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April  2018, 11(2): 293-307. doi: 10.3934/dcdss.2018016

Location of Nodal solutions for quasilinear elliptic equations with gradient dependence

Dumitru Motreanu 1,, , Viorica V. Motreanu 2, and  Abdelkrim Moussaoui 3,

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

* Corresponding author: Dumitru Motreanu

Received  August 2016 Revised  April 2017 Published  January 2018

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.

Keywords: Quasilinear elliptic equations, (p, q)-Laplacian, gradient dependence, nodal solutions, constant-sign solutions, location, sub-supersolutions, extremal solutions.
Mathematics Subject Classification: Primary: 35J62; Secondary: 35J92.
Citation: Dumitru Motreanu, Viorica V. Motreanu, Abdelkrim Moussaoui. Location of Nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 293-307. doi: 10.3934/dcdss.2018016
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[5]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory Interscience Publishers, Inc. , New York; Interscience Publishers, Ltd. , London, 1958.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar
[11] P. Pucci and J. Serrin, The Maximum Principle, Springer, New York, 2007.  doi: 10.1007/978-3-7643-8145-5.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[5]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory Interscience Publishers, Inc. , New York; Interscience Publishers, Ltd. , London, 1958.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar
[11] P. Pucci and J. Serrin, The Maximum Principle, Springer, New York, 2007.  doi: 10.1007/978-3-7643-8145-5.  Google Scholar
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