# American Institute of Mathematical Sciences

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

## Supercritical problems in domains with thin toroidal holes

 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.
Citation: Seunghyeok Kim, Angela Pistoia. Supercritical problems in domains with thin toroidal holes. Discrete & 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.  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), 253-294. 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), 67-93. 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), 265-282. 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), 611-623. 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), 209-212.  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), 113-145. 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), 1553-1605. 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), 507-520. 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), 493-511. 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), 1419-1457. 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), 2302-2339. 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), 227-250. 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), 567-597. 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), 510-528. 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), 1-40. 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), 97-105. 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), 577-586.  Google Scholar [20] S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  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), 325-340. 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), 307-333. doi: 10.1016/j.matpur.2011.01.006.  Google Scholar

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##### 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.  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), 253-294. 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), 67-93. 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), 265-282. 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), 611-623. 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), 209-212.  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), 113-145. 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), 1553-1605. 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), 507-520. 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), 493-511. 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), 1419-1457. 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), 2302-2339. 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), 227-250. 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), 567-597. 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), 510-528. 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), 1-40. 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), 97-105. 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), 577-586.  Google Scholar [20] S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  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), 325-340. 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), 307-333. doi: 10.1016/j.matpur.2011.01.006.  Google Scholar
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