Infinitely many sign-changing solutions for the Brézis-Nirenberg problem

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November 2014, 13(6): 2317-2330. doi: 10.3934/cpaa.2014.13.2317

Infinitely many sign-changing solutions for the Brézis-Nirenberg problem

Jijiang Sun ^1, and Shiwang Ma ^1,

1.	School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071

Received September 2013 Revised March 2014 Published July 2014

Abstract
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In this paper, we present a new proof on the existence of infinitely many sign-changing solutions for the following Brézis-Nirenberg problem \begin{eqnarray} -\Delta u=\lambda u+|u|^{2^{*}-2}u \quad \textrm{in}\, \Omega, \qquad u=0 \quad \textrm{on}\,\partial\Omega, \end{eqnarray} for each fixed $\lambda>0$, under the assumptions that $N\geq 7$, where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent. In order to construct sign-changing solutions, we will use a combination of invariant sets method and Ljusternik-Schnirelman type minimax method, which is much simpler than the proof of [20] depending on the estimates of Morse indices of nodal solutions to obtain the same result.

Keywords: minimax method., sign-changing solutions, Critical exponent, invariant sets.

Mathematics Subject Classification: Primary: 35J60; Secondary: 47J30, 58E0.

Citation: Jijiang Sun, Shiwang Ma. Infinitely many sign-changing solutions for the Brézis-Nirenberg problem. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2317-2330. doi: 10.3934/cpaa.2014.13.2317

References:

[1]	G. Arioli, F. Gazzola, H. C. Grunau and E. Sassone, The second bifurcation branch for radial solutions of the Brézis-Nirenberg problem in dimension four}, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 69-90. doi: 10.1007/s00030-007-6034-8.
[2]	F. V. Atkinson, H. Brézis and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differential Equations, 85 (1990), 151-170. doi: 10.1016/0022-0396(90)90093-5.
[3]	T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42. doi: 10.1081/PDE-120028842.
[4]	T. Bartsch, Z. L. Liu and T. Weth, Nodal solutions of a $p$-Laplcain equation, Proc. London Math. Soc., 91 (2005), 129-152. doi: 10.1112/S0024611504015187.
[5]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[6]	D. M. Cao and S. S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations, 38 (2010), 471-501. doi: 10.1007/s00526-009-0295-5.
[7]	D. M. Cao, S. J. Peng and S. S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902. doi: 10.1016/j.jfa.2012.01.006.
[8]	A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéarier, 2 (1985), 463-470.
[9]	G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéarier, 1 (1984), 341-350.
[10]	G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306. doi: 10.1016/0022-1236(86)90094-7.
[11]	Z. J. Chen and W. M. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987. doi: 10.1016/j.jde.2011.09.042.
[12]	M. Clapp and T. Weth, Multiple solutions for the Brézis-Nirenberg problem, Adv. Differ. Equ., 10 (2005), 463-480.
[13]	G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differ. Equ., 7 (2002), 1257-1280.
[14]	G. Devillanova and S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension, Comm. Contemp. Math., 5 (2003), 171-177. doi: 10.1142/S0219199703000938.
[15]	D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 205-213. doi: 10.1017/S0308210500022046.
[16]	S. J. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227. doi: 10.1090/S0002-9947-02-03031-3.
[17]	P. L. Lions, The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Iberoamericana, 1 (1985), 45-121, 145-201. doi: 10.4171/RMI/6.
[18]	Z. L. Liu, F. A. van Heerden and Z.-Q. Wang, Nodal type bound states of Schrödinger equations via invariant set and minimax methods, J. Differential Equations, 214 (2005), 358-390. doi: 10.1016/j.jde.2004.08.023.
[19]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986.
[20]	M. Schechter and W. Zou, On the Brézis-Nirenberg problem, Arch. Rational Mech. Anal., 197 (2010), 337-356. doi: 10.1007/s00205-009-0288-8.
[21]	S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sovolev space, Ann. Inst. H. Poincaré Anal. Non Linéarier, 12 (1995), 319-337.
[22]	M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.
[23]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04194-9.
[24]	A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differential Integral Equations, 22 (2009), 913-926.
[25]	S. S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents, Chinese Ann. Math. Ser. A, 16 (1995), 227-234.

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References:

[1]	G. Arioli, F. Gazzola, H. C. Grunau and E. Sassone, The second bifurcation branch for radial solutions of the Brézis-Nirenberg problem in dimension four}, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 69-90. doi: 10.1007/s00030-007-6034-8.
[2]	F. V. Atkinson, H. Brézis and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differential Equations, 85 (1990), 151-170. doi: 10.1016/0022-0396(90)90093-5.
[3]	T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42. doi: 10.1081/PDE-120028842.
[4]	T. Bartsch, Z. L. Liu and T. Weth, Nodal solutions of a $p$-Laplcain equation, Proc. London Math. Soc., 91 (2005), 129-152. doi: 10.1112/S0024611504015187.
[5]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[6]	D. M. Cao and S. S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations, 38 (2010), 471-501. doi: 10.1007/s00526-009-0295-5.
[7]	D. M. Cao, S. J. Peng and S. S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902. doi: 10.1016/j.jfa.2012.01.006.
[8]	A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéarier, 2 (1985), 463-470.
[9]	G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéarier, 1 (1984), 341-350.
[10]	G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306. doi: 10.1016/0022-1236(86)90094-7.
[11]	Z. J. Chen and W. M. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987. doi: 10.1016/j.jde.2011.09.042.
[12]	M. Clapp and T. Weth, Multiple solutions for the Brézis-Nirenberg problem, Adv. Differ. Equ., 10 (2005), 463-480.
[13]	G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differ. Equ., 7 (2002), 1257-1280.
[14]	G. Devillanova and S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension, Comm. Contemp. Math., 5 (2003), 171-177. doi: 10.1142/S0219199703000938.
[15]	D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 205-213. doi: 10.1017/S0308210500022046.
[16]	S. J. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227. doi: 10.1090/S0002-9947-02-03031-3.
[17]	P. L. Lions, The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Iberoamericana, 1 (1985), 45-121, 145-201. doi: 10.4171/RMI/6.
[18]	Z. L. Liu, F. A. van Heerden and Z.-Q. Wang, Nodal type bound states of Schrödinger equations via invariant set and minimax methods, J. Differential Equations, 214 (2005), 358-390. doi: 10.1016/j.jde.2004.08.023.
[19]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986.
[20]	M. Schechter and W. Zou, On the Brézis-Nirenberg problem, Arch. Rational Mech. Anal., 197 (2010), 337-356. doi: 10.1007/s00205-009-0288-8.
[21]	S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sovolev space, Ann. Inst. H. Poincaré Anal. Non Linéarier, 12 (1995), 319-337.
[22]	M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.
[23]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04194-9.
[24]	A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differential Integral Equations, 22 (2009), 913-926.
[25]	S. S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents, Chinese Ann. Math. Ser. A, 16 (1995), 227-234.

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