# American Institute of Mathematical Sciences

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:
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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
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