereΩs a smooth bounded domain in R^N, Q(x) changes signs. Suppose 1＜p,g＜2^*-1,α,β＞1, (?)＜1,0＜μ,v＜λ₁, whereλ₁ is the first eigenvalue of (-Δ,H₀¹(Ω)), we obtain at least three nontrivial solutions of problem (2) by a Linking theorem.In chapter 3, we consider the existence of solutions of problem (2) in the critical case p = q = 2^*-1. We show that there existsλ₀∈(0,λ₁) such that for 0＜μ,v＜λ₀ problem (2) has at least three nontrivial solutions by a Linking theorem. The main results of this paper have been published in Differential Integral Equations 3-4(2009).239-250.