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January  2019, 18(1): 227-236. doi: 10.3934/cpaa.2019012

On a p-Laplacian eigenvalue problem with supercritical exponent

Yansheng Zhong , and  Yongqing Li

School of Mathematics and Computer Science & FJKLMAA, Fujian Normal University, Fuzhou, 350117, China

* Corresponding author

Received  October 2017 Revised  April 2018 Published  August 2018

Fund Project: Partially supported by NSFC Grant(11401100, 11671085), the Science foundation of Fujian province(2017J01552), and the innovation foundation of Fujian Normal University(IRTL1206).

In this paper, we prove the existence of the positive and negative solutions to p-Laplacian eigenvalue problems with supercritical exponent. This extends previous results on the problems with subcritical and critical exponents.

Keywords: Constraint functional, igenvalue problem, p-Laplacian.
Mathematics Subject Classification: 35A15; 35B09.
Citation: Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
References:
[1]

H. Amann, Lusternik-Schnirelman theory and nonlinear eigenvalue problems, Math. Ann., 199 (1972), 55-72.  doi: 10.1007/BF01419576.

[2]

J. Benedikt and P. Drábek, Asymptotics for the principal eigenvalue of the p-Laplacian on the ball as p approaches 1, Nonlinear Anal. TMA, 93 (2013), 23-29.  doi: 10.1016/j.na.2013.07.026.

[3]

J. Q. Chen, S. W. Chen and Y. Q. Li, On a quasilinear elliptic eigenvalue problem with constraint, Sci. China, Ser. A: Math., 47 (2004), 523-537.  doi: 10.1360/02ys0324.

[4]

D. G. De Figueiredo, J. P. Gossez and P. Ubilla, Local "superlinearity" and "sublinearity" for the p-Laplacian, J. Funct. Anal., 257 (2009), 721-752.  doi: 10.1016/j.jfa.2009.04.001.

[5]

J. Fleckinger, E. M. Harrell II and F. de Thélin, On the fundamental eigenvalue ratio of the p-Laplacian, Bull. Sci. Math., 131 (2007), 613-619.  doi: 10.1016/j.bulsci.2006.03.016.

[6]

B. L. Guo, Q. X. Li and Y. Q. Li, Sign-changing solutions of a p-Laplacian elliptic problem with constraint in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 451 (2017), 604-622.  doi: 10.1016/j.jmaa.2017.01.091.

[7]

S. C. Hu and N. S. Papageorgiou, Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian, Nonlinear Anal. TMA, 69 (2008), 4286-4300.  doi: 10.1016/j.na.2007.10.053.

[8]

Y. Q. Li, On a nonlinear elliptic eigenvalue problem, J. Differ. Equ., 117 (1995), 151-164 doi: 10.1006/jdeq.1995.1051.

[9]

Y. Q. Li, Three solutions of a semilinear elliptic eigenvalue problem, Acta Math. Sin., New Ser., 11 (1995), 142-152. 

[10]

Y. Q. Li and Z. L. Liu, Multiple and sign-changing solutions of an elliptic eigenvalue problem with constraint, Sci. China, Ser. A., 44 (2001), 48-57.  doi: 10.1007/BF02872282.

[11]

A. Lê, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. TMA, 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056.

[12]

J. Q. Liu and X. Q. Liu, On the eigenvalue problem for the p-Laplacian operator in $R^N$, J. Math. Anal. Appl., 379 (2011), 861-869.  doi: 10.1016/j.jmaa.2011.01.075.

[13]

E. H. Lieb and M. Loss, Analysis, second edition, Americal Mathematical sociaty, provedince Rhode Island, 2001.

[14]

R. E. Megginson, An introduction to Banach Space Theory, Springer, 1998. doi: 10.1007/978-1-4612-0603-3.

[15]

A. Szulkin, Ljusternik-Schnirelman Theory on $C^1$-manifolds, Ann. Inst. Henri Poincaré, 5 (1988), 119-139. 

[16]

S. Sakaguchi, Concavity properties of solutions to some degerate quasilinear elliptic Dirichlet Problems, Ann. Scuola Normale Sup. di Pisa Serie 4, 14 (1987), 403-421. 

[17]

D. Valtorta, Sharp estimate on the first eigenvalue of the p-Laplacian, Nonlinear Anal., 75 (2012), 4974-4994.  doi: 10.1016/j.na.2012.04.012.

[18]

M. Xu and X. P. Yang, Remark on solvability of p-laplacian equtions in large dimension, Israel J. Math., 172 (2009), 349-356.  doi: 10.1007/s11856-009-0077-y.

[19]

E. Zeidler, Ljusternik-Schnirelman theory on general level sets, Math. Nachr., 129 (1986), 235-259.  doi: 10.1002/mana.19861290121.

[20]

E. Zeidler, Nonlinear Functional Analysis and Its Applications III, New-York: Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5020-3.

[21]

Y. S. Zhong and Y. Q. Li, A new form for the differential of the constraint functional in strictly convex reflexive Banach spaces, J. Math. Anal. Appl., 455 (2017), 1783-1800.  doi: 10.1016/j.jmaa.2017.06.080.

show all references

References:
[1]

H. Amann, Lusternik-Schnirelman theory and nonlinear eigenvalue problems, Math. Ann., 199 (1972), 55-72.  doi: 10.1007/BF01419576.

[2]

J. Benedikt and P. Drábek, Asymptotics for the principal eigenvalue of the p-Laplacian on the ball as p approaches 1, Nonlinear Anal. TMA, 93 (2013), 23-29.  doi: 10.1016/j.na.2013.07.026.

[3]

J. Q. Chen, S. W. Chen and Y. Q. Li, On a quasilinear elliptic eigenvalue problem with constraint, Sci. China, Ser. A: Math., 47 (2004), 523-537.  doi: 10.1360/02ys0324.

[4]

D. G. De Figueiredo, J. P. Gossez and P. Ubilla, Local "superlinearity" and "sublinearity" for the p-Laplacian, J. Funct. Anal., 257 (2009), 721-752.  doi: 10.1016/j.jfa.2009.04.001.

[5]

J. Fleckinger, E. M. Harrell II and F. de Thélin, On the fundamental eigenvalue ratio of the p-Laplacian, Bull. Sci. Math., 131 (2007), 613-619.  doi: 10.1016/j.bulsci.2006.03.016.

[6]

B. L. Guo, Q. X. Li and Y. Q. Li, Sign-changing solutions of a p-Laplacian elliptic problem with constraint in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 451 (2017), 604-622.  doi: 10.1016/j.jmaa.2017.01.091.

[7]

S. C. Hu and N. S. Papageorgiou, Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian, Nonlinear Anal. TMA, 69 (2008), 4286-4300.  doi: 10.1016/j.na.2007.10.053.

[8]

Y. Q. Li, On a nonlinear elliptic eigenvalue problem, J. Differ. Equ., 117 (1995), 151-164 doi: 10.1006/jdeq.1995.1051.

[9]

Y. Q. Li, Three solutions of a semilinear elliptic eigenvalue problem, Acta Math. Sin., New Ser., 11 (1995), 142-152. 

[10]

Y. Q. Li and Z. L. Liu, Multiple and sign-changing solutions of an elliptic eigenvalue problem with constraint, Sci. China, Ser. A., 44 (2001), 48-57.  doi: 10.1007/BF02872282.

[11]

A. Lê, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. TMA, 64 (2006), 1057-1099.  doi: 10.1016/j.na.2005.05.056.

[12]

J. Q. Liu and X. Q. Liu, On the eigenvalue problem for the p-Laplacian operator in $R^N$, J. Math. Anal. Appl., 379 (2011), 861-869.  doi: 10.1016/j.jmaa.2011.01.075.

[13]

E. H. Lieb and M. Loss, Analysis, second edition, Americal Mathematical sociaty, provedince Rhode Island, 2001.

[14]

R. E. Megginson, An introduction to Banach Space Theory, Springer, 1998. doi: 10.1007/978-1-4612-0603-3.

[15]

A. Szulkin, Ljusternik-Schnirelman Theory on $C^1$-manifolds, Ann. Inst. Henri Poincaré, 5 (1988), 119-139. 

[16]

S. Sakaguchi, Concavity properties of solutions to some degerate quasilinear elliptic Dirichlet Problems, Ann. Scuola Normale Sup. di Pisa Serie 4, 14 (1987), 403-421. 

[17]

D. Valtorta, Sharp estimate on the first eigenvalue of the p-Laplacian, Nonlinear Anal., 75 (2012), 4974-4994.  doi: 10.1016/j.na.2012.04.012.

[18]

M. Xu and X. P. Yang, Remark on solvability of p-laplacian equtions in large dimension, Israel J. Math., 172 (2009), 349-356.  doi: 10.1007/s11856-009-0077-y.

[19]

E. Zeidler, Ljusternik-Schnirelman theory on general level sets, Math. Nachr., 129 (1986), 235-259.  doi: 10.1002/mana.19861290121.

[20]

E. Zeidler, Nonlinear Functional Analysis and Its Applications III, New-York: Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5020-3.

[21]

Y. S. Zhong and Y. Q. Li, A new form for the differential of the constraint functional in strictly convex reflexive Banach spaces, J. Math. Anal. Appl., 455 (2017), 1783-1800.  doi: 10.1016/j.jmaa.2017.06.080.

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