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
$2<p<\frac{2N}{N-2}$
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.
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. BenciC. 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.

[2]

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

[3]

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

[4]

M. Del PinoF. 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.

[5]

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

[6]

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

[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.

[8]

M. GrossiA. 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.

[9]

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

[10]

C. GuiJ. 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.

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[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.

show all references

References:
[1]

V. BenciC. 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.

[2]

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

[3]

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

[4]

M. Del PinoF. 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.

[5]

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

[6]

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

[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.

[8]

M. GrossiA. 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.

[9]

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

[10]

C. GuiJ. 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.

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[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.

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