Abstract
Let Ω ⊂ ℝ2 be a bounded domain with smooth boundary and b(x) > 0 a smooth function defined on ∂Ω. We study the following Robin boundary value problem:
$$\begin{array}{}
\displaystyle
\left\{
\begin{alignedat}{2}
&{\it\Delta} u+u^p=0 &\quad& \text{in }{\it\Omega},\\
&u>0 &\quad& \text{in }{\it\Omega},\\
&\frac{\partial u}{\partial\nu} +\lambda b(x)u=0
&\quad& \text{on }
\partial{\it\Omega},
\end{alignedat}
\right.
\end{array}$$
where ν denotes the exterior unit vector normal to ∂Ω, 0 < λ < +∞ and p > 1 is a large exponent. We construct solutions of this problem which exhibit concentration as p → +∞ and simultaneously as λ → +∞ at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.
Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
