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April  2018, 11(2): 309-321. doi: 10.3934/dcdss.2018017

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

Dumitru Motreanu 1, , Calogero Vetro 2, and  Francesca Vetro 3,

1. 

Department of Mathematics, University of Perpignan, 52, Avenue Paul Alduy, 66860 Perpignan, France
2. 

Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34,90123 Palermo, Italy
3. 

Department of Energy, Information Engineering and Mathematical Models, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy

Received  February 2017 Revised  May 2017 Published  January 2018

For the homogeneous Dirichlet problem involving a system of equations driven by $(p,q)$-Laplacian operators and general gradient dependence we prove the existence of solutions in the ordered rectangle determined by a subsolution-supersolution. This extends the preceding results based on the method of subsolution-supersolution for systems of elliptic equations. Positive and negative solutions are obtained.

Keywords: Dirichlet problem, system of elliptic equations, (p, q)-Laplacian, subsolution-supersolution and gradient dependence.
Mathematics Subject Classification: Primary: 35J92; Secondary: 35J57.
Citation: Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017
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, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer-Verlag, New York, 2007.  doi: 10.1007/978-0-387-46252-3.
[3]

S. Carl and D. Motreanu, Extremal solutions for nonvariational quasilinear elliptic systems via expanding trapping regions, Monatsh. Math., 182 (2017), 801-821.  doi: 10.1007/s00605-015-0874-9.

[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]

D. De Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differ. Integr. Equ., 17 (2004), 119-126. 

[6]

F. Faraci, D. Motreanu and D. Puglisi, Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var. Partial Differential Equations, 54 (2015), 525-538.  doi: 10.1007/s00526-014-0793-y.

[7]

L. F. O. Faria, O. H. Miyagaki and D. Motreanu, Comparison and positive solutions for problems with $(p, q)$-Laplacian and convection term, Proc. Edinb. Math. Soc., 57 (2014), 687-698.  doi: 10.1017/S0013091513000576.

[8]

L. F. O. Faria, O. H. Miyagaki, D. Motreanu and M. Tanaka, Existence results for nonlinear elliptic equations with Leray-Lions operator and dependence on the gradient, Nonlinear Anal., 96 (2014), 154-166.  doi: 10.1016/j.na.2013.11.006.

[9] D. Motreanu, V. V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer-Verlag, New York, 2014.  doi: 10.1007/978-1-4614-9323-5.
[10]

D. Motreanu and M. Tanaka, On a positive solution for $(p, q)$-Laplace equation with indefinite weight, Minimax Theory Appl., 1 (2016), 1-20. 

[11]

D. Motreanu, C. Vetro and F. Vetro, A parametric Dirichlet problem for systems of quasilinear elliptic equations with gradient dependence, Numer. Funct. Anal. Optim., 37 (2016), 1551-1561.  doi: 10.1080/01630563.2016.1219866.

[12] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser Verlag, Basel, 2007. 
[13]

D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Differ. Equ., 199 (2004), 96-114.  doi: 10.1016/j.jde.2003.10.021.

[14]

M. Tanaka, Existence of a positive solution for quasilinear elliptic equations with a nonlinearity including the gradient Bound. Value Probl. 2013 (2013), 11 pp. doi: 10.1186/1687-2770-2013-173.

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, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer-Verlag, New York, 2007.  doi: 10.1007/978-0-387-46252-3.
[3]

S. Carl and D. Motreanu, Extremal solutions for nonvariational quasilinear elliptic systems via expanding trapping regions, Monatsh. Math., 182 (2017), 801-821.  doi: 10.1007/s00605-015-0874-9.

[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]

D. De Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differ. Integr. Equ., 17 (2004), 119-126. 

[6]

F. Faraci, D. Motreanu and D. Puglisi, Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var. Partial Differential Equations, 54 (2015), 525-538.  doi: 10.1007/s00526-014-0793-y.

[7]

L. F. O. Faria, O. H. Miyagaki and D. Motreanu, Comparison and positive solutions for problems with $(p, q)$-Laplacian and convection term, Proc. Edinb. Math. Soc., 57 (2014), 687-698.  doi: 10.1017/S0013091513000576.

[8]

L. F. O. Faria, O. H. Miyagaki, D. Motreanu and M. Tanaka, Existence results for nonlinear elliptic equations with Leray-Lions operator and dependence on the gradient, Nonlinear Anal., 96 (2014), 154-166.  doi: 10.1016/j.na.2013.11.006.

[9] D. Motreanu, V. V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer-Verlag, New York, 2014.  doi: 10.1007/978-1-4614-9323-5.
[10]

D. Motreanu and M. Tanaka, On a positive solution for $(p, q)$-Laplace equation with indefinite weight, Minimax Theory Appl., 1 (2016), 1-20. 

[11]

D. Motreanu, C. Vetro and F. Vetro, A parametric Dirichlet problem for systems of quasilinear elliptic equations with gradient dependence, Numer. Funct. Anal. Optim., 37 (2016), 1551-1561.  doi: 10.1080/01630563.2016.1219866.

[12] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser Verlag, Basel, 2007. 
[13]

D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Differ. Equ., 199 (2004), 96-114.  doi: 10.1016/j.jde.2003.10.021.

[14]

M. Tanaka, Existence of a positive solution for quasilinear elliptic equations with a nonlinearity including the gradient Bound. Value Probl. 2013 (2013), 11 pp. doi: 10.1186/1687-2770-2013-173.

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