Existence and asymptotic behavior of solutions for a class of semilinear subcritical elliptic systems

Abstract

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W ( x ) u = Q u ( u , v ) in R N , − ε 2 Δ v + V ( x ) v = Q v ( u , v ) in R N , u , v ∈ H 1 ( R N ) , u ( x ) , v ( x ) > 0 for each x ∈ R N , where ε > 0, W and V are positive potentials of C 2 class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme.