Limit of nonlinear elliptic equations with concentrated terms and varying domains: the non uniformly Lipschitz case

Abstract

In this article, we analyze the limit of the solutions of nonlinear elliptic equations with Neumann boundary conditions, when nonlinear terms are concentrated in a region which neighbors the boundary of domain and this boundary presents a highly oscillatory behavior which is non uniformly Lipschitz. More precisely, if the Neumann boundary conditions are nonlinear and the nonlinearity in the boundary is dissipative, then we obtain a limit equation with homogeneous Dirichlet boundary conditions. Moreover, if the Neumann boundary conditions are homogeneous, then we obtain a limit equation with nonlinear Neumann boundary conditions, which captures the behavior of the concentration's region. We also prove the upper semicontinuity of the families of solutions for both cases.

Keywords

• Nonlinear elliptic equation
• boundary value problem
• varying boundary
• oscillatory behavior
• concentrating term
• upper semicontinuity