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

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.
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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