On singular perturbation problems with Robin boundary condition
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, p. 199-230

We consider the following singularly perturbed elliptic problem $\begin{array}{cc}\hfill {ϵ}^{2}\Delta u-u+f\left(u\right)& =0,\phantom{\rule{4pt}{0ex}}u>0\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\hfill \\ \hfill ϵ\frac{\partial u}{\partial \nu }+\lambda u& =0\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega ,\hfill \end{array}$ where $f$ satisfies some growth conditions, $0\le \lambda \le +\infty$, and $\Omega \subset {ℝ}^{N}$ ($N>1$) is a smooth and bounded domain. The cases $\lambda =0$ (Neumann problem) and $\lambda =+\infty$ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant ${\lambda }_{*}>1$ such that, as $ϵ\to 0$, the least energy solution has a spike near the boundary if $\lambda \le {\lambda }_{*}$, and has an interior spike near the innermost part of the domain if $\lambda >{\lambda }_{*}$. Central to our study is the corresponding problem on the half space.

Classification:  35B35,  35J40,  92C40
@article{ASNSP_2003_5_2_1_199_0,
author = {Berestycki, Henri and Wei, Juncheng},
title = {On singular perturbation problems with Robin boundary condition},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {1},
year = {2003},
pages = {199-230},
zbl = {1121.35008},
mrnumber = {1990979},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0}
}

Berestycki, Henri; Wei, Juncheng. On singular perturbation problems with Robin boundary condition. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, pp. 199-230. http://www.numdam.org/item/ASNSP_2003_5_2_1_199_0/

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