On a p-Laplacian eigenvalue problem with supercritical exponent

Yansheng Zhong; Yongqing Li

doi:10.3934/cpaa.2019012

Article Contents

2019, Volume 18, Issue 1: 227-236. Doi: 10.3934/cpaa.2019012

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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
* Corresponding author

Received: September 30, 2017

Revised: March 31, 2018

Published: August 2018

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

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Abstract

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.

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Mathematics Subject Classification: 35A15; 35B09.

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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.