Location of Nodal solutions for quasilinear elliptic equations with gradient dependence

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Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method

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.
[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.

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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