DCDS
Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents
Shengbing Deng Fethi Mahmoudi Monica Musso
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.
DCDS
An improved Hardy inequality for a nonlocal operator
Boumediene Abdellaoui Fethi Mahmoudi
Let $0 < s < 1$ and $1< p < 2$ be such that $ps < N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality:
    Given $1 \le q < p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality (1.1).
keywords: Fractional Sobolev spaces nonlocal problems. weighted Hardy inequality

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