# American Institute of Mathematical Sciences

## Journals

CPAA
Let
 $(M, g)$
be a smooth compact riemannian manifold of dimension
 $N≥2$
with constant scalar curvature. We are concerned with the following elliptic problem
 $\begin{eqnarray*}-{\varepsilon}^2Δ_g u+ u = u^{p-1}, ~~~~u>0,\ \ \ \ \ in \ \M.\end{eqnarray*}$
where
 $Δ_g$
is the Laplace-Beltrami operator on
 $M$
,
 $p>2$
if
 $N = 2$
and
 $2 if $N≥q3$, $\varepsilon$is a small real parameter. We prove that there exist a function $Ξ$such that if $ξ_0$is a stable critical point of $Ξ(ξ)$there exists ${\varepsilon}_0>0$such that for any ${\varepsilon}∈(0,{\varepsilon}_0)$, problem (1) has a solution $u_{\varepsilon}$which concentrates near $ξ_0$as ${\varepsilon}$tends to zero. This result generalizes previous works which handle the case where the scalar curvature function of $(M,g)$has non-degenerate critical points. keywords: Singular perturbation problems concentration phenomena finite dimensional reduction DCDS 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 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