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

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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

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

Received March 2010 Revised October 2010 Published December 2010

Abstract
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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 and 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.
[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.
[3]	W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, preprint 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), 1-69.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[32]	J. Wei, On the effect of the domain geometry in singular perturbatation problems, Diff. Int. Eqns., 13 (2000), 15-45.
[33]	J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.
[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.

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.
[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.
[3]	W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, preprint 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), 1-69.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[32]	J. Wei, On the effect of the domain geometry in singular perturbatation problems, Diff. Int. Eqns., 13 (2000), 15-45.
[33]	J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.
[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.

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