American Institute of Mathematical Sciences

September  2018, 17(5): 2063-2084. doi: 10.3934/cpaa.2018098

On spike solutions for a singularly perturbed problem in a compact riemannian manifold

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China 2 University of Tunis El Manar Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire 2092 Tunis El Manar, Tunisia 3 Centro de Modelamiento Matemático, Universidad de Chile, Beauchef 851, Edificio Norte-Piso 7, Santiago de Chile

* Corresponding author

Received  September 2017 Revised  December 2017 Published  April 2018

Fund Project: S. Deng has been partly supported by National Natural Science Foundation of China 11501469 and the Basic Science and Advanced Technology Research of Chongqing cstc2016jcyA0032 and XDJK2017B014. F. Mahmoudi has been supported by Fondecyt Grant 1140311, fondo Basal PFB03 C.C. 2420 CMM and "Millennium Nucleus Center for Analysis of PDE NC130017"

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 $N≥3$, $\varepsilon$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.
Citation: Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098
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