Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 3035 - 3076, Volume 36, Issue 6, June 2016      doi:10.3934/dcds.2016.36.3035

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Shengbing Deng - School of Mathematics and Statistics, Southwest University, Chongqing 400715, China (email)
Fethi Mahmoudi - Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago, Chile (email)
Monica Musso - Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago, Chile (email)

Abstract: 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.

Keywords:  Critical Sobolev exponent, blowing-up solution, non degenerate minimal submanifolds.
Mathematics Subject Classification:  35J20, 35J60.

Received: March 2015;      Revised: October 2015;      Available Online: December 2015.