-
Previous Article
Concentration phenomena for critical fractional Schrödinger systems
- CPAA Home
- This Issue
-
Next Article
Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping
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 |
$(M, g)$ |
$N≥2$ |
$\begin{eqnarray*}-{\varepsilon}^2Δ_g u+ u = u^{p-1}, ~~~~u>0,\ \ \ \ \ in \ M.\end{eqnarray*}$ |
$Δ_g$ |
$M$ |
$p>2$ |
$N = 2$ |
$2<p<\frac{2N}{N-2}$ |
$N≥3$ |
$\varepsilon$ |
$Ξ$ |
$ξ_0$ |
$Ξ(ξ)$ |
${\varepsilon}_0>0$ |
${\varepsilon}∈(0,{\varepsilon}_0)$ |
$u_{\varepsilon}$ |
$ξ_0$ |
${\varepsilon}$ |
$(M,g)$ |
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.
|
[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. 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.
|
[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.
|
[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. 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.
|
[9] |
C. Gui,
Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.
|
[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.
|
[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. Lee, John 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. Lin, W. 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. 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.
|
[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. 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.
|
[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. 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.
|
[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.
|
[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. 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.
|
[9] |
C. Gui,
Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.
|
[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.
|
[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. Lee, John 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. Lin, W. 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. 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.
|
[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. |
[1] |
Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355 |
[2] |
Marc Massot. Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 433-456. doi: 10.3934/dcdsb.2002.2.433 |
[3] |
Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657 |
[4] |
Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807 |
[5] |
Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099 |
[6] |
Chao Ji, Vicenţiu D. Rădulescu. Concentration phenomena for magnetic Kirchhoff equations with critical growth. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5551-5577. doi: 10.3934/dcds.2021088 |
[7] |
Shan Jiang, Li Liang, Meiling Sun, Fang Su. Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28 (2) : 935-949. doi: 10.3934/era.2020049 |
[8] |
Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039 |
[9] |
Jean-François Babadjian, Francesca Prinari, Elvira Zappale. Dimensional reduction for supremal functionals. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1503-1535. doi: 10.3934/dcds.2012.32.1503 |
[10] |
Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279 |
[11] |
Monica Motta, Caterina Sartori. Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 513-535. doi: 10.3934/dcds.2008.21.513 |
[12] |
Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks and Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655 |
[13] |
Nikolaos S. Papageorgiou, Vicenšiu D. Rădulescu, Dušan D. Repovš. Robin problems with indefinite linear part and competition phenomena. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1293-1314. doi: 10.3934/cpaa.2017063 |
[14] |
Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783 |
[15] |
Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 |
[16] |
Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006 |
[17] |
Manuel del Pino. Supercritical elliptic problems from a perturbation viewpoint. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 69-89. doi: 10.3934/dcds.2008.21.69 |
[18] |
Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201 |
[19] |
Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049 |
[20] |
Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks and Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]