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Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations
Location of Nodal solutions for quasilinear elliptic equations with gradient dependence
| 1. | Université de Perpignan, Département de Mathématiques, 66860 Perpignan, France |
| 2. | Lycée Polyvalent Franklin Roosevelt, 10 Rue du Président Franklin Roosevelt, 51100 Reims, France |
| 3. | Université A. Mira de Bejaia, Laboratoire de Mathématiques Appliquées (LMA), Faculté des Sciences Exactes, Targa Ouzemour 06000 Bejaia, Algeria |
Existence and regularity results for quasilinear elliptic equations driven by $(p, q)$-Laplacian and with gradient dependence are presented. A location principle for nodal (i.e., sign-changing) solutions is obtained by means of constant-sign solutions whose existence is also derived. Criteria for the existence of extremal solutions are finally established.
References:
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D. Averna, D. Motreanu and E. Tornatore,
Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), 102-107.
doi: 10.1016/j.aml.2016.05.009. |
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S. Carl,
Barrier solutions of elliptic variational inequalities, Nonlinear Anal. Real World Appl., 26 (2015), 75-92.
doi: 10.1016/j.nonrwa.2015.04.004. |
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S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer, New York, 2007.
doi: 10.1007/978-0-387-46252-3.
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A. Cianchi and V. Maz'ya,
Global gradient estimates in elliptic problems under minimal data and domain regularity, Commun. Pure Appl. Anal., 14 (2015), 285-311.
doi: 10.3934/cpaa.2015.14.285. |
| [5] |
N. Dunford and J. T. Schwartz,
Linear Operators. I. General Theory Interscience Publishers, Inc. , New York; Interscience Publishers, Ltd. , London, 1958. |
| [6] |
G. M. Lieberman,
Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
| [7] |
G. M. Lieberman,
The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.
doi: 10.1080/03605309108820761. |
| [8] |
S. Miyajima, D. Motreanu and M. Tanaka,
Multiple existence results of solutions for the Neumann problems via super- and sub-solutions, J. Funct. Anal., 262 (2012), 1921-1953.
doi: 10.1016/j.jfa.2011.11.028. |
| [9] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou,
A unified approach for multiple constant sign and nodal solutions, Adv. Differential Equations, 12 (2007), 1363-1392.
|
| [10] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5.
|
| [11] |
P. Pucci and J. Serrin, The Maximum Principle, Springer, New York, 2007.
doi: 10.1007/978-3-7643-8145-5.
|
show all references
References:
| [1] |
D. Averna, D. Motreanu and E. Tornatore,
Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), 102-107.
doi: 10.1016/j.aml.2016.05.009. |
| [2] |
S. Carl,
Barrier solutions of elliptic variational inequalities, Nonlinear Anal. Real World Appl., 26 (2015), 75-92.
doi: 10.1016/j.nonrwa.2015.04.004. |
| [3] |
S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer, New York, 2007.
doi: 10.1007/978-0-387-46252-3.
|
| [4] |
A. Cianchi and V. Maz'ya,
Global gradient estimates in elliptic problems under minimal data and domain regularity, Commun. Pure Appl. Anal., 14 (2015), 285-311.
doi: 10.3934/cpaa.2015.14.285. |
| [5] |
N. Dunford and J. T. Schwartz,
Linear Operators. I. General Theory Interscience Publishers, Inc. , New York; Interscience Publishers, Ltd. , London, 1958. |
| [6] |
G. M. Lieberman,
Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: 10.1016/0362-546X(88)90053-3. |
| [7] |
G. M. Lieberman,
The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.
doi: 10.1080/03605309108820761. |
| [8] |
S. Miyajima, D. Motreanu and M. Tanaka,
Multiple existence results of solutions for the Neumann problems via super- and sub-solutions, J. Funct. Anal., 262 (2012), 1921-1953.
doi: 10.1016/j.jfa.2011.11.028. |
| [9] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou,
A unified approach for multiple constant sign and nodal solutions, Adv. Differential Equations, 12 (2007), 1363-1392.
|
| [10] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4614-9323-5.
|
| [11] |
P. Pucci and J. Serrin, The Maximum Principle, Springer, New York, 2007.
doi: 10.1007/978-3-7643-8145-5.
|
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