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

作者：Abreu Emerson; do O Joao Marcos^*; Medeiros Everaldo

来源：Calculus of Variations and Partial Differential Equations, 2014, 49(1-2): 491-516.

DOI：10.1007/s00526-012-0590-4

We consider the problem of finding a positive harmonic function u(epsilon) in a bounded domain Omega subset of R-N (N %26gt;= 3) satisfying a nonlinear boundary condition of the form epsilon partial derivative(nu)u + u = vertical bar u vertical bar(p-2)u, x is an element of partial derivative Omega, where e is a positive parameter and 2 %26lt; p %26lt; 2(*) := 2(N - 1)/(N - 2). To be more precise, by using min-max methods, we study the existence of least energy solution u(epsilon) of the problem depending on the parameter (epsilon). We provide a detailed description of the shape of u(epsilon) and prove that the maximum of u(epsilon) is achieved at a point z(epsilon), which lies on the boundary partial derivative Omega and concentrates at the mean curvature maximum point of the boundary. partial derivative 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.

• 出版日期2014-1