American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

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