Advances in Differential Equations

Nodal bubbling solutions to a weighted sinh-Poisson equation

Yang Wang and Long Wei

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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 $\mathbb R^2$.

Article information

Adv. Differential Equations, Volume 13, Number 9-10 (2008), 881-906.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations


Wang, Yang; Wei, Long. Nodal bubbling solutions to a weighted sinh-Poisson equation. Adv. Differential Equations 13 (2008), no. 9-10, 881--906.

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