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Existence of solutions for quasilinear Dirichlet problems with gradient terms
Robin problems for the p-Laplacian with gradient dependence
1. | Dipartimento di Matematica, Università di Bari, Via E. Orabona 4, 70125 Bari, Italy |
2. | Dipartimento di Scienze Ecologiche e Biologiche (DEB), Università della Tuscia, Largo dell'Università, 01100 Viterbo, Italy |
3. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece |
We consider a nonlinear elliptic equation with Robin boundary condition driven by the p-Laplacian and with a reaction term which depends also on the gradient. By using a topological approach based on the Leray-Schauder alternative principle, we show the existence of a smooth solution.
References:
[1] |
F. Faraci, D. Motreanu and D. Puglisi,
Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var., 54 (2015), 525-538.
doi: 10.1007/s00526-014-0793-y. |
[2] |
D. de Figueiredo, M. Girardi and M. Matzeu,
Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Diff. Integral Equ., 17 (2004), 119-126.
|
[3] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.
doi: MR2168068. |
[4] |
L. Gasinski and N. S. Papageorgiou,
Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differential Equations, 263 (2017), 1451-1476.
doi: 10.1016/j.jde.2017.03.021. |
[5] |
M. Girardi and M. Matzeu,
Positive and negative solutions of a quasilinear elliptic equation by a mountain pass method and truncature techniques, Nonlinear Anal., 59 (2004), 199-210.
doi: 10.1016/j.na.2004.04.014. |
[6] |
N. B. Huy, B. T. Quan and N. H. Khanh,
Existence and multiplicity results for generalized logistic equations, Nonlinear Anal., 144 (2016), 77-92.
doi: 10.1016/j.na.2016.06.001. |
[7] |
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. |
[8] |
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. |
[9] |
D. Mugnai and N. S. Papageorgiou,
Resonant nonlinear Neumann problems with indefinite weight, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 11 (2012), 729-788.
doi: 10.2422/2036-2145.201012_003. |
[10] |
N. S. Papageorgiou and V. D. Radulescu,
Multiple solutions with precise sign information for nonlinear Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
[11] |
N. S. Papageorgiou and V. D. Radulescu,
Nonlinear, nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737-764.
doi: 10.1515/ans-2016-0023. |
[12] |
N. S. Papageorgiou, V. D. Radulescu and D. D. Repovs, Nonlinear elliptic inclusions with unilateral constraint and dependence on the gradient, Appl. Math. Optim., (2016), 1-23.
doi: 10.1007/s00245-016-9392-y. |
[13] |
D. Ruiz,
A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Differential Equations, 199 (2004), 96-114.
doi: 10.1016/j.jde.2003.10.021. |
show all references
Dedicated to Vicentiu, on the occasion of his 60th birthday, with sincere friendship and esteem
References:
[1] |
F. Faraci, D. Motreanu and D. Puglisi,
Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var., 54 (2015), 525-538.
doi: 10.1007/s00526-014-0793-y. |
[2] |
D. de Figueiredo, M. Girardi and M. Matzeu,
Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Diff. Integral Equ., 17 (2004), 119-126.
|
[3] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.
doi: MR2168068. |
[4] |
L. Gasinski and N. S. Papageorgiou,
Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differential Equations, 263 (2017), 1451-1476.
doi: 10.1016/j.jde.2017.03.021. |
[5] |
M. Girardi and M. Matzeu,
Positive and negative solutions of a quasilinear elliptic equation by a mountain pass method and truncature techniques, Nonlinear Anal., 59 (2004), 199-210.
doi: 10.1016/j.na.2004.04.014. |
[6] |
N. B. Huy, B. T. Quan and N. H. Khanh,
Existence and multiplicity results for generalized logistic equations, Nonlinear Anal., 144 (2016), 77-92.
doi: 10.1016/j.na.2016.06.001. |
[7] |
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. |
[8] |
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. |
[9] |
D. Mugnai and N. S. Papageorgiou,
Resonant nonlinear Neumann problems with indefinite weight, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 11 (2012), 729-788.
doi: 10.2422/2036-2145.201012_003. |
[10] |
N. S. Papageorgiou and V. D. Radulescu,
Multiple solutions with precise sign information for nonlinear Robin problems, J. Differential Equations, 256 (2014), 2449-2479.
doi: 10.1016/j.jde.2014.01.010. |
[11] |
N. S. Papageorgiou and V. D. Radulescu,
Nonlinear, nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737-764.
doi: 10.1515/ans-2016-0023. |
[12] |
N. S. Papageorgiou, V. D. Radulescu and D. D. Repovs, Nonlinear elliptic inclusions with unilateral constraint and dependence on the gradient, Appl. Math. Optim., (2016), 1-23.
doi: 10.1007/s00245-016-9392-y. |
[13] |
D. Ruiz,
A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Differential Equations, 199 (2004), 96-114.
doi: 10.1016/j.jde.2003.10.021. |
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