On the Hollman McKenna conjecture: Interior concentration near curves
Bhakti Bhusan Manna, Sanjiban Santra · Discrete and Continuous Dynamical Systems · 2016
Consider the problem \begin{equation} otag\left\{\begin{aligned} -\epsilon^2\Delta u&=|u|^p-\Phi_{1} &&\text{in } \Omega\\ u &= 0 &&\text{on }\partial \Omega \end{aligned} \right.\end{equation}where $\epsilon>0$ is a parameter, $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ and $p>2$. Let $\Gamma$ be a stationarynon-degenerate closed curve relative to the weighted arc-length $\int_{\Gamma} \Phi_{1}^{\frac{p+3}{2p}}.$ We prove that for $\epsilon>0$ sufficiently small, there exists a solution $u_{\epsilon}$of the problem, which concentrates near the curve$\Gamma$ whenever $d(\Gamma, \partial \Omega)>c_{0}>0.$ As a result, we prove the higher dimensional concentration for a Ambrosetti-Prodi problem,thereby proving an affirmative result to the conjecture by Hollman-McKenna [9] in two dimensions.