# American Institute of Mathematical Sciences

April  2019, 12(2): 287-295. doi: 10.3934/dcdss.2019020

## 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

* Corresponding author: Dimitri Mugnai

Dedicated to Vicentiu, on the occasion of his 60th birthday, with sincere friendship and esteem

Received  June 2017 Revised  November 2017 Published  August 2018

Fund Project: The first author is member of the INDAM Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA). Her research is supported by the 2017 INdAM-GNAMPA Project Comportamento asintotico e controllo di equazioni di evoluzione non lineari. The second author is member of the INDAM Gruppo Nazionale per l'Analisi Matematica, la Probabilità a e le loro Applicazioni (GNAMPA). His research is supported by the 2017 INdAM-GNAMPA Project Equazioni Differenziali Non Lineari and by the M.I.U.R. project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT 009).

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.

Citation: Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020
##### 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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