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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

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

show all references

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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