Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions
The authors consider positive solutions of the Neumann boundary value problem for the elliptic system \(-\varepsilon^2\Delta u+u=g(v)\), \(-\varepsilon^2\Delta v+v=f(u)\) in \(\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\), \(N\geq3\), and \(f,g\) are power-type nonlinearities with superlinear and subcritical growth at infinity. They prove that the least energy solutions concentrate, as \(\varepsilon\) goes to zero, at a point of \(\partial\Omega\) which maximizes the mean curvature of \(\partial\Omega\). This statement generalizes and improves the result by \textit{A. I.~Ávila} and \textit{J.~Yang} [J. Differ. Equations 191, No. 2, 348--376 (2003; Zbl 1109.35325)] who considered the power nonlinearities \(f(u)=| u| ^{p-1}u\) and \(g(v)=| v| ^{q-1}v\) with \(p,q>1\) satisfying the subcriticality condition \(1/(p+1)+1/(q+1)>(N-2)/N\). In fact, Ávila and Yang had to assume \(p,q<(N+2)/(N-2)\) and they proved a weaker concentration result.
scientific article; zbMATH DE number 2104530