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

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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

Yuxin Ge ^1, , Monica Musso ^2, , A. Pistoia ^3, and Daniel Pollack ^4,

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

Abstract
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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 $i=1,\cdots,m$.

Keywords: multiple blow up., Green function, Critical elliptic problem.

Mathematics Subject Classification: Primary: 35J60, 35J2.

Citation: Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure and 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), 253-294. doi: 10.1002/cpa.3160410302.
[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), 67-93. doi: 10.1007/BF01190892.
[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), 1-14.
[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), 265-282. doi: 10.1007/s00526-006-0004-6.
[5]	H. Brezis and L. Nirenberg, Positive solutions of non linear elliptic equations involving critical Sobolev expronents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[6]	A. Byde, Gluing theorems for constant scalar curvature manifolds, Indiana Univ. Math. J., 52 (2003), 1147-1199. doi: 10.1512/iumj.2003.52.2109.
[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), 1-14. doi: 10.1007/s00526-003-0241-x.
[8]	E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGill-Hill, New York, 1955.
[9]	J. M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sr. I Math., 299 (1984), 209-212.
[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), 280-306. doi: 10.1016/S0022-0396(03)00151-7.
[11]	M. del Pino, J. Dolbeault and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl., 83 (2004), 1405-1456. doi: 10.1016/j.matpur.2004.02.007.
[12]	M. del Pino, M. Musso and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 45-82. doi: 10.1016/j.anihpc.2004.05.001.
[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), 511-540. doi: 10.1006/jdeq.2001.4098.
[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), 513-521. doi: 10.1112/S0024609303001942.
[15]	Y. Ge, R. Jing and F. Pacard, Bubble towers for supercritical semilinear elliptic equations, J. Funct. Anal., 221 (2005), 251-302. doi: 10.1016/j.jfa.2004.09.011.
[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), 751-770. doi: 10.3934/dcds.2007.17.751.
[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), 1419-1457. doi: 10.1080/03605302.2010.490286.
[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), 298-318. doi: 10.1006/jfan.1994.1012.
[19]	J. Kazdan, F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597. doi: 10.1002/cpa.3160280502.
[20]	Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I., J. Differ. Equations, 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.
[21]	R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities, J. Duke Math., 99 (1999), 353-418. doi: 10.1215/S0012-7094-99-09913-1.
[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), 851-866. doi: 10.1088/0951-7715/17/3/007.
[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), 510-528. doi: 10.1016/j.matpur.2006.10.006.
[24]	M. Musso and A. Pistoia, Sign changing solutions to a Bahri-Coron's problem in pierced domains, Discrete Contin. Dyn. Syst., 21 (2008), 295-306. doi: 10.3934/dcds.2008.21.295.
[25]	M. Musso and A. Pistoia, Persistence of Coron's solution in nearly critical problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 331-357.
[26]	M. Musso and A. Pistoia, Tower of Bubbles for almost critical problems in general domains, J. Math. Pure Appl., 93 (2010), 1-40. doi: 10.1016/j.matpur.2009.08.001.
[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 Boston, Birkhäuser, 2000. doi: 10.1007/978-1-4612-1386-4.
[28]	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.
[29]	S. I. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
[30]	O. Rey, On a variational problem with lack of compactness: the effect of small holes in the domain, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 349-352.
[31]	O. Rey, The role of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponetion, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.

show all references

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), 253-294. doi: 10.1002/cpa.3160410302.
[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), 67-93. doi: 10.1007/BF01190892.
[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), 1-14.
[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), 265-282. doi: 10.1007/s00526-006-0004-6.
[5]	H. Brezis and L. Nirenberg, Positive solutions of non linear elliptic equations involving critical Sobolev expronents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[6]	A. Byde, Gluing theorems for constant scalar curvature manifolds, Indiana Univ. Math. J., 52 (2003), 1147-1199. doi: 10.1512/iumj.2003.52.2109.
[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), 1-14. doi: 10.1007/s00526-003-0241-x.
[8]	E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGill-Hill, New York, 1955.
[9]	J. M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sr. I Math., 299 (1984), 209-212.
[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), 280-306. doi: 10.1016/S0022-0396(03)00151-7.
[11]	M. del Pino, J. Dolbeault and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl., 83 (2004), 1405-1456. doi: 10.1016/j.matpur.2004.02.007.
[12]	M. del Pino, M. Musso and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 45-82. doi: 10.1016/j.anihpc.2004.05.001.
[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), 511-540. doi: 10.1006/jdeq.2001.4098.
[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), 513-521. doi: 10.1112/S0024609303001942.
[15]	Y. Ge, R. Jing and F. Pacard, Bubble towers for supercritical semilinear elliptic equations, J. Funct. Anal., 221 (2005), 251-302. doi: 10.1016/j.jfa.2004.09.011.
[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), 751-770. doi: 10.3934/dcds.2007.17.751.
[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), 1419-1457. doi: 10.1080/03605302.2010.490286.
[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), 298-318. doi: 10.1006/jfan.1994.1012.
[19]	J. Kazdan, F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597. doi: 10.1002/cpa.3160280502.
[20]	Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I., J. Differ. Equations, 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.
[21]	R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities, J. Duke Math., 99 (1999), 353-418. doi: 10.1215/S0012-7094-99-09913-1.
[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), 851-866. doi: 10.1088/0951-7715/17/3/007.
[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), 510-528. doi: 10.1016/j.matpur.2006.10.006.
[24]	M. Musso and A. Pistoia, Sign changing solutions to a Bahri-Coron's problem in pierced domains, Discrete Contin. Dyn. Syst., 21 (2008), 295-306. doi: 10.3934/dcds.2008.21.295.
[25]	M. Musso and A. Pistoia, Persistence of Coron's solution in nearly critical problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 331-357.
[26]	M. Musso and A. Pistoia, Tower of Bubbles for almost critical problems in general domains, J. Math. Pure Appl., 93 (2010), 1-40. doi: 10.1016/j.matpur.2009.08.001.
[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 Boston, Birkhäuser, 2000. doi: 10.1007/978-1-4612-1386-4.
[28]	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.
[29]	S. I. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
[30]	O. Rey, On a variational problem with lack of compactness: the effect of small holes in the domain, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 349-352.
[31]	O. Rey, The role of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponetion, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.

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