CPAA
On spike solutions for a singularly perturbed problem in a compact riemannian manifold
Shengbing Deng Zied Khemiri Fethi Mahmoudi
Communications on Pure & Applied Analysis 2018, 17(5): 2063-2084 doi: 10.3934/cpaa.2018098
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
$2<p<\frac{2N}{N-2}$
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.
keywords: Singular perturbation problems concentration phenomena finite dimensional reduction

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