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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

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 & Continuous Dynamical Systems - A, 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. doi: 10.1016/j.jde.2013.02.015. Google Scholar

[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. doi: 10.1002/cpa.3160410302. Google Scholar

[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. doi: 10.1007/BF01190892. Google Scholar

[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. doi: 10.1007/s00526-006-0004-6. Google Scholar

[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. doi: 10.1007/s00526-012-0564-6. Google Scholar

[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., (). Google Scholar

[7]

J. M. Coron, Topologie et cas limite des injections de sobolev,, C. R. Acad. Sci. Paris Ser. I Math., 299 (1984), 209. Google Scholar

[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. doi: 10.1007/s005260100142. Google Scholar

[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. doi: 10.4171/JEMS/241. Google Scholar

[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. doi: 10.1016/j.anihpc.2006.03.001. Google Scholar

[11]

E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes,, Manuscripta Math, 123 (2007), 493. doi: 10.1007/s00229-007-0110-6. Google Scholar

[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. doi: 10.1080/03605302.2010.490286. Google Scholar

[13]

S. Kim and A. Pistoia, Boundary towers of layers for some supercritical problems,, J. Differential Equations, 255 (2013), 2302. doi: 10.1016/j.jde.2013.06.017. Google Scholar

[14]

S. Kim and A. Pistoia, Clustered boundary layer sign changing solutions for a supercritical problem,, J. London Math. Soc., 88 (2013), 227. doi: 10.1112/jlms/jdt006. Google Scholar

[15]

J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations,, Comm. Pure Appl. Math., 28 (1975), 567. doi: 10.1002/cpa.3160280502. Google Scholar

[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. doi: 10.1016/j.matpur.2006.10.006. Google Scholar

[17]

M. Musso and A. Pistoia, Tower of bubbles for almost critical problems in general domains,, J. Math. Pures Appl., 93 (2010), 1. doi: 10.1016/j.matpur.2009.08.001. Google Scholar

[18]

D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains,, J. Func. Anal., 114 (1993), 97. doi: 10.1006/jfan.1993.1064. Google Scholar

[19]

D. Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity,, Diff. Int. Equat., 8 (1995), 577. Google Scholar

[20]

S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar

[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. doi: 10.1016/j.anihpc.2006.03.002. Google Scholar

[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. doi: 10.1016/j.matpur.2011.01.006. Google Scholar

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. doi: 10.1016/j.jde.2013.02.015. Google Scholar

[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. doi: 10.1002/cpa.3160410302. Google Scholar

[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. doi: 10.1007/BF01190892. Google Scholar

[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. doi: 10.1007/s00526-006-0004-6. Google Scholar

[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. doi: 10.1007/s00526-012-0564-6. Google Scholar

[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., (). Google Scholar

[7]

J. M. Coron, Topologie et cas limite des injections de sobolev,, C. R. Acad. Sci. Paris Ser. I Math., 299 (1984), 209. Google Scholar

[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. doi: 10.1007/s005260100142. Google Scholar

[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. doi: 10.4171/JEMS/241. Google Scholar

[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. doi: 10.1016/j.anihpc.2006.03.001. Google Scholar

[11]

E. N. Dancer and J. Wei, Sign-changing solutions for supercritical elliptic problems in domains with small holes,, Manuscripta Math, 123 (2007), 493. doi: 10.1007/s00229-007-0110-6. Google Scholar

[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. doi: 10.1080/03605302.2010.490286. Google Scholar

[13]

S. Kim and A. Pistoia, Boundary towers of layers for some supercritical problems,, J. Differential Equations, 255 (2013), 2302. doi: 10.1016/j.jde.2013.06.017. Google Scholar

[14]

S. Kim and A. Pistoia, Clustered boundary layer sign changing solutions for a supercritical problem,, J. London Math. Soc., 88 (2013), 227. doi: 10.1112/jlms/jdt006. Google Scholar

[15]

J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations,, Comm. Pure Appl. Math., 28 (1975), 567. doi: 10.1002/cpa.3160280502. Google Scholar

[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. doi: 10.1016/j.matpur.2006.10.006. Google Scholar

[17]

M. Musso and A. Pistoia, Tower of bubbles for almost critical problems in general domains,, J. Math. Pures Appl., 93 (2010), 1. doi: 10.1016/j.matpur.2009.08.001. Google Scholar

[18]

D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains,, J. Func. Anal., 114 (1993), 97. doi: 10.1006/jfan.1993.1064. Google Scholar

[19]

D. Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity,, Diff. Int. Equat., 8 (1995), 577. Google Scholar

[20]

S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar

[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. doi: 10.1016/j.anihpc.2006.03.002. Google Scholar

[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. doi: 10.1016/j.matpur.2011.01.006. Google Scholar

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