New solutions for critical Neumann problems in ℝ2

Abstract

We consider the elliptic equation -Δ⁢u+u=0{-\Delta u+u=0} in a bounded, smooth domain Ω in ℝ2{\mathbb{R}^{2}} subject to the nonlinear Neumann boundary condition ∂⁡u∂⁡ν=λ⁢u⁢eu2{\frac{\partial u}{\partial\nu}=\lambda ue^{u^{2}}}, where ν denotes the outer normal vector of ∂⁡Ω{\partial\Omega}. Here λ>0{\lambda>0} is a small parameter. For any λ small we construct positive solutions concentrating, as λ→0{\lambda\to 0}, around points of the boundary of Ω.

Keywords

• elliptic partial differential equations
• nonlinear neumann boundary condition, trudinger–moser trace embedding
• 35j05
• 35j08
• 35j25
• 35j60
• 35j61