Mixed interior and boundary peak solutions of the Neumann problem for the Henon equation in R^2
Yibin Zhang, Haitao Yang · DOAJ (DOAJ: Directory of Open Access Journals) · 2015
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 $$\displaylines{ -\Delta u+u=|x|^{2\alpha}u^p,\quad u>0 \quad \text{in } \Omega,\cr \frac{\partial u}{\partial u}=0\quad \text{on } \partial\Omega, }$$ where $ u$ denotes the outer unit normal vector to $\partial\Omega$, $-1<\alpha ot\in\mathbb{N}\cup\{0\}$ 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.