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

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

Shengbing Deng 1, , Zied Khemiri 2, and  Fethi Mahmoudi 3,,

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
$2<p<\frac{2N}{N-2}$
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.
Mathematics Subject Classification: Primary: 35J20, 35J60, 35B33; Secondary: 35B40.
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
References:
[1]

V. Benci, C. Bonanno and A. M. Micheletti, On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds, J. Funct. Anal., 252 (2007), 464-489.   Google Scholar

[2]

J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calculus of Variations and Partial Differential Equations, 24 (2005), 459-477.   Google Scholar

[3]

E. N. Dancer, A. M. Micheletti and A. Pistoia, Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifold, Manuscripta Math., 128 (2009), 163-193.   Google Scholar

[4]

M. Del Pino, F. L. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79.   Google Scholar

[5]

S. Deng, Multipeak solutions for asymptotically critical elliptic equations on Riemannian manifolds, Nonlinear Analysis., 74 (2011), 859-881.   Google Scholar

[6]

P. Esposito and A. Pistoia, Blowing-up solutions for the Yamabe equation, Portugal. Math. (N.S.), 71 (2014), 249-276.   Google Scholar

[7]

M. Grossi and A. Pistoia, On the effect of critical points of distance function in superlinear elliptic problems, Adv. Differ. Equ., 5 (2000), 1397-1420.   Google Scholar

[8]

M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differ. Equ., 11 (2000), 143-175.   Google Scholar

[9]

C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.   Google Scholar

[10]

C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincare Anal. Non Linéaire, 17 (2000), 47-82.   Google Scholar

[11]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differ. Equ., 158 (1999), 1-27.   Google Scholar

[12]

J. M. Lee, John and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc., 17 (1987), 37-91.   Google Scholar

[13]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differ. Equ., 23 (1998), 487-545.   Google Scholar

[14]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equ., 72 (1988), 1-27.   Google Scholar

[15]

A. M. Micheletti and A. Pistoia, The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds, Calc. Var. Partial Differential Equations, 34 (2009), 233-265.   Google Scholar

[16]

A. M. Micheletti and A. Pistoia, Nodal solutions for a singularly perturbed nonlinear elliptic problem on Riemannian manifolds, Advanced Nonlinear Studies, 9 (2009), 565-577.   Google Scholar

[17]

A. M. Micheletti, A. Pistoia and J. Vétois, Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds, Indiana University Math. Journal, 58 (2009), 1719-1746.   Google Scholar

[18]

F. Mahmoudi, Constant k-curvature hypersurfaces in Riemannian manifolds, Differential Geom. Appl., 28 (2010), 1-11.   Google Scholar

[19]

W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.   Google Scholar

[20]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.   Google Scholar

[21]

F. Pacard and X. Xu, Constant mean curvature spheres in Riemannian manifolds, Manuscripta Math., 128 (2009), 275-295.   Google Scholar

[22]

S. Schoen, Conformal deformation of a Riemannian metric to a constant scalar curvature, J. Differential Geom., 20 (1984), 479-496.   Google Scholar

[23]

J. Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differ. Equ., 134 (1997), 104-133.   Google Scholar

[24]

J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J., 50 (1998), 159-178.   Google Scholar

[25]

J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. Lond. Math. Soc., 59 (1999), 585-606.   Google Scholar

[26]

H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960), 21-37.   Google Scholar

[27]

R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math., 147 (1991), 381-396.   Google Scholar

[28]

R. Ye, Foliation by constant mean curvature spheres on asymptotically flat manifolds, Geometric analysis and the calculus of variations, 369-383, Int. Press, Cambridge, MA, 1996. Google Scholar

show all references

References:
[1]

V. Benci, C. Bonanno and A. M. Micheletti, On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds, J. Funct. Anal., 252 (2007), 464-489.   Google Scholar

[2]

J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calculus of Variations and Partial Differential Equations, 24 (2005), 459-477.   Google Scholar

[3]

E. N. Dancer, A. M. Micheletti and A. Pistoia, Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifold, Manuscripta Math., 128 (2009), 163-193.   Google Scholar

[4]

M. Del Pino, F. L. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79.   Google Scholar

[5]

S. Deng, Multipeak solutions for asymptotically critical elliptic equations on Riemannian manifolds, Nonlinear Analysis., 74 (2011), 859-881.   Google Scholar

[6]

P. Esposito and A. Pistoia, Blowing-up solutions for the Yamabe equation, Portugal. Math. (N.S.), 71 (2014), 249-276.   Google Scholar

[7]

M. Grossi and A. Pistoia, On the effect of critical points of distance function in superlinear elliptic problems, Adv. Differ. Equ., 5 (2000), 1397-1420.   Google Scholar

[8]

M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differ. Equ., 11 (2000), 143-175.   Google Scholar

[9]

C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.   Google Scholar

[10]

C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincare Anal. Non Linéaire, 17 (2000), 47-82.   Google Scholar

[11]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differ. Equ., 158 (1999), 1-27.   Google Scholar

[12]

J. M. Lee, John and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc., 17 (1987), 37-91.   Google Scholar

[13]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differ. Equ., 23 (1998), 487-545.   Google Scholar

[14]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equ., 72 (1988), 1-27.   Google Scholar

[15]

A. M. Micheletti and A. Pistoia, The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds, Calc. Var. Partial Differential Equations, 34 (2009), 233-265.   Google Scholar

[16]

A. M. Micheletti and A. Pistoia, Nodal solutions for a singularly perturbed nonlinear elliptic problem on Riemannian manifolds, Advanced Nonlinear Studies, 9 (2009), 565-577.   Google Scholar

[17]

A. M. Micheletti, A. Pistoia and J. Vétois, Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds, Indiana University Math. Journal, 58 (2009), 1719-1746.   Google Scholar

[18]

F. Mahmoudi, Constant k-curvature hypersurfaces in Riemannian manifolds, Differential Geom. Appl., 28 (2010), 1-11.   Google Scholar

[19]

W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.   Google Scholar

[20]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.   Google Scholar

[21]

F. Pacard and X. Xu, Constant mean curvature spheres in Riemannian manifolds, Manuscripta Math., 128 (2009), 275-295.   Google Scholar

[22]

S. Schoen, Conformal deformation of a Riemannian metric to a constant scalar curvature, J. Differential Geom., 20 (1984), 479-496.   Google Scholar

[23]

J. Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differ. Equ., 134 (1997), 104-133.   Google Scholar

[24]

J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J., 50 (1998), 159-178.   Google Scholar

[25]

J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. Lond. Math. Soc., 59 (1999), 585-606.   Google Scholar

[26]

H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 12 (1960), 21-37.   Google Scholar

[27]

R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math., 147 (1991), 381-396.   Google Scholar

[28]

R. Ye, Foliation by constant mean curvature spheres on asymptotically flat manifolds, Geometric analysis and the calculus of variations, 369-383, Int. Press, Cambridge, MA, 1996. Google Scholar

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