Supercritical problems in domains with thin toroidal holes

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November 2014, 34(11): 4671-4688. doi: 10.3934/dcds.2014.34.4671

Supercritical problems in domains with thin toroidal holes

Seunghyeok Kim ^1, and Angela Pistoia ^2,

1.	Departamento de Matemática, Pontificia Universidad Catóica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
2.	Dipartimento SBAI, Università di Roma "La Sapienza", via Antonio Scarpa 16, 00161 Roma

Received September 2013 Revised December 2013 Published May 2014

Abstract
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In this paper we study the Lane-Emden-Fowler equation $$ (P)_ \epsilon \quad \left\{ \begin{aligned} &\Delta u+|u|^{q-2}u=0\ &\hbox{in}\ \mathcal D_ \epsilon,\\ & u=0\ &\hbox{on}\ \partial\mathcal D_ \epsilon.\\ \end{aligned}\right. $$ Here $\mathcal D_ \epsilon=\mathcal D\setminus \left\{x\in \mathcal D\ :\ \mathrm{dist}(x,\Gamma_l)\le \epsilon \right\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_l$ is an $l-$dimensional closed manifold such that $\Gamma_l\subset\mathcal D$ with $1\le l\le N-3$ and $q={2(N-l)\over N-l-2} .$ We prove that, under some symmetry assumptions, the number of sign changing solutions to $ (P)_ \epsilon$ increases as $\epsilon$ goes to zero.

Keywords: concentration on $l-$dimensional manifolds., Supercritical problem.

Mathematics Subject Classification: Primary: 35J60; Secondary: 35J2.

Citation: Seunghyeok Kim, Angela Pistoia. Supercritical problems in domains with thin toroidal holes. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4671-4688. doi: 10.3934/dcds.2014.34.4671

References:

[1]	N. Ackermann, M. Clápp and A. Pistoia, Boundary clustered layers near the higher critical exponents, J. Differential Equations, 254 (2013), 4168-4193. doi: 10.1016/j.jde.2013.02.015.
[2]	A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.
[3]	A. Bahri, Y.-Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Diff. Eq., 3 (1995), 67-93. doi: 10.1007/BF01190892.
[4]	T. Bartsch, A. M. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Diff. Eq., 26 (2006), 265-282. doi: 10.1007/s00526-006-0004-6.
[5]	M. Clapp, J. Faya and A. Pistoia, Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents, Calc. Var. Partial Diff. Eq., 48 (2013), 611-623. doi: 10.1007/s00526-012-0564-6.
[6]	M. Clapp, J. Faya and A. Pistoia, Positive solutions to a supercritical elliptic problem which concentrate along a think spherical hole, J. Anal. Math., in press.
[7]	J. M. Coron, Topologie et cas limite des injections de sobolev, C. R. Acad. Sci. Paris Ser. I Math., 299 (1984), 209-212.
[8]	M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Diff. Eq., 16 (2003), 113-145. doi: 10.1007/s005260100142.
[9]	M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent, J. Eur. Math. Soc., 12 (2010), 1553-1605. doi: 10.4171/JEMS/241.
[10]	M. del Pino and J. Wei, Supercritical elliptic problems in domains with small holes, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 507-520. doi: 10.1016/j.anihpc.2006.03.001.
[11]	E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscripta Math, 123 (2007), 493-511. doi: 10.1007/s00229-007-0110-6.
[12]	Y. Ge, M. Musso and A. Pistoia, Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains, Commun. Partial Differ. Equ., 35 (2010), 1419-1457. doi: 10.1080/03605302.2010.490286.
[13]	S. Kim and A. Pistoia, Boundary towers of layers for some supercritical problems, J. Differential Equations, 255 (2013), 2302-2339. doi: 10.1016/j.jde.2013.06.017.
[14]	S. Kim and A. Pistoia, Clustered boundary layer sign changing solutions for a supercritical problem, J. London Math. Soc., 88 (2013), 227-250. doi: 10.1112/jlms/jdt006.
[15]	J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597. doi: 10.1002/cpa.3160280502.
[16]	M. Musso and A. Pistoia, Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains, J. Math. Pures Appl., 86 (2006), 510-528. doi: 10.1016/j.matpur.2006.10.006.
[17]	M. Musso and A. Pistoia, Tower of bubbles for almost critical problems in general domains, J. Math. Pures Appl., 93 (2010), 1-40. doi: 10.1016/j.matpur.2009.08.001.
[18]	D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Func. Anal., 114 (1993), 97-105. doi: 10.1006/jfan.1993.1064.
[19]	D. Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity, Diff. Int. Equat., 8 (1995), 577-586.
[20]	S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
[21]	A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340. doi: 10.1016/j.anihpc.2006.03.002.
[22]	S. Yan and J. Wei, Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math Pures Appl., 96 (2011), 307-333. doi: 10.1016/j.matpur.2011.01.006.

show all references

References:

[1]	N. Ackermann, M. Clápp and A. Pistoia, Boundary clustered layers near the higher critical exponents, J. Differential Equations, 254 (2013), 4168-4193. doi: 10.1016/j.jde.2013.02.015.
[2]	A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.
[3]	A. Bahri, Y.-Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Diff. Eq., 3 (1995), 67-93. doi: 10.1007/BF01190892.
[4]	T. Bartsch, A. M. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Diff. Eq., 26 (2006), 265-282. doi: 10.1007/s00526-006-0004-6.
[5]	M. Clapp, J. Faya and A. Pistoia, Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents, Calc. Var. Partial Diff. Eq., 48 (2013), 611-623. doi: 10.1007/s00526-012-0564-6.
[6]	M. Clapp, J. Faya and A. Pistoia, Positive solutions to a supercritical elliptic problem which concentrate along a think spherical hole, J. Anal. Math., in press.
[7]	J. M. Coron, Topologie et cas limite des injections de sobolev, C. R. Acad. Sci. Paris Ser. I Math., 299 (1984), 209-212.
[8]	M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Diff. Eq., 16 (2003), 113-145. doi: 10.1007/s005260100142.
[9]	M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent, J. Eur. Math. Soc., 12 (2010), 1553-1605. doi: 10.4171/JEMS/241.
[10]	M. del Pino and J. Wei, Supercritical elliptic problems in domains with small holes, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 507-520. doi: 10.1016/j.anihpc.2006.03.001.
[11]	E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes, Manuscripta Math, 123 (2007), 493-511. doi: 10.1007/s00229-007-0110-6.
[12]	Y. Ge, M. Musso and A. Pistoia, Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains, Commun. Partial Differ. Equ., 35 (2010), 1419-1457. doi: 10.1080/03605302.2010.490286.
[13]	S. Kim and A. Pistoia, Boundary towers of layers for some supercritical problems, J. Differential Equations, 255 (2013), 2302-2339. doi: 10.1016/j.jde.2013.06.017.
[14]	S. Kim and A. Pistoia, Clustered boundary layer sign changing solutions for a supercritical problem, J. London Math. Soc., 88 (2013), 227-250. doi: 10.1112/jlms/jdt006.
[15]	J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597. doi: 10.1002/cpa.3160280502.
[16]	M. Musso and A. Pistoia, Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains, J. Math. Pures Appl., 86 (2006), 510-528. doi: 10.1016/j.matpur.2006.10.006.
[17]	M. Musso and A. Pistoia, Tower of bubbles for almost critical problems in general domains, J. Math. Pures Appl., 93 (2010), 1-40. doi: 10.1016/j.matpur.2009.08.001.
[18]	D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Func. Anal., 114 (1993), 97-105. doi: 10.1006/jfan.1993.1064.
[19]	D. Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity, Diff. Int. Equat., 8 (1995), 577-586.
[20]	S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
[21]	A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340. doi: 10.1016/j.anihpc.2006.03.002.
[22]	S. Yan and J. Wei, Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math Pures Appl., 96 (2011), 307-333. doi: 10.1016/j.matpur.2011.01.006.

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