Boundary concentrating solutions for an anisotropic planar nonlinear Neumann problem with large exponent

Abstract

Let Ω be a bounded domain in R2 with smooth boundary, we study the following anisotropic elliptic problem{−∇(a(x)∇u)+a(x)u=0in Ω,u>0in Ω,∂u∂ν=upon ∂Ω, where ν denotes the outer unit normal vector to ∂Ω, p>1 is a large exponent and a(x) is a positive smooth function. We construct solutions of this problem which exhibit the accumulation of arbitrarily many boundary peaks at any isolated local maximum point of a(x) on the boundary.