American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130
[2]

Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371
[3]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[4]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[5]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[6]

Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020
[7]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587
[8]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033
[9]

CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004
[10]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922
[11]

Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
[12]

Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063
[13]

Carlo Mercuri, Michel Willem. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 469-493. doi: 10.3934/dcds.2010.28.469
[14]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[15]

Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743
[16]

Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593
[17]

Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191
[18]

Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219
[19]

Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299
[20]

Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (59)
  • HTML views (107)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences