March  2011, 10(2): 731-744. doi: 10.3934/cpaa.2011.10.731

The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem

1. 

Institute of Mathematics, Hangzhou Dianzi Universitye, Xiasha Hangzhou Zhejiang 310018, China

Received  March 2010 Revised  October 2010 Published  December 2010

We consider the following singularly perturbed elliptic problem

$\varepsilon^2 \Delta u-u+f(u)=0, u>0 $ in $B_1$,

$\frac{\partial u}{\partial \nu}=0 $ on $\partial B_1,$

where $\Delta = \sum_{i=1}^N \frac{\partial^2}{\partial x_i^2}$ is the Laplace operator, $B_1$ is the unit ball centered at the origin in $R^N$ $(N\ge 3)$, $\nu$ denotes the unit outer normal to $\partial B_1$, $\varepsilon > 0$ is a constant, and $f$ is a superlinear, subcritical nonlinearity . We will show that when $\e$ is sufficiently small there exists a solution with K interior peaks located on a hyperplane, where $1\le K \varepsilon\frac{C}{(\varepsilon)^{N-1}}$ with $C$ a positive constant depending on $N$ and $f$ only. As a consequence, we obtain that there exists at least $[\frac{C}{(\varepsilon)^{N-1}}]$ number of solutions for $\varepsilon$ sufficiently small.

Citation: Yang Wang. The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem. Communications on Pure & Applied Analysis, 2011, 10 (2) : 731-744. doi: 10.3934/cpaa.2011.10.731
References:
[1]

A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I,, Comm. Math. Phys, 235 (2003), 427. doi: doi:10.1007/s00220-003-0811-y. Google Scholar

[2]

A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part II,, Indiana Univ. Math. J, 53 (2004), 297. doi: doi:10.1512/iumj.2004.53.2400. Google Scholar

[3]

W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem,, preprint 2010., (2010). Google Scholar

[4]

P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability,, Adv. Diff. Eqns, 4 (1999), 1. Google Scholar

[5]

P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation,, J. Diff. Eqns, 160 (2000), 283. doi: doi:10.1006/jdeq.1999.3660. Google Scholar

[6]

M. del Pino and D. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: doi:10.1512/iumj.1999.48.1596. Google Scholar

[7]

M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (1999), 63. doi: doi:10.1137/S0036141098332834. Google Scholar

[8]

M. del Pino, P. Felmer and J. Wei, On the role of distance function in some singularly perturbed problems,, Comm. P.D.E., 25 (2000), 155. doi: doi:10.1080/03605300008821511. Google Scholar

[9]

M. del Pino, P. Felmer and J. Wei, Multiple-peak solutions for some singular perturbation problems, , Cal. Var. P.D.E., 10 (2000), 119. doi: doi:10.1007/s005260050147. Google Scholar

[10]

E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem,, Pacific. J. Math., 189 (1999), 241. doi: doi:10.2140/pjm.1999.189.241. Google Scholar

[11]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^N$,, in, (1981), 369. Google Scholar

[12]

R. Gardner and L. A. Peletier, The set of positive solutions of semilinear equations in large balls,, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 53. Google Scholar

[13]

C. Gui and J. Wei, Multiple interior spike solutions for some singularly perturbed Neumann problems,, J. Diff. Eqns., 158 (1999), 1. doi: doi:10.1016/S0022-0396(99)80016-3. Google Scholar

[14]

C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems,, Can. J. Math., 52 (2000), 522. doi: doi:10.4153/CJM-2000-024-x. Google Scholar

[15]

C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 17 (2000), 47. Google Scholar

[16]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition,, Comm. P.D.E., 23 (1998), 487. doi: doi:10.1080/03605309808821354. Google Scholar

[17]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445. doi: doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. Google Scholar

[18]

C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems,, J. Diff. Eqns., 72 (1988), 1. doi: doi:10.1016/0022-0396(88)90147-7. Google Scholar

[19]

F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem,, Comm. Pure Appl. Math., 60 (2007), 252. doi: doi:10.1002/cpa.20139. Google Scholar

[20]

A. Malchiodi, Solutions concentrating at curves for some singularly perturbed elliptic problems,, C. R. Math. Acad. Sci. Paris, 338 (2004), 775. doi: doi:10.1016/j.crma.2004.03.023. Google Scholar

[21]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: doi:10.1002/cpa.10049. Google Scholar

[22]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: doi:10.1215/S0012-7094-04-12414-5. Google Scholar

[23]

A. Malchiodi, W.-M. Ni and J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 143. Google Scholar

[24]

W.-M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem,, Comm. Pure Appl. Math., 41 (1991), 819. doi: doi:10.1002/cpa.3160440705. Google Scholar

[25]

W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247. doi: doi:10.1215/S0012-7094-93-07004-4. Google Scholar

[26]

W.-M. Ni, I. Takagi and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems: intermediate solutions,, Duke Math. J., 94 (1998), 597. doi: doi:10.1215/S0012-7094-98-09424-8. Google Scholar

[27]

W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 48 (1995), 731. doi: doi:10.1002/cpa.3160480704. Google Scholar

[28]

Yang Wang, Concentration phenomena of solutions for some singularly perturbed elliptic equations,, J. Math. Anal. Appl., 331 (2007), 927. doi: doi:10.1016/j.jmaa.2006.09.029. Google Scholar

[29]

J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problems,, J. Diff. Eqns., 129 (1996), 315. doi: doi:10.1006/jdeq.1996.0120. Google Scholar

[30]

J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem,, J. Diff. Eqns., 134 (1997), 104. doi: doi:10.1006/jdeq.1996.3218. Google Scholar

[31]

J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem,, Tohoku Math. J., 50 (1998), 159. doi: doi:10.2748/tmj/1178224971. Google Scholar

[32]

J. Wei, On the effect of the domain geometry in singular perturbatation problems,, Diff. Int. Eqns., 13 (2000), 15. Google Scholar

[33]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 15 (1998), 459. Google Scholar

[34]

J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems,, J. London Math. Soc., 59 (1999), 585. doi: doi:10.1112/S002461079900719X. Google Scholar

show all references

References:
[1]

A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I,, Comm. Math. Phys, 235 (2003), 427. doi: doi:10.1007/s00220-003-0811-y. Google Scholar

[2]

A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part II,, Indiana Univ. Math. J, 53 (2004), 297. doi: doi:10.1512/iumj.2004.53.2400. Google Scholar

[3]

W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem,, preprint 2010., (2010). Google Scholar

[4]

P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability,, Adv. Diff. Eqns, 4 (1999), 1. Google Scholar

[5]

P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation,, J. Diff. Eqns, 160 (2000), 283. doi: doi:10.1006/jdeq.1999.3660. Google Scholar

[6]

M. del Pino and D. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: doi:10.1512/iumj.1999.48.1596. Google Scholar

[7]

M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (1999), 63. doi: doi:10.1137/S0036141098332834. Google Scholar

[8]

M. del Pino, P. Felmer and J. Wei, On the role of distance function in some singularly perturbed problems,, Comm. P.D.E., 25 (2000), 155. doi: doi:10.1080/03605300008821511. Google Scholar

[9]

M. del Pino, P. Felmer and J. Wei, Multiple-peak solutions for some singular perturbation problems, , Cal. Var. P.D.E., 10 (2000), 119. doi: doi:10.1007/s005260050147. Google Scholar

[10]

E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem,, Pacific. J. Math., 189 (1999), 241. doi: doi:10.2140/pjm.1999.189.241. Google Scholar

[11]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^N$,, in, (1981), 369. Google Scholar

[12]

R. Gardner and L. A. Peletier, The set of positive solutions of semilinear equations in large balls,, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 53. Google Scholar

[13]

C. Gui and J. Wei, Multiple interior spike solutions for some singularly perturbed Neumann problems,, J. Diff. Eqns., 158 (1999), 1. doi: doi:10.1016/S0022-0396(99)80016-3. Google Scholar

[14]

C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems,, Can. J. Math., 52 (2000), 522. doi: doi:10.4153/CJM-2000-024-x. Google Scholar

[15]

C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 17 (2000), 47. Google Scholar

[16]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition,, Comm. P.D.E., 23 (1998), 487. doi: doi:10.1080/03605309808821354. Google Scholar

[17]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445. doi: doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. Google Scholar

[18]

C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems,, J. Diff. Eqns., 72 (1988), 1. doi: doi:10.1016/0022-0396(88)90147-7. Google Scholar

[19]

F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem,, Comm. Pure Appl. Math., 60 (2007), 252. doi: doi:10.1002/cpa.20139. Google Scholar

[20]

A. Malchiodi, Solutions concentrating at curves for some singularly perturbed elliptic problems,, C. R. Math. Acad. Sci. Paris, 338 (2004), 775. doi: doi:10.1016/j.crma.2004.03.023. Google Scholar

[21]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: doi:10.1002/cpa.10049. Google Scholar

[22]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: doi:10.1215/S0012-7094-04-12414-5. Google Scholar

[23]

A. Malchiodi, W.-M. Ni and J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 143. Google Scholar

[24]

W.-M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem,, Comm. Pure Appl. Math., 41 (1991), 819. doi: doi:10.1002/cpa.3160440705. Google Scholar

[25]

W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247. doi: doi:10.1215/S0012-7094-93-07004-4. Google Scholar

[26]

W.-M. Ni, I. Takagi and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems: intermediate solutions,, Duke Math. J., 94 (1998), 597. doi: doi:10.1215/S0012-7094-98-09424-8. Google Scholar

[27]

W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 48 (1995), 731. doi: doi:10.1002/cpa.3160480704. Google Scholar

[28]

Yang Wang, Concentration phenomena of solutions for some singularly perturbed elliptic equations,, J. Math. Anal. Appl., 331 (2007), 927. doi: doi:10.1016/j.jmaa.2006.09.029. Google Scholar

[29]

J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problems,, J. Diff. Eqns., 129 (1996), 315. doi: doi:10.1006/jdeq.1996.0120. Google Scholar

[30]

J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem,, J. Diff. Eqns., 134 (1997), 104. doi: doi:10.1006/jdeq.1996.3218. Google Scholar

[31]

J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem,, Tohoku Math. J., 50 (1998), 159. doi: doi:10.2748/tmj/1178224971. Google Scholar

[32]

J. Wei, On the effect of the domain geometry in singular perturbatation problems,, Diff. Int. Eqns., 13 (2000), 15. Google Scholar

[33]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 15 (1998), 459. Google Scholar

[34]

J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems,, J. London Math. Soc., 59 (1999), 585. doi: doi:10.1112/S002461079900719X. Google Scholar

[1]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237

[2]

Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823

[3]

Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465

[4]

Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure & Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771

[5]

Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351

[6]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1001-1030. doi: 10.3934/dcds.2011.29.1001

[7]

Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure & Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341

[8]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[9]

Yi He, Gongbao Li. Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity. Mathematical Control & Related Fields, 2016, 6 (4) : 551-593. doi: 10.3934/mcrf.2016016

[10]

J. W. Choi, D. S. Lee, S. H. Oh, S. M. Sun, S. I. Whang. Multi-hump solutions of some singularly-perturbed equations of KdV type. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5181-5209. doi: 10.3934/dcds.2014.34.5181

[11]

Nara Bobko, Jorge P. Zubelli. A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 1-21. doi: 10.3934/mbe.2015.12.1

[12]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499

[13]

Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041

[14]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709

[15]

Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211

[16]

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

[17]

Adam Bobrowski, Radosław Bogucki. Two theorems on singularly perturbed semigroups with applications to models of applied mathematics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 735-757. doi: 10.3934/dcdsb.2012.17.735

[18]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737

[19]

Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058

[20]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]