American Institute of Mathematical Sciences

January  2013, 12(1): 125-155. doi: 10.3934/cpaa.2013.12.125

A refined result on sign changing solutions for a critical elliptic problem

 1 Laboratoire d'Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Département de Mathématiques, Université Paris Est Créteil, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France 2 Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago 3 Dipartimento di Metodi e Modelli Matematici, Università di Roma "La Sapienza", Via Scarpa, 16 - 00166 Roma 4 University of Washington

Received  January 2011 Revised  April 2012 Published  September 2012

In this work, we consider sign changing solutions to the critical elliptic problem $\Delta u + |u|^{\frac{4}{N-2}}u = 0$ in $\Omega_\varepsilon$ and $u=0$ on $\partial\Omega_\varepsilon$, where $\Omega_\varepsilon:=\Omega-\left(\bigcup_{i=1}^m (a_i+\varepsilon\Omega_i)\right)$ for small parameter $\varepsilon>0$ is a perforated domain, $\Omega$ and $\Omega_i$ with $0\in \Omega_i$ ($\forall i=1,\cdots,m$) are bounded regular general domains without symmetry in $\mathbb{R}^N$ and $a_i$ are points in $\Omega$ for all $i=1,\cdots,m$. As $\varepsilon$ goes to zero, we construct by gluing method solutions with multiple blow up at each point $a_i$ for all $i=1,\cdots,m$.
Citation: Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure & Applied Analysis, 2013, 12 (1) : 125-155. doi: 10.3934/cpaa.2013.12.125
References:
 [1] 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 [2] A. Bahri, 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 Differ. Equ., 3 (1995), 67. doi: 10.1007/BF01190892. Google Scholar [3] T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, Top. Meth. Nonlin. Anal., 22 (2003), 1. 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 Differential Equationsm, 26 (2006), 265. doi: 10.1007/s00526-006-0004-6. Google Scholar [5] H. Brezis and L. Nirenberg, Positive solutions of non linear elliptic equations involving critical Sobolev expronents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [6] A. Byde, Gluing theorems for constant scalar curvature manifolds,, Indiana Univ. Math. J., 52 (2003), 1147. doi: 10.1512/iumj.2003.52.2109. Google Scholar [7] M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain,, Calc. Var. Partial Differential Equations, 21 (2004), 1. doi: 10.1007/s00526-003-0241-x. Google Scholar [8] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGill-Hill, (1955). Google Scholar [9] J. M. Coron, Topologie et cas limite des injections de Sobolev,, C. R. Acad. Sci. Paris Sr. I Math., 299 (1984), 209. Google Scholar [10] M. del Pino, J. Dolbeault and M. Musso, "Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem,, J. Differ. Equation, 193 (2003), 280. doi: 10.1016/S0022-0396(03)00151-7. Google Scholar [11] M. del Pino, J. Dolbeault and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3,, J. Math. Pures Appl., 83 (2004), 1405. doi: 10.1016/j.matpur.2004.02.007. Google Scholar [12] M. del Pino, M. Musso and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 45. doi: 10.1016/j.anihpc.2004.05.001. Google Scholar [13] M. del Pino, P. Felmer and M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes,, J. Differ. Equation, 182 (2002), 511. doi: 10.1006/jdeq.2001.4098. Google Scholar [14] M. del Pino, P. Felmer and M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries,, Bull. London Math. Soc., 35 (2003), 513. doi: 10.1112/S0024609303001942. Google Scholar [15] Y. Ge, R. Jing and F. Pacard, Bubble towers for supercritical semilinear elliptic equations,, J. Funct. Anal., 221 (2005), 251. doi: 10.1016/j.jfa.2004.09.011. Google Scholar [16] Y. Ge, R. Jing and F. Zhou, Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains,, DCDS-A, 17 (2007), 751. doi: 10.3934/dcds.2007.17.751. Google Scholar [17] 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,, Communications in Partial Differential Equations, 35 (2010), 1419. doi: 10.1080/03605302.2010.490286. Google Scholar [18] E. Hebey and M. Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth,, J. Funct. Anal., 119 (1994), 298. doi: 10.1006/jfan.1994.1012. Google Scholar [19] J. Kazdan, F. W. Warner, Remarks on some quasilinear elliptic equations,, Comm. Pure Appl. Math., 28 (1975), 567. doi: 10.1002/cpa.3160280502. Google Scholar [20] Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I.,, J. Differ. Equations, 120 (1995), 319. doi: 10.1006/jdeq.1995.1115. Google Scholar [21] R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities,, J. Duke Math., 99 (1999), 353. doi: 10.1215/S0012-7094-99-09913-1. Google Scholar [22] A. M. Micheletti and A. Pistoia, On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth,, Nonlinearity, 17 (2004), 851. doi: 10.1088/0951-7715/17/3/007. Google Scholar [23] 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 [24] M. Musso and A. Pistoia, Sign changing solutions to a Bahri-Coron's problem in pierced domains,, Discrete Contin. Dyn. Syst., 21 (2008), 295. doi: 10.3934/dcds.2008.21.295. Google Scholar [25] M. Musso and A. Pistoia, Persistence of Coron's solution in nearly critical problems,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 331. Google Scholar [26] M. Musso and A. Pistoia, Tower of Bubbles for almost critical problems in general domains,, J. Math. Pure Appl., 93 (2010), 1. doi: 10.1016/j.matpur.2009.08.001. Google Scholar [27] F. Pacard and T. Rivière, "Linear and Nonlinear Aspects of Vortices. The Ginzburg-Landau Model,", Progress in Nonlinear Differential Equations and their Applications, 39 (2000). doi: 10.1007/978-1-4612-1386-4. Google Scholar [28] A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 24 (2007), 325. doi: 10.1016/j.anihpc.2006.03.002. Google Scholar [29] S. I. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar [30] O. Rey, On a variational problem with lack of compactness: the effect of small holes in the domain,, C. R. Acad. Sci. Paris S\'er. I Math., 308 (1989), 349. Google Scholar [31] O. Rey, The role of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponetion,, J. Funct. Anal., 89 (1990), 1. doi: 10.1016/0022-1236(90)90002-3. Google Scholar

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References:
 [1] 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 [2] A. Bahri, 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 Differ. Equ., 3 (1995), 67. doi: 10.1007/BF01190892. Google Scholar [3] T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, Top. Meth. Nonlin. Anal., 22 (2003), 1. 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 Differential Equationsm, 26 (2006), 265. doi: 10.1007/s00526-006-0004-6. Google Scholar [5] H. Brezis and L. Nirenberg, Positive solutions of non linear elliptic equations involving critical Sobolev expronents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [6] A. Byde, Gluing theorems for constant scalar curvature manifolds,, Indiana Univ. Math. J., 52 (2003), 1147. doi: 10.1512/iumj.2003.52.2109. Google Scholar [7] M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain,, Calc. Var. Partial Differential Equations, 21 (2004), 1. doi: 10.1007/s00526-003-0241-x. Google Scholar [8] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGill-Hill, (1955). Google Scholar [9] J. M. Coron, Topologie et cas limite des injections de Sobolev,, C. R. Acad. Sci. Paris Sr. I Math., 299 (1984), 209. Google Scholar [10] M. del Pino, J. Dolbeault and M. Musso, "Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem,, J. Differ. Equation, 193 (2003), 280. doi: 10.1016/S0022-0396(03)00151-7. Google Scholar [11] M. del Pino, J. Dolbeault and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3,, J. Math. Pures Appl., 83 (2004), 1405. doi: 10.1016/j.matpur.2004.02.007. Google Scholar [12] M. del Pino, M. Musso and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 45. doi: 10.1016/j.anihpc.2004.05.001. Google Scholar [13] M. del Pino, P. Felmer and M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes,, J. Differ. Equation, 182 (2002), 511. doi: 10.1006/jdeq.2001.4098. Google Scholar [14] M. del Pino, P. Felmer and M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries,, Bull. London Math. Soc., 35 (2003), 513. doi: 10.1112/S0024609303001942. Google Scholar [15] Y. Ge, R. Jing and F. Pacard, Bubble towers for supercritical semilinear elliptic equations,, J. Funct. Anal., 221 (2005), 251. doi: 10.1016/j.jfa.2004.09.011. Google Scholar [16] Y. Ge, R. Jing and F. Zhou, Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains,, DCDS-A, 17 (2007), 751. doi: 10.3934/dcds.2007.17.751. Google Scholar [17] 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,, Communications in Partial Differential Equations, 35 (2010), 1419. doi: 10.1080/03605302.2010.490286. Google Scholar [18] E. Hebey and M. Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth,, J. Funct. Anal., 119 (1994), 298. doi: 10.1006/jfan.1994.1012. Google Scholar [19] J. Kazdan, F. W. Warner, Remarks on some quasilinear elliptic equations,, Comm. Pure Appl. Math., 28 (1975), 567. doi: 10.1002/cpa.3160280502. Google Scholar [20] Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I.,, J. Differ. Equations, 120 (1995), 319. doi: 10.1006/jdeq.1995.1115. Google Scholar [21] R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities,, J. Duke Math., 99 (1999), 353. doi: 10.1215/S0012-7094-99-09913-1. Google Scholar [22] A. M. Micheletti and A. Pistoia, On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth,, Nonlinearity, 17 (2004), 851. doi: 10.1088/0951-7715/17/3/007. Google Scholar [23] 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 [24] M. Musso and A. Pistoia, Sign changing solutions to a Bahri-Coron's problem in pierced domains,, Discrete Contin. Dyn. Syst., 21 (2008), 295. doi: 10.3934/dcds.2008.21.295. Google Scholar [25] M. Musso and A. Pistoia, Persistence of Coron's solution in nearly critical problems,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 331. Google Scholar [26] M. Musso and A. Pistoia, Tower of Bubbles for almost critical problems in general domains,, J. Math. Pure Appl., 93 (2010), 1. doi: 10.1016/j.matpur.2009.08.001. Google Scholar [27] F. Pacard and T. Rivière, "Linear and Nonlinear Aspects of Vortices. The Ginzburg-Landau Model,", Progress in Nonlinear Differential Equations and their Applications, 39 (2000). doi: 10.1007/978-1-4612-1386-4. Google Scholar [28] A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 24 (2007), 325. doi: 10.1016/j.anihpc.2006.03.002. Google Scholar [29] S. I. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar [30] O. Rey, On a variational problem with lack of compactness: the effect of small holes in the domain,, C. R. Acad. Sci. Paris S\'er. I Math., 308 (1989), 349. Google Scholar [31] O. Rey, The role of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponetion,, J. Funct. Anal., 89 (1990), 1. doi: 10.1016/0022-1236(90)90002-3. Google Scholar
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