Asymptotic behavior of least energy solutions for a singularly perturbed problem with nonlinear boundary condition

Abstract

We consider the problem of finding a positive harmonic function \(u_\varepsilon \) in a bounded domain \(\Omega \subset \mathbb R ^N (N\ge 3)\) satisfying a nonlinear boundary condition of the form \(\varepsilon \partial _{\nu } u +u =|u|^{p-2}u,\,x\in \partial \Omega \), where \(\varepsilon \) is a positive parameter and \(2<p<2_*:=2(N-1)/(N-2)\). To be more precise, by using min-max methods, we study the existence of least energy solution \(u_\varepsilon \) of the problem depending on the parameter \(\varepsilon \). We provide a detailed description of the shape of \(u_\varepsilon \) and prove that the maximum of \(u_\varepsilon \) is achieved at a point \(z_{\varepsilon }\), which lies on the boundary \(\partial \Omega \) and concentrates at the mean curvature maximum point of the boundary \(\partial \Omega \). This problem is related to the existence of extremals for a Sobolev inequality involving the trace embedding and the asymptotic behavior of the best constants in expanding domains.