The estimates on the energy functional of an elliptic system with Neumann boundary conditions

We consider an elliptic system of the form − e 2 Δ u + u = f ( v ) , − e 2 Δ v + v = g ( u ) in Ω with Neumann boundary conditions, where Ω is a C 2 domain in R N , f and g are nonlinearities having superlinear and subcritical growth at infinity. We prove the existence of nonconstant positive solutions of the system, and estimate the energy functional on a configuration space H ¯ by a different technique, which is an important step in the proof of the solution’s concentrative property. We conclude that the least energy solutions of the system concentrate at the point of boundary, which maximizes the mean curvature of ∂ Ω.