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Location of Nodal solutions for quasilinear elliptic equations with gradient dependence
Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method
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 |
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.
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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