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

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 & 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},, \emph{NoDEA Nonlinear Differential Equations Appl.}, 15 (2008), 69. doi: 10.1007/s00030-007-6034-8. Google Scholar
[2]	F. V. Atkinson, H. Brézis and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents,, \emph{J. Differential Equations}, 85 (1990), 151. doi: 10.1016/0022-0396(90)90093-5. Google Scholar
[3]	T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations,, \emph{Commun. Partial Differential Equations}, 29 (2004), 25. doi: 10.1081/PDE-120028842. Google Scholar
[4]	T. Bartsch, Z. L. Liu and T. Weth, Nodal solutions of a $p$-Laplcain equation,, \emph{Proc. London Math. Soc.}, 91 (2005), 129. doi: 10.1112/S0024611504015187. Google Scholar
[5]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar
[6]	D. M. Cao and S. S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential,, \emph{Calc. Var. Partial Differential Equations}, 38 (2010), 471. doi: 10.1007/s00526-009-0295-5. Google Scholar
[7]	D. M. Cao, S. J. Peng and S. S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth,, \emph{J. Funct. Anal.}, 262 (2012), 2861. doi: 10.1016/j.jfa.2012.01.006. Google Scholar
[8]	A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'earier}, 2 (1985), 463. Google Scholar
[9]	G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'earier}, 1 (1984), 341. Google Scholar
[10]	G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents,, \emph{J. Funct. Anal.}, 69 (1986), 289. doi: 10.1016/0022-1236(86)90094-7. Google Scholar
[11]	Z. J. Chen and W. M. Zou, On an elliptic problem with critical exponent and Hardy potential,, \emph{J. Differential Equations}, 252 (2012), 969. doi: 10.1016/j.jde.2011.09.042. Google Scholar
[12]	M. Clapp and T. Weth, Multiple solutions for the Brézis-Nirenberg problem,, \emph{Adv. Differ. Equ.}, 10 (2005), 463. Google Scholar
[13]	G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth,, \emph{Adv. Differ. Equ.}, 7 (2002), 1257. Google Scholar
[14]	G. Devillanova and S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension,, \emph{Comm. Contemp. Math.}, 5 (2003), 171. doi: 10.1142/S0219199703000938. Google Scholar
[15]	D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 105 (1987), 205. doi: 10.1017/S0308210500022046. Google Scholar
[16]	S. J. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces,, \emph{Trans. Amer. Math. Soc.}, 354 (2002), 3207. doi: 10.1090/S0002-9947-02-03031-3. Google Scholar
[17]	P. L. Lions, The concentration-compactness principle in the calculus of variations: the limit case,, \emph{Rev. Mat. Iberoamericana}, 1 (1985), 45. doi: 10.4171/RMI/6. Google Scholar
[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,, \emph{J. Differential Equations}, 214 (2005), 358. doi: 10.1016/j.jde.2004.08.023. Google Scholar
[19]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conference Series in Mathematics, 65 (1986). Google Scholar
[20]	M. Schechter and W. Zou, On the Brézis-Nirenberg problem,, \emph{Arch. Rational Mech. Anal.}, 197 (2010), 337. doi: 10.1007/s00205-009-0288-8. Google Scholar
[21]	S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sovolev space,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'earier}, 12 (1995), 319. Google Scholar
[22]	M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, \emph{Math. Z.}, 187 (1984), 511. doi: 10.1007/BF01174186. Google Scholar
[23]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, 3$^{rd}$ Edition, (2000). doi: 10.1007/978-3-662-04194-9. Google Scholar
[24]	A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent,, \emph{Differential Integral Equations}, 22 (2009), 913. Google Scholar
[25]	S. S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents,, \emph{Chinese Ann. Math. Ser. A}, 16 (1995), 227. Google Scholar

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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},, \emph{NoDEA Nonlinear Differential Equations Appl.}, 15 (2008), 69. doi: 10.1007/s00030-007-6034-8. Google Scholar
[2]	F. V. Atkinson, H. Brézis and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents,, \emph{J. Differential Equations}, 85 (1990), 151. doi: 10.1016/0022-0396(90)90093-5. Google Scholar
[3]	T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations,, \emph{Commun. Partial Differential Equations}, 29 (2004), 25. doi: 10.1081/PDE-120028842. Google Scholar
[4]	T. Bartsch, Z. L. Liu and T. Weth, Nodal solutions of a $p$-Laplcain equation,, \emph{Proc. London Math. Soc.}, 91 (2005), 129. doi: 10.1112/S0024611504015187. Google Scholar
[5]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar
[6]	D. M. Cao and S. S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential,, \emph{Calc. Var. Partial Differential Equations}, 38 (2010), 471. doi: 10.1007/s00526-009-0295-5. Google Scholar
[7]	D. M. Cao, S. J. Peng and S. S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth,, \emph{J. Funct. Anal.}, 262 (2012), 2861. doi: 10.1016/j.jfa.2012.01.006. Google Scholar
[8]	A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'earier}, 2 (1985), 463. Google Scholar
[9]	G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'earier}, 1 (1984), 341. Google Scholar
[10]	G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents,, \emph{J. Funct. Anal.}, 69 (1986), 289. doi: 10.1016/0022-1236(86)90094-7. Google Scholar
[11]	Z. J. Chen and W. M. Zou, On an elliptic problem with critical exponent and Hardy potential,, \emph{J. Differential Equations}, 252 (2012), 969. doi: 10.1016/j.jde.2011.09.042. Google Scholar
[12]	M. Clapp and T. Weth, Multiple solutions for the Brézis-Nirenberg problem,, \emph{Adv. Differ. Equ.}, 10 (2005), 463. Google Scholar
[13]	G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth,, \emph{Adv. Differ. Equ.}, 7 (2002), 1257. Google Scholar
[14]	G. Devillanova and S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension,, \emph{Comm. Contemp. Math.}, 5 (2003), 171. doi: 10.1142/S0219199703000938. Google Scholar
[15]	D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 105 (1987), 205. doi: 10.1017/S0308210500022046. Google Scholar
[16]	S. J. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces,, \emph{Trans. Amer. Math. Soc.}, 354 (2002), 3207. doi: 10.1090/S0002-9947-02-03031-3. Google Scholar
[17]	P. L. Lions, The concentration-compactness principle in the calculus of variations: the limit case,, \emph{Rev. Mat. Iberoamericana}, 1 (1985), 45. doi: 10.4171/RMI/6. Google Scholar
[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,, \emph{J. Differential Equations}, 214 (2005), 358. doi: 10.1016/j.jde.2004.08.023. Google Scholar
[19]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conference Series in Mathematics, 65 (1986). Google Scholar
[20]	M. Schechter and W. Zou, On the Brézis-Nirenberg problem,, \emph{Arch. Rational Mech. Anal.}, 197 (2010), 337. doi: 10.1007/s00205-009-0288-8. Google Scholar
[21]	S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sovolev space,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'earier}, 12 (1995), 319. Google Scholar
[22]	M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, \emph{Math. Z.}, 187 (1984), 511. doi: 10.1007/BF01174186. Google Scholar
[23]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, 3$^{rd}$ Edition, (2000). doi: 10.1007/978-3-662-04194-9. Google Scholar
[24]	A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent,, \emph{Differential Integral Equations}, 22 (2009), 913. Google Scholar
[25]	S. S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents,, \emph{Chinese Ann. Math. Ser. A}, 16 (1995), 227. Google Scholar

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