Construction of Nodal Bubbling Solutions for the Weighted Sinh-Poisson Equation

Abstract

We consider the weighted sinh-Poisson equation $\mathrm{\Delta }u+2{\epsilon }^2{|x|}^{2\alpha }\text{sinh\hspace{0.17em}}u=0$ in $\mathrm{}{B}_{\mathrm{1}}(\mathrm{0})$, $u=\mathrm{0}$ on $\mathrm{}\mathrm{}\partial B_{\mathrm{1}}(\mathrm{0})$, where $\epsilon >\mathrm{0}$ is a small parameter, $\alpha \in (-\mathrm{1},+\mathrm{\infty })\setminus \{\mathrm{0}\}$, and $B_{\mathrm{1}}(\mathrm{0})$ is a unit ball in ${ℝ}^{\mathrm{2}}$. By a constructive way, we prove that for any positive integer $m$, there exists a nodal bubbling solution $u_{\epsilon }$ which concentrates at the origin and the other $m$-points ${\tilde{q}}_l=(\lambda \text{\hspace{0.17em}cos\hspace{0.17em}}(2\pi (l-1)/m),\lambda \text{\hspace{0.17em}sin\hspace{0.17em}}(2\pi (l-1)/m))$, $l=2,\dots ,m+1$, such that as $\epsilon \to 0$, $2{\epsilon }^2{|x|}^{2\alpha }\text{sinh\hspace{0.17em}}{u}_{\epsilon }\rightharpoonup 8\pi (1+\alpha ){\delta }_0+{\sum }_{l=2}^{m+1}8\pi {(-1)}^{l-1}{\delta }_{{\tilde{q}}_l}$, where $\lambda \in (\mathrm{0,1})$ and $m$ is an odd integer with $(\mathrm{1}+\alpha )(m+\mathrm{2})-\mathrm{1}>\mathrm{0}$, or $m$ is an even integer. The same techniques lead also to a more general result on general domains.