-
Previous Article
A loop type component in the non-negative solutions set of an indefinite elliptic problem
- CPAA Home
- This Issue
-
Next Article
Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential
May
2018, 17(3): 1219-1253.
doi: 10.3934/cpaa.2018059
Remarks on minimizers for (p, q)-Laplace equations with two parameters
| 1. | Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8,306 14 Plzeň, Czech Republic |
| 2. | Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo 162-8601, Japan |
We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p, q)$ -Laplace equation $-Δ_p u -Δ_q u = α |u|^{p-2}u + β |u|^{q-2}u$ in a bounded domain $Ω \subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $α, β ∈ \mathbb{R}$ . A curve on the $(α, β)$ -plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the p-and q-Laplacians under zero Dirichlet boundary condition are linearly independent.
Keywords:
p-Laplacian,
(p,q)-Laplacian,
nonlinear eigenvalue problem,
global minimizer,
ground states,
Nehari manifold,
fibered functional,
improved Poincaré inequality.
Mathematics Subject Classification:
Primary: 35J60, 35J20; Secondary: 35P30.
Citation:
Vladimir Bobkov, Mieko Tanaka. Remarks on minimizers for (p, q)-Laplace equations with two parameters. Communications on Pure and Applied Analysis,
2018, 17
(3)
: 1219-1253.
doi: 10.3934/cpaa.2018059
![]()
References:
| [1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu,
Nodal solutions for (p, 2)-equations, Transactions of the American Mathematical Society, 367 (2015), 7343-7372.
doi: 10.1090/S0002-9947-2014-06324-1. |
| [2] |
M. J. Alves, R. B. Assunç ao and O. H. Miyagaki,
Existence result for a class of quasilinear elliptic equations with (p − q)-Laplacian and vanishing potentials, Illinois Journal of Mathematics, 59 (2015), 545-575.
|
| [3] |
A. Anane,
Simplicité et isolation de la premiere valeur propre du p-laplacien avec poids, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 305 (1987), 725-728.
|
| [4] |
J. Bellazzini and N. Visciglia,
Max-min characterization of the mountain pass energy level
for a class of variational problems, Proceedings of the American Mathematical Society, 138 (2010), 3335-3343.
doi: 10.1090/S0002-9939-10-10415-8. |
| [5] |
V. Benci, P. D'Avenia, D. Fortunato and L. Pisani,
Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Archive for Rational Mechanics and Analysis, 154 (2000), 297-324.
doi: 10.1007/s002050000101. |
| [6] |
V. Bobkov and M. Tanaka,
On positive solutions for (p, q)-Laplace equations with two parameters, Calculus of Variations and Partial Differential Equations, 54 (2015), 3277-3301.
doi: 10.1007/s00526-015-0903-5. |
| [7] |
V. Bobkov and M. Tanaka, On sign-changing solutions for (p, q)-Laplace equations with two parameters Advances in Nonlinear Analysis.
doi: 10.1515/anona-2016-0172. |
| [8] |
P. J. Bushell and D. E. Edmunds,
Remarks on generalised trigonometric functions, Rocky
Mountain Journal of Mathematics, 42 (2012), 25-57.
doi: 10.1216/rmj-2012-42-1-25. |
| [9] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. I. Interfacial free energy, The Journal of chemical physics, 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
| [10] |
L. Cherfils and Y. Il'yasov,
On the stationary solutions of generalized reaction diffusion equations with p&q-laplacian, Communications on Pure and Applied Mathematics, 4 (2005), 9-22.
doi: 10.3934/cpaa.2005.4.9. |
| [11] |
I. Chueshov and I. Lasiecka,
Existence, uniqueness of weak solutions and global attractors
for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete and Continuous Dynamical
Systems, 15 (2006), 777-809.
doi: 10.3934/dcds.2006.15.777. |
| [12] |
M. Colombo and M. Colombo,
Regularity for double phase variational problems, Archive for
Rational Mechanics and Analysis, 215 (2015), 443-496.
doi: 10.1007/s00205-014-0785-2. |
| [13] |
M. Cuesta, D. de Figueiredo and J.-P. Gossez,
The beginning of the Fučik spectrum for the
p-Laplacian, Journal of Differential Equations, 159 (1999), 212-238.
doi: 10.1006/jdeq.1999.3645. |
| [14] |
L. Damascelli and B. Sciunzi,
Regularity, monotonicity and symmetry of positive solutions of
m-Laplace equations, Journal of Differential Equations, 206 (2004), 483-515.
doi: 10.1016/j.jde.2004.05.012. |
| [15] |
P. Drábek,
Geometry of the energy functional and the Fredholm alternative for the p-Laplacian
in higher dimensions, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electronic Journal of Differential Equations, Conference 08 (2002), 103-120.
|
| [16] |
P. Drábek, A. Kufner and F. Nicolosi,
Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997. |
| [17] |
P. Drábek and J. Milota,
Methods of Nonlinear Analysis: Applications to Differential Equations, 2nd edition, Springer, 2013.
doi: 10.1007/978-3-0348-0387-8. |
| [18] |
L. F. Faria, O. H. Miyagaki and D. Motreanu,
Comparison and positive solutions for problems
with the (p, q)-Laplacian and a convection term, Proceedings of the Edinburgh Mathematical
Society (Series 2), 57 (2014), 687-698.
doi: 10.1017/S0013091513000576. |
| [19] |
G. M. Figueiredo,
Existence of positive solutions for a class of p&q elliptic problems with critical growth on $\mathbb{R}^N$, Journal of Mathematical Analysis and Applications, 378 (2011), 507-518.
doi: 10.1016/j.jmaa.2011.02.017. |
| [20] |
J. García-Melián,
On the behavior of the first eigenfunction of the p-Laplacian near its critical points, Bulletin of the London Mathematical Society, 35 (2003), 391-400.
doi: 10.1112/S0024609303001966. |
| [21] |
J. Fleckinger-Pellé and P. Takáč,
An improved Poincaré inequality and the p-Laplacian at resonance for p>2, Advances in Differential Equations, 7 (2002), 951-971.
|
| [22] |
Y. S. Il'yasov,
Bifurcation calculus by the extended functional method, Functional Analysis and Its Applications, 41 (2007), 18-30.
doi: 10.1007/s10688-007-0002-2. |
| [23] |
Y. Il'yasov and K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method, preprint,arXiv: 1704.02477. |
| [24] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $\mathbb{R}^N$, Proceedings of the American Mathematical Society, 131 (2003), 2399-2408.
|
| [25] |
R. Kajikiya, M. Tanaka and S. Tanaka,
Bifurcation of positive solutions for the one dimensional (p, q)-Laplace equation, Electronic Journal of Differential Equations, 2017 (2017), 1-37.
|
| [26] |
G. M. Lieberman,
Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
| [27] |
G. M. Lieberman,
The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Communications in Partial Differential Equations, 16 (1991), 311-361.
doi: 10.1080/03605309108820761. |
| [28] |
S. Marano and S. Mosconi,
Some recent results on the Dirichlet problem for (p, q)-Laplace equations, Discrete and Continuous Dynamical Systems -Series S, 11 (2018), 279-291.
doi: 10.3934/dcdss.2018015. |
| [29] |
S. A. Marano and N. S. Papageorgiou,
Constant-sign and nodal solutions of coercive $(p, q)$-Laplacian problems, Nonlinear Analysis: Theory, Methods & Applications, 77 (2013), 118-129.
doi: 10.1016/j.na.2012.09.007. |
| [30] |
D. Motreanu and M. Tanaka,
On a positive solution for $(p, q)$-Laplace equation with indefinite weight, Minimax Theory and its Applications, 1 (2016), 1-20.
|
| [31] |
S. I. Pohozaev, Nonlinear variational problems via the fibering method, in Handbook of differential equations: stationary partial differential equations (ed. M. Chipot), 5 (2008), 49–209.
doi: 10.1016/S1874-5733(08)80009-5. |
| [32] |
P. Pucci and J. Serrin,
The Maximum Principle, Springer, 2007.
doi: 10.1007/978-3-7643-8145-5. |
| [33] |
J. Sun, J. Chu and T. F. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian Journal of Differential Equations,
262 (2017), 945-977.
doi: 10.016/j.jde.2016.10.001. |
| [34] |
P. Takáč,
On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue, Indiana University Mathematics Journal, 51 (2002), 187-238.
doi: 10.1512/iumj.2002.51.2156. |
| [35] |
M. Tanaka, Uniqueness of a positive solution and existence of a sign-changing solution for (p, q)-Laplace equation, Journal of Nonlinear Functional Analysis, 2014 (2014), 1–15. |
| [36] |
M. Tanaka,
Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight, Journal of Mathematical Analysis and Applications, 419 (2014), 1181-1192.
doi: 10.1016/j.jmaa.2014.05.044. |
| [37] |
P. Tolksdorf,
Regularity for a more general class of quasilinear elliptic equations, Journal of Differential equations, 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
| [38] |
H. Yin and Z. Yang,
A class of p-q-Laplacian type equation with concave-convex nonlinearities in bounded domain, Journal of Mathematical Analysis and Applications, 382 (2011), 843-855.
doi: 10.1016/j.jmaa.2011.04.090. |
| [39] |
V. E. Zakharov, Collapse of Langmuir waves,
Soviet Journal of Experimental and Theoretical Physics,
35 (1972), 908-914. |
show all references
References:
| [1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu,
Nodal solutions for (p, 2)-equations, Transactions of the American Mathematical Society, 367 (2015), 7343-7372.
doi: 10.1090/S0002-9947-2014-06324-1. |
| [2] |
M. J. Alves, R. B. Assunç ao and O. H. Miyagaki,
Existence result for a class of quasilinear elliptic equations with (p − q)-Laplacian and vanishing potentials, Illinois Journal of Mathematics, 59 (2015), 545-575.
|
| [3] |
A. Anane,
Simplicité et isolation de la premiere valeur propre du p-laplacien avec poids, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 305 (1987), 725-728.
|
| [4] |
J. Bellazzini and N. Visciglia,
Max-min characterization of the mountain pass energy level
for a class of variational problems, Proceedings of the American Mathematical Society, 138 (2010), 3335-3343.
doi: 10.1090/S0002-9939-10-10415-8. |
| [5] |
V. Benci, P. D'Avenia, D. Fortunato and L. Pisani,
Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Archive for Rational Mechanics and Analysis, 154 (2000), 297-324.
doi: 10.1007/s002050000101. |
| [6] |
V. Bobkov and M. Tanaka,
On positive solutions for (p, q)-Laplace equations with two parameters, Calculus of Variations and Partial Differential Equations, 54 (2015), 3277-3301.
doi: 10.1007/s00526-015-0903-5. |
| [7] |
V. Bobkov and M. Tanaka, On sign-changing solutions for (p, q)-Laplace equations with two parameters Advances in Nonlinear Analysis.
doi: 10.1515/anona-2016-0172. |
| [8] |
P. J. Bushell and D. E. Edmunds,
Remarks on generalised trigonometric functions, Rocky
Mountain Journal of Mathematics, 42 (2012), 25-57.
doi: 10.1216/rmj-2012-42-1-25. |
| [9] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. I. Interfacial free energy, The Journal of chemical physics, 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
| [10] |
L. Cherfils and Y. Il'yasov,
On the stationary solutions of generalized reaction diffusion equations with p&q-laplacian, Communications on Pure and Applied Mathematics, 4 (2005), 9-22.
doi: 10.3934/cpaa.2005.4.9. |
| [11] |
I. Chueshov and I. Lasiecka,
Existence, uniqueness of weak solutions and global attractors
for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete and Continuous Dynamical
Systems, 15 (2006), 777-809.
doi: 10.3934/dcds.2006.15.777. |
| [12] |
M. Colombo and M. Colombo,
Regularity for double phase variational problems, Archive for
Rational Mechanics and Analysis, 215 (2015), 443-496.
doi: 10.1007/s00205-014-0785-2. |
| [13] |
M. Cuesta, D. de Figueiredo and J.-P. Gossez,
The beginning of the Fučik spectrum for the
p-Laplacian, Journal of Differential Equations, 159 (1999), 212-238.
doi: 10.1006/jdeq.1999.3645. |
| [14] |
L. Damascelli and B. Sciunzi,
Regularity, monotonicity and symmetry of positive solutions of
m-Laplace equations, Journal of Differential Equations, 206 (2004), 483-515.
doi: 10.1016/j.jde.2004.05.012. |
| [15] |
P. Drábek,
Geometry of the energy functional and the Fredholm alternative for the p-Laplacian
in higher dimensions, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electronic Journal of Differential Equations, Conference 08 (2002), 103-120.
|
| [16] |
P. Drábek, A. Kufner and F. Nicolosi,
Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997. |
| [17] |
P. Drábek and J. Milota,
Methods of Nonlinear Analysis: Applications to Differential Equations, 2nd edition, Springer, 2013.
doi: 10.1007/978-3-0348-0387-8. |
| [18] |
L. F. Faria, O. H. Miyagaki and D. Motreanu,
Comparison and positive solutions for problems
with the (p, q)-Laplacian and a convection term, Proceedings of the Edinburgh Mathematical
Society (Series 2), 57 (2014), 687-698.
doi: 10.1017/S0013091513000576. |
| [19] |
G. M. Figueiredo,
Existence of positive solutions for a class of p&q elliptic problems with critical growth on $\mathbb{R}^N$, Journal of Mathematical Analysis and Applications, 378 (2011), 507-518.
doi: 10.1016/j.jmaa.2011.02.017. |
| [20] |
J. García-Melián,
On the behavior of the first eigenfunction of the p-Laplacian near its critical points, Bulletin of the London Mathematical Society, 35 (2003), 391-400.
doi: 10.1112/S0024609303001966. |
| [21] |
J. Fleckinger-Pellé and P. Takáč,
An improved Poincaré inequality and the p-Laplacian at resonance for p>2, Advances in Differential Equations, 7 (2002), 951-971.
|
| [22] |
Y. S. Il'yasov,
Bifurcation calculus by the extended functional method, Functional Analysis and Its Applications, 41 (2007), 18-30.
doi: 10.1007/s10688-007-0002-2. |
| [23] |
Y. Il'yasov and K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method, preprint,arXiv: 1704.02477. |
| [24] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $\mathbb{R}^N$, Proceedings of the American Mathematical Society, 131 (2003), 2399-2408.
|
| [25] |
R. Kajikiya, M. Tanaka and S. Tanaka,
Bifurcation of positive solutions for the one dimensional (p, q)-Laplace equation, Electronic Journal of Differential Equations, 2017 (2017), 1-37.
|
| [26] |
G. M. Lieberman,
Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
| [27] |
G. M. Lieberman,
The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Communications in Partial Differential Equations, 16 (1991), 311-361.
doi: 10.1080/03605309108820761. |
| [28] |
S. Marano and S. Mosconi,
Some recent results on the Dirichlet problem for (p, q)-Laplace equations, Discrete and Continuous Dynamical Systems -Series S, 11 (2018), 279-291.
doi: 10.3934/dcdss.2018015. |
| [29] |
S. A. Marano and N. S. Papageorgiou,
Constant-sign and nodal solutions of coercive $(p, q)$-Laplacian problems, Nonlinear Analysis: Theory, Methods & Applications, 77 (2013), 118-129.
doi: 10.1016/j.na.2012.09.007. |
| [30] |
D. Motreanu and M. Tanaka,
On a positive solution for $(p, q)$-Laplace equation with indefinite weight, Minimax Theory and its Applications, 1 (2016), 1-20.
|
| [31] |
S. I. Pohozaev, Nonlinear variational problems via the fibering method, in Handbook of differential equations: stationary partial differential equations (ed. M. Chipot), 5 (2008), 49–209.
doi: 10.1016/S1874-5733(08)80009-5. |
| [32] |
P. Pucci and J. Serrin,
The Maximum Principle, Springer, 2007.
doi: 10.1007/978-3-7643-8145-5. |
| [33] |
J. Sun, J. Chu and T. F. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian Journal of Differential Equations,
262 (2017), 945-977.
doi: 10.016/j.jde.2016.10.001. |
| [34] |
P. Takáč,
On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue, Indiana University Mathematics Journal, 51 (2002), 187-238.
doi: 10.1512/iumj.2002.51.2156. |
| [35] |
M. Tanaka, Uniqueness of a positive solution and existence of a sign-changing solution for (p, q)-Laplace equation, Journal of Nonlinear Functional Analysis, 2014 (2014), 1–15. |
| [36] |
M. Tanaka,
Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight, Journal of Mathematical Analysis and Applications, 419 (2014), 1181-1192.
doi: 10.1016/j.jmaa.2014.05.044. |
| [37] |
P. Tolksdorf,
Regularity for a more general class of quasilinear elliptic equations, Journal of Differential equations, 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
| [38] |
H. Yin and Z. Yang,
A class of p-q-Laplacian type equation with concave-convex nonlinearities in bounded domain, Journal of Mathematical Analysis and Applications, 382 (2011), 843-855.
doi: 10.1016/j.jmaa.2011.04.090. |
| [39] |
V. E. Zakharov, Collapse of Langmuir waves,
Soviet Journal of Experimental and Theoretical Physics,
35 (1972), 908-914. |
| [1] |
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 |
| [2] |
Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 |
| [3] |
Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure and Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 |
| [4] |
Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130 |
| [5] |
Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743 |
| [6] |
Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 |
| [7] |
Carlo Mercuri, Michel Willem. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 469-493. doi: 10.3934/dcds.2010.28.469 |
| [8] |
Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 |
| [9] |
Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations and Control Theory, 2022, 11 (3) : 975-1000. doi: 10.3934/eect.2021033 |
| [10] |
Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171 |
| [11] |
Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 |
| [12] |
Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations and Control Theory, 2022, 11 (2) : 399-414. doi: 10.3934/eect.2021005 |
| [13] |
Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173 |
| [14] |
Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020 |
| [15] |
Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure and Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675 |
| [16] |
Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 |
| [17] |
Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 |
| [18] |
Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055 |
| [19] |
Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475 |
| [20] |
Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]