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On a p-Laplacian eigenvalue problem with supercritical exponent
School of Mathematics and Computer Science & FJKLMAA, Fujian Normal University, Fuzhou, 350117, China |
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.
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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