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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

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