Multi-peak solutions for a planar Robin nonlinear elliptic problemwith large exponent

Electronic Journal of Differential Equations

卷:2016

Lei Shi¹  Yibin Zhang¹ 

[1] Nanjing Agricultural Univ., Nanjing, China ;

关键词: Concentrating solutions;    largeexponent;    Robin boundary condition;    finite-dimensional reduction;   

DOI  :  

来源: DOAJ

【 摘 要 】

We consider the elliptic equation $\Delta u+u^p=0$ in a bounded smooth domain $\Omega$ in $\mathbb{R}^2$ subject to the Robin boundary condition $\frac{\partial u}{\partial\nu} +\lambda b(x)u=0$. Here $\nu$ denotes the unit outward normal vector on $\partial\Omega$, $b(x)$ is a smooth positive function defined on $\partial\Omega$, $0<\lambda<+\infty$, and p is a largeexponent. For any fixed $\lambda$ largewe find topological conditions on$\Omega$ which ensure the existence of a positive solution with exactly m peaks separated by a uniform positive distance from the boundary and each from other as $p\to+\infty$ and $\lambda\to+\infty$. In particular, for a nonsimply connected domain such solution exists for any $m\geq 1$.

【 授权许可】

Unknown