Advances in Differential Equations

Multiple boundary bubbling phenomenon of solutions to a Neumann problem

Yang Wang and Long Wei

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We consider the following anisotropic problem $$-\div\big( a(x)\nabla u\big)+a(x)u=0 \quad \mbox{in ${\Omega }$,}\qquad \frac{{\partial} u} {{\partial}\nu}={\varepsilon } e^u \quad\mbox{on ${\partial\Omega }$,} $$ where ${\Omega }\subseteq \mathbb{R}^2$ is a bounded smooth domain, ${\varepsilon }$ is a small parameter and $a(x)$ is a positive smooth function. First, we establish a decomposition result for the regular part of a relative Green's function, which yields its Hölder continuous character and the smoothness of its diagonal. Next, we employ this result to derive the accumulation of bubbles at given local maximum points of $a(x)$ on the boundary, which verifies the existence of large energy solutions to the problem in [17].

Article information

Adv. Differential Equations, Volume 13, Number 9-10 (2008), 829-856.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35A08: Fundamental solutions 35J67: Boundary values of solutions to elliptic equations


Wang, Yang; Wei, Long. Multiple boundary bubbling phenomenon of solutions to a Neumann problem. Adv. Differential Equations 13 (2008), no. 9-10, 829--856.

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