Advances in Differential Equations

Multiple boundary bubbling phenomenon of solutions to a Neumann problem

Yang Wang and Long Wei

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Abstract

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

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

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867321

Mathematical Reviews number (MathSciNet)
MR2482577

Zentralblatt MATH identifier
1178.35181

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

Citation

Wang, Yang; Wei, Long. Multiple boundary bubbling phenomenon of solutions to a Neumann problem. Adv. Differential Equations 13 (2008), no. 9-10, 829--856. https://projecteuclid.org/euclid.ade/1355867321.


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