New solutions for critical Neumann problems in ℝ2

Abstract

Abstract We consider the elliptic equation {-\Delta u+u=0} in a bounded, smooth domain Ω in {\mathbb{R}^{2}} subject to the nonlinear Neumann boundary condition {\frac{\partial u}{\partial\nu}=\lambda ue^{u^{2}}} , where ν denotes the outer normal vector of {\partial\Omega} . Here {\lambda>0} is a small parameter. For any λ small we construct positive solutions concentrating, as {\lambda\to 0} , around points of the boundary of Ω.