Nodal bubbling solutions to a weighted sinh-Poisson equation

Abstract

We consider the existence of multiple nodal bubbling solutions to the equation $-{\Delta } u=2\ {\varepsilon }^2|x|^{2\alpha}\sinh u$ posed on a bounded smooth domain ${\Omega }$ in $\mathbb{R}^2$ with homogeneous Dirichlet boundary conditions. By construction, we show that there exists a solution such that $2{\varepsilon }^2|x|^{2\alpha}\sinh u_{{\varepsilon }}$ develops not only many positive and negative Dirac deltas with weights $8\pi$ and $-8\pi$ respectively, but also a Dirac delta with weight $8\pi(1+\alpha)$ at the origin, where $\alpha\not\in \mathbb{N}$. In particular, we provide explicit examples to show the existence of nodal bubbling solutions to our problem in the unit disc in $\mathbb R^2$.