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On the Hollman McKenna conjecture: Interior concentration near curves

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  • Consider the problem \begin{equation} \notag \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 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.
    Mathematics Subject Classification: Primary: 35B25, 35B40; Secondary: 35J65.

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