Existence of solutions to a Neumann boundary value problem with exponential nonlinearity

Abstract

This paper is concerned with the solutions to the following sinh-Poisson equation with Hénon term{−Δu+u=ε2|x−q1|2α1⋯|x−qn|2αn(eu−e−u),u>0,inΩ,∂u∂ν=0,on∂Ω, where Ω⊂R2 is a bounded, smooth domain, ε>0, α1,...,αn∈(0,∞)∖N, and q1,...,qn∈Ω are fixed. Given any two non-negative integers k,l with k+l⩾1, it is shown that, for sufficiently small ε>0, there exists a solution uε for which ε2|x−q1|2α1⋯|x−qn|2αn(eu−e−u) asymptotically (i.e. the limit as ε→0) develops k+n interior Dirac measures and l boundary Dirac measures. The location of blow-up points is characterized explicitly in terms of Green's function of Neumann problem and the function k(x)=|x−q1|2α1⋯|x−qn|2αn.