Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 76, pp. 1-28.
Title: Mixed interior and boundary peak solutions of the Neumann problem
for the Henon equation in R^2
Authors: Yibin Zhang (Nanjing Agricultural Univ., Nanjing, China)
Haitao Yang (Zhejiang Univ., Hangzhou, China)
Abstract:
Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth
boundary and $0\in\overline{\Omega}$, we study the Neumann problem
for the Henon equation
$$\displaylines{
-\Delta u+u=|x|^{2\alpha}u^p,\quad u>0 \quad \text{in } \Omega,\cr
\frac{\partial u}{\partial\nu}=0\quad \text{on } \partial\Omega,
}$$
where $\nu$ denotes the outer unit normal vector to $\partial\Omega$,
$-1<\alpha\not\in\mathbb{N}\cup\{0\}$ and p is a large exponent.
In a constructive way, we show that, as p approaches $+\infty$,
such a problem has a family of positive solutions with arbitrarily
many interior and boundary spikes involving the origin.
The same techniques lead also to a more general result on
Henon-type weights.
Submitted October 14, 2014. Published March 26, 2015.
Math Subject Classifications: 35J25, 35J20, 58K05.
Key Words: Mixed interior and boundary peak solutions; Henon-type weight;
large exponent; Lyapunov-Schmidt reduction process.