Spike-layered solutions for an elliptic system with Neumann boundary conditions

Abstract

We prove the existence of nonconstant positive solutions for a system of the form − ε 2 Δ u + u = g ( v ) -\varepsilon ^2\Delta u + u = g(v) , − ε 2 Δ v + v = f ( u ) -\varepsilon ^2\Delta v + v = f(u) in Ω \Omega , with Neumann boundary conditions on ∂ Ω \partial \Omega , where Ω \Omega is a smooth bounded domain and f f , g g are power-type nonlinearities having superlinear and subcritical growth at infinity. For small values of ε \varepsilon , the corresponding solutions u ε u_{\varepsilon } and v ε v_{\varepsilon } admit a unique maximum point which is located at the boundary of Ω \Omega .