Positive solutions for weakly coupled nonlinear elliptic systems

Abstract

In this article, we consider the existence of positive solutions for weakly coupled nonlinear elliptic systems (0.1) { - Δ u + u = ( 1 + a ( x ) ) | u | p - 1 u + μ | u | α - 2 u | v | β + λ v i n ℝ N , - Δ v + v = ( 1 + b ( x ) ) | v | p - 1 v + μ | u | α | v | β - 2 v + λ u ​ ​ ​ ​ ​ i n ℝ N , To find nontrivial solutions, we first investigate autonomous systems. In this case, results of bifurcation from semi-trivial solutions are obtained by the implicit function theorem. Next, the existence of positive solutions of problem (0.1) is obtained by variational methods.