Concentrating solutions for an anisotropic elliptic problem with large exponent

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August 2015, 35(8): 3771-3797. doi: 10.3934/dcds.2015.35.3771

Concentrating solutions for an anisotropic elliptic problem with large exponent

Liping Wang ^1, and Dong Ye ^2,

1.	Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
2.	IECL, UMR 7502, Département de Mathématiques, Université de Lorraine, Bât. A, Ile de Saulcy, 57045 Metz Cedex 1, France

Received September 2014 Revised October 2014 Published February 2015

Abstract
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We consider the following anisotropic boundary value problem $$\nabla (a(x)\nabla u) + a(x)u^p = 0, \;\; u>0 \ \ \mbox{in} \ \Omega, \quad u = 0 \ \ \mbox{on} \ \partial\Omega,$$ where $\Omega \subset \mathbb{R}^2$ is a bounded smooth domain, $p$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of anisotropic coefficient $a(x)$ on the existence of concentrating solutions. We show that at a given strict local maximum point of $a(x)$, there exist arbitrarily many concentrating solutions.

Keywords: large exponent, multi-bubble concentration., Anisotropic elliptic problem.

Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J4.

Citation: Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771

References:

[1]	Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2004), 1013-1019. doi: 10.1090/S0002-9939-03-07301-5.
[2]	T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère equations, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.
[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 Differential Equations, 3 (1995), 67-93. doi: 10.1007/BF01190892.
[4]	P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Diff. Equations, 4 (1999), 1-69.
[5]	D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142. doi: 10.1007/s002200000302.
[6]	C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (2002), 728-771. doi: 10.1002/cpa.3014.
[7]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[8]	E. N. Dancer and S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241.
[9]	M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 16 (2003), 113-145. doi: 10.1007/s005260100142.
[10]	M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81. doi: 10.1007/s00526-004-0314-5.
[11]	P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001.
[12]	P. Esposito, M. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Diff. Eqns., 227 (2006), 29-68. doi: 10.1016/j.jde.2006.01.023.
[13]	P. Esposito, M. Musso and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. math. Soc., 94 (2007), 497-519. doi: 10.1112/plms/pdl020.
[14]	P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbbR^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.
[15]	M. Flucher and J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math., 94 (1997), 337-346. doi: 10.1007/BF02677858.
[16]	I. M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29 (1963), 295-381.
[17]	Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Analyse Non Linéaire, 8 (1991), 159-174.
[18]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269.
[19]	S. Khenissy, Y. Rébaï and D. Ye, Expansion of the Green's function for divergence form operators, C. R. Math. Acad. Sci. Paris, 348 (2010), 891-896. doi: 10.1016/j.crma.2010.06.024.
[20]	L. Ma and J. Wei, Convergence for a Liouville equation, Comm. Math. Helv., 76 (2001), 506-514. doi: 10.1007/PL00013216.
[21]	K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlineat Stud., 4 (2004), 15-36.
[22]	F. Mignot, F. Murat and J. P. Puel, Variation d'un point retourment par rapport au domaine, Comm. Part. Diff. Equations, 4 (1979), 1263-1297. doi: 10.1080/03605307908820128.
[23]	X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763. doi: 10.1090/S0002-9947-1994-1232190-7.
[24]	X. Ren and J. Wei, Singular point condensation and least energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120. doi: 10.1090/S0002-9939-96-03156-5.
[25]	O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.
[26]	O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonlinear Anal., 13 (1989), 1241-1249. doi: 10.1016/0362-546X(89)90009-6.
[27]	J. Wei, D. Ye and F. Zhou, Bubbling slutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differential Equations, 28 (2007), 217-247. doi: 10.1007/s00526-006-0044-y.
[28]	D. Ye, Une remarque sur le comportement asymptotique des solutions de $- \Delta u = \lambda f(u)$, Comp. Rend. Acad. Sci. Paris, 325 (1997), 1279-1282. doi: 10.1016/S0764-4442(97)82353-1.

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References:

[1]	Adimurthi and M. Grossi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity, Proc. Amer. Math. Soc., 132 (2004), 1013-1019. doi: 10.1090/S0002-9939-03-07301-5.
[2]	T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère equations, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.
[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 Differential Equations, 3 (1995), 67-93. doi: 10.1007/BF01190892.
[4]	P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Diff. Equations, 4 (1999), 1-69.
[5]	D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142. doi: 10.1007/s002200000302.
[6]	C. C. Chen and C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (2002), 728-771. doi: 10.1002/cpa.3014.
[7]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.
[8]	E. N. Dancer and S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241.
[9]	M. del Pino, P. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 16 (2003), 113-145. doi: 10.1007/s005260100142.
[10]	M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81. doi: 10.1007/s00526-004-0314-5.
[11]	P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 22 (2005), 227-257. doi: 10.1016/j.anihpc.2004.12.001.
[12]	P. Esposito, M. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Diff. Eqns., 227 (2006), 29-68. doi: 10.1016/j.jde.2006.01.023.
[13]	P. Esposito, M. Musso and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. math. Soc., 94 (2007), 497-519. doi: 10.1112/plms/pdl020.
[14]	P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbbR^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.
[15]	M. Flucher and J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math., 94 (1997), 337-346. doi: 10.1007/BF02677858.
[16]	I. M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29 (1963), 295-381.
[17]	Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Analyse Non Linéaire, 8 (1991), 159-174.
[18]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973), 241-269.
[19]	S. Khenissy, Y. Rébaï and D. Ye, Expansion of the Green's function for divergence form operators, C. R. Math. Acad. Sci. Paris, 348 (2010), 891-896. doi: 10.1016/j.crma.2010.06.024.
[20]	L. Ma and J. Wei, Convergence for a Liouville equation, Comm. Math. Helv., 76 (2001), 506-514. doi: 10.1007/PL00013216.
[21]	K. El Mehdi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, Adv. Nonlineat Stud., 4 (2004), 15-36.
[22]	F. Mignot, F. Murat and J. P. Puel, Variation d'un point retourment par rapport au domaine, Comm. Part. Diff. Equations, 4 (1979), 1263-1297. doi: 10.1080/03605307908820128.
[23]	X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763. doi: 10.1090/S0002-9947-1994-1232190-7.
[24]	X. Ren and J. Wei, Singular point condensation and least energy solutions, Proc. Amer. Math. Soc., 124 (1996), 111-120. doi: 10.1090/S0002-9939-96-03156-5.
[25]	O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.
[26]	O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonlinear Anal., 13 (1989), 1241-1249. doi: 10.1016/0362-546X(89)90009-6.
[27]	J. Wei, D. Ye and F. Zhou, Bubbling slutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differential Equations, 28 (2007), 217-247. doi: 10.1007/s00526-006-0044-y.
[28]	D. Ye, Une remarque sur le comportement asymptotique des solutions de $- \Delta u = \lambda f(u)$, Comp. Rend. Acad. Sci. Paris, 325 (1997), 1279-1282. doi: 10.1016/S0764-4442(97)82353-1.

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