American Institute of Mathematical Sciences

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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

Yang Wang 1,

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.

Keywords: Multiple interior peaks, singularly perturbed equations..
Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J4.
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), 427-466. 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), 297-329. doi: doi:10.1512/iumj.2004.53.2400.  Google Scholar

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W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, preprint 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), 1-69.  Google Scholar

[5]

P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns, 160 (2000), 283-356. doi: doi:10.1006/jdeq.1999.3660.  Google Scholar

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M. del Pino and D. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898. doi: doi:10.1512/iumj.1999.48.1596.  Google Scholar

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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), 63-79. doi: doi:10.1137/S0036141098332834.  Google Scholar

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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), 155-177. doi: doi:10.1080/03605300008821511.  Google Scholar

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M. del Pino, P. Felmer and J. Wei, Multiple-peak solutions for some singular perturbation problems, Cal. Var. P.D.E., 10 (2000), 119-134. doi: doi:10.1007/s005260050147.  Google Scholar

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E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific. J. Math., 189 (1999), 241-262. doi: doi:10.2140/pjm.1999.189.241.  Google Scholar

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B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^N$, in "Mathematical Analysis and Applications," Part A, Adv. Math. Suppl. Studies, Vol. 7A, pp.369-402, Academic Press, New York, 1981. 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), 53-72.  Google Scholar

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C. Gui and J. Wei, Multiple interior spike solutions for some singularly perturbed Neumann problems, J. Diff. Eqns., 158 (1999), 1-27. 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), 522-538. 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. Poincaré Anal. Non Linéaire, 17 (2000), 47-82.  Google Scholar

[16]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. P.D.E., 23 (1998), 487-545. 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), 1445-1490. doi: doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z.  Google Scholar

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C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Diff. Eqns., 72 (1988), 1-27. doi: doi:10.1016/0022-0396(88)90147-7.  Google Scholar

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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), 252-281. 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), 775-780. 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), 1507-1568. 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), 105-143. 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. Poincaré Anal. Non Linéaire, 22 (2005), 143-163.  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), 819-851. 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), 247-281. 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), 597-618. 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), 731-768. 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) 927-946. 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), 315-333. 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), 104-133. 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), 159-178. 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), 15-45.  Google Scholar

[33]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.  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), 585-606. 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), 427-466. 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), 297-329. 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. 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), 1-69.  Google Scholar

[5]

P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns, 160 (2000), 283-356. 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), 883-898. 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), 63-79. 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), 155-177. 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), 119-134. 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), 241-262. 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 "Mathematical Analysis and Applications," Part A, Adv. Math. Suppl. Studies, Vol. 7A, pp.369-402, Academic Press, New York, 1981. 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), 53-72.  Google Scholar

[13]

C. Gui and J. Wei, Multiple interior spike solutions for some singularly perturbed Neumann problems, J. Diff. Eqns., 158 (1999), 1-27. 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), 522-538. 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. Poincaré Anal. Non Linéaire, 17 (2000), 47-82.  Google Scholar

[16]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. P.D.E., 23 (1998), 487-545. 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), 1445-1490. 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), 1-27. 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), 252-281. 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), 775-780. 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), 1507-1568. 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), 105-143. 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. Poincaré Anal. Non Linéaire, 22 (2005), 143-163.  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), 819-851. 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), 247-281. 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), 597-618. 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), 731-768. 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) 927-946. 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), 315-333. 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), 104-133. 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), 159-178. 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), 15-45.  Google Scholar

[33]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.  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), 585-606. doi: doi:10.1112/S002461079900719X.  Google Scholar

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