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

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