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
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show all references

References:
[1]

J. Differential Equations, 254 (2013), 4168-4193. doi: 10.1016/j.jde.2013.02.015.  Google Scholar

[2]

Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.  Google Scholar

[3]

Calc. Var. Partial Diff. Eq., 3 (1995), 67-93. doi: 10.1007/BF01190892.  Google Scholar

[4]

Calc. Var. Partial Diff. Eq., 26 (2006), 265-282. doi: 10.1007/s00526-006-0004-6.  Google Scholar

[5]

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]

C. R. Acad. Sci. Paris Ser. I Math., 299 (1984), 209-212.  Google Scholar

[8]

Calc. Var. Partial Diff. Eq., 16 (2003), 113-145. doi: 10.1007/s005260100142.  Google Scholar

[9]

J. Eur. Math. Soc., 12 (2010), 1553-1605. doi: 10.4171/JEMS/241.  Google Scholar

[10]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 507-520. doi: 10.1016/j.anihpc.2006.03.001.  Google Scholar

[11]

Manuscripta Math, 123 (2007), 493-511. doi: 10.1007/s00229-007-0110-6.  Google Scholar

[12]

Commun. Partial Differ. Equ., 35 (2010), 1419-1457. doi: 10.1080/03605302.2010.490286.  Google Scholar

[13]

J. Differential Equations, 255 (2013), 2302-2339. doi: 10.1016/j.jde.2013.06.017.  Google Scholar

[14]

J. London Math. Soc., 88 (2013), 227-250. doi: 10.1112/jlms/jdt006.  Google Scholar

[15]

Comm. Pure Appl. Math., 28 (1975), 567-597. doi: 10.1002/cpa.3160280502.  Google Scholar

[16]

J. Math. Pures Appl., 86 (2006), 510-528. doi: 10.1016/j.matpur.2006.10.006.  Google Scholar

[17]

J. Math. Pures Appl., 93 (2010), 1-40. doi: 10.1016/j.matpur.2009.08.001.  Google Scholar

[18]

J. Func. Anal., 114 (1993), 97-105. doi: 10.1006/jfan.1993.1064.  Google Scholar

[19]

Diff. Int. Equat., 8 (1995), 577-586.  Google Scholar

[20]

Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  Google Scholar

[21]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340. doi: 10.1016/j.anihpc.2006.03.002.  Google Scholar

[22]

J. Math Pures Appl., 96 (2011), 307-333. doi: 10.1016/j.matpur.2011.01.006.  Google Scholar

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