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Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents

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  • In this paper we consider the following problem \begin{eqnarray} \label{abstract} \quad \left\{ \begin{array}{ll}-\Delta u +u= u^{{n-k+2\over n-k-2} \pm\epsilon} & \mbox{ in } \Omega \\ u>0& \mbox{ in }\Omega                                  (0.1)\\ {\partial u\over\partial\nu}=0 & \mbox{ on } \partial\Omega \end{array} \right. \end{eqnarray} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\ge 7$, $k$ is an integer with $k\ge 1$, and $\epsilon >0$ is a small parameter. Assume there exists a $k$-dimensional closed, embedded, non degenerate minimal submanifold $K$ in $\partial \Omega$. Under a sign condition on a certain weighted avarage of sectional curvatures of $\partial \Omega$ along $K$, we prove the existence of a sequence $\epsilon = \epsilon_j \to 0$ and of solutions $u_\epsilon$ to (0.1) such that $$ |\nabla u_\epsilon |^2 \, \rightharpoonup \, S \delta_K , \quad {\mbox {as}} \quad \epsilon \to 0 $$ in the sense of measure, where $\delta_K$ denotes a Dirac delta along $K$ and $S$ is a universal positive constant.
    Mathematics Subject Classification: 35J20, 35J60.

    Citation:

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