American Institute of Mathematical Sciences

October  2016, 36(10): 5595-5626. doi: 10.3934/dcds.2016046

On the Hollman McKenna conjecture: Interior concentration near curves

 1 Instituto de Matemáticas, UNAM, Circuito Exterior S/N, Coyoácan, Cd, Universitaria, 04510 Ciudad de México, D.F., Mexico 2 Department of Basic Mathematics, Centro de Investigacióne en Mathematicás, Guanajuato, Mexico

Received  September 2015 Revised  December 2015 Published  July 2016

Consider the problem \notag \left\{\begin{aligned} -\epsilon^2\Delta u&=|u|^p-\Phi_{1} &&\text{in } \Omega\\ u &= 0 &&\text{on }\partial \Omega \end{aligned} \right. where $\epsilon>0$ is a parameter, $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ and $p>2$. Let $\Gamma$ be a stationary non-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.
Citation: Bhakti Bhusan Manna, Sanjiban Santra. On the Hollman McKenna conjecture: Interior concentration near curves. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5595-5626. doi: 10.3934/dcds.2016046
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