# 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 ##### 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  [1] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in$ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020447 [2] Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020349 [3] Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270 [4] Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020452 [5] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020351 [6] Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274 [7] Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular$ p $-Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020442 [8] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of$ L^2- \$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [9] Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460 [10] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [11] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [12] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [13] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [14] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385 [15] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [16] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [17] Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031 [18] Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446 [19] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [20] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

2019 Impact Factor: 1.105