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

Authors：Wang, Chang-Jian; Zheng, Gao-Feng^*

Source：Journal of Mathematical Analysis and Applications, 2022, 505(1): 125458.

DOI：10.1016/j.jmaa.2021.125458

• Exponential nonlinearity
• Henon term
• Finite-dimensional reduction
• Interior and boundary concentrating
• solutions

This paper is concerned with the solutions to the following sinh-Poisson equation with Henon term @@@ {-Delta u + u = epsilon(2)|x - q(1)|(2 alpha 1) ...|x - q(n)|(2 alpha n) (e(u) - e(-u)), u > 0, in Omega, @@@ partial derivative u/partial derivative v = 0, on partial derivative Omega, @@@ where Omega C R-2 is a bounded, smooth domain, epsilon > 0, alpha(1), ..., alpha(n) is an element of (0, infinity) \ N, and q(1), ..., q(n) is an element of Omega are fixed. Given any two non-negative integers k, l with k + l >= 1, it is shown that, for sufficiently small epsilon > 0, there exists a solution u(epsilon) for which epsilon(2)|x -q(1)|(2 alpha 1) ... |x -q(n)|(2 alpha n) (e(u)-e(-u)) asymptotically (i.e. the limit as epsilon -> 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 -q(1)|(2 alpha 1) ... |x - q(n)|2 alpha(n) .