Multi-peak solutions for a planar Robin nonlinear elliptic problem with large exponent
Yibin Zhang, Lei Shi · DOAJ (DOAJ: Directory of Open Access Journals) · 2016
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 $\frac{\partial u}{\partial u} +\lambda b(x)u=0$. Here $ u$ 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 large exponent. For any fixed $\lambda$ large we 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 $m\geq 1$.