On superquadratic elliptic systems

Abstract

In this article we study the existence of solutions for the elliptic system \[ − Δ u = ∂ H ∂ v ( u , v , x ) in Ω , − Δ v = ∂ H ∂ u ( u , v , x ) in Ω , u = 0 , v = 0 on ∂ Ω . \begin {array}{*{20}{c}} { - \Delta u = \frac {{\partial H}}{{\partial v}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ { - \Delta v = \frac {{\partial H}}{{\partial u}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ {u = 0,\quad v = 0\quad {\text {on}}\;\partial \Omega .} \\ \end {array} \] where Ω \Omega is a bounded open subset of R N {\mathbb {R}^N} with smooth boundary ∂ Ω \partial \Omega , and the function H : R 2 × Ω ¯ → R H:{\mathbb {R}^2} \times \bar \Omega \to \mathbb {R} , is of class C 1 {C^1} . We assume the function H has a superquadratic behavior that includes a Hamiltonian of the form \[ H ( u , v ) = | u | α + | v | β where 1 − 2 N > 1 α + 1 β > 1 with α > 1 , β > 1. H(u,v) = |u{|^\alpha } + |v{|^\beta }\quad {\text {where}}\;1 - \frac {2}{N} > \frac {1}{\alpha } + \frac {1}{\beta } > 1\;{\text {with}}\;\alpha > 1,\beta > 1. \] We obtain existence of nontrivial solutions using a variational approach through a version of the Generalized Mountain Pass Theorem. Existence of positive solutions is also discussed.