April  2019, 12(2): 267-286. doi: 10.3934/dcdss.2019019

Existence of solutions for quasilinear Dirichlet problems with gradient terms

Department of Mathematics, University of Perugia, via Vanvitelli 1, 06123 Perugia, Italy

* Corresponding author: Roberta Filippucci

Dedicated to Professor Vicentiu Radulescu on the occasion of his 60th birthday, with deep feelings of esteem and affection.

Received  May 2017 Revised  December 2017 Published  August 2018

In this paper we prove an existence theorem for positive solutions of a nonlinear Dirichlet problem involving the p-Laplacian operator on a smooth bounded domain when a nonlinearity depending on the gradient is considered. Our main theorem extends a previous result by Ruiz in [19], in which a slight modification of the celebrated blowup technique due to Gidas and Spruck, [11] and [12] is introduced.

Citation: Roberta Filippucci, Chiara Lini. Existence of solutions for quasilinear Dirichlet problems with gradient terms. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 267-286. doi: 10.3934/dcdss.2019019
References:
[1]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0.

[2]

C. Azizieh and P. Clement, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Diff. Eq., 179 (2002), 213-245.  doi: 10.1006/jdeq.2001.4029.

[3]

H. Brezis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations, 2 (1977), 601-614.  doi: 10.1080/03605307708820041.

[4]

P. ClementR. Manasevich and E. Mitidieri, Positive solutions for a quasilinear system via blow-up, Comm. Partial Differential Equations, 18 (1993), 2071-2106.  doi: 10.1080/03605309308821005.

[5]

L. Damascelli and F. Pascella, Monotonicity and symmetry of solutions of $p$-Laplace equations, $1< p ≤ 2$, via the moving plane method, Ann. Scuola Norm. Pisa, Cl. Sci, 26 (1998), 689-707. 

[6]

E. Di Benedetto, $C^{1, α}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.

[7]

D. De FigueiredoP. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. 

[8]

L. DupaigneM. Ghergu and V. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl., 87 (2007), 563-581.  doi: 10.1016/j.matpur.2007.03.002.

[9]

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.

[10]

M. Ghergu and V. Rădulescu, On a class of sublinear elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646.  doi: 10.1016/j.jmaa.2005.03.012.

[11]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math, 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[12]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[13]

M. A. Krasnoselskii, Fixed point of cone-compressing or cone-extending operators, Soviet. Math. Dokl., 1 (1960), 1285-1288. 

[14]

G. M. Lieberman, Boundary regulary for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[15]

E. Mitidieri and S. I. Pohozaev, The absence of global positive solutions to quasilinear elliptic inequalities, Dokl. Math., 57 (1998), 250--253. 

[16]

D. Motreanu and M. Tanaka, Existence of positive solutions for nonlinear elliptic equations with convection terms, Nonlinear Anal., 152 (2017), 38-59.  doi: 10.1016/j.na.2016.12.011.

[17]

P. Pucci and J. Serrin, The Strong Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhauser Publ., Switzerland, 2007, X, 234 pages.

[18]

V. RădulescuM. Xiang and B. Zhang, Existence of solutions for a bi-nonlocal fractional p-Kirchhoff type problem, Comput. Math. Appl., 71 (2016), 255-266.  doi: 10.1016/j.camwa.2015.11.017.

[19]

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.

[20]

J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.

[21]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.

[22]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.

[23]

N. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.

[24]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.

show all references

Dedicated to Professor Vicentiu Radulescu on the occasion of his 60th birthday, with deep feelings of esteem and affection.

References:
[1]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.  doi: 10.1016/S0362-546X(97)00530-0.

[2]

C. Azizieh and P. Clement, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Diff. Eq., 179 (2002), 213-245.  doi: 10.1006/jdeq.2001.4029.

[3]

H. Brezis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations, 2 (1977), 601-614.  doi: 10.1080/03605307708820041.

[4]

P. ClementR. Manasevich and E. Mitidieri, Positive solutions for a quasilinear system via blow-up, Comm. Partial Differential Equations, 18 (1993), 2071-2106.  doi: 10.1080/03605309308821005.

[5]

L. Damascelli and F. Pascella, Monotonicity and symmetry of solutions of $p$-Laplace equations, $1< p ≤ 2$, via the moving plane method, Ann. Scuola Norm. Pisa, Cl. Sci, 26 (1998), 689-707. 

[6]

E. Di Benedetto, $C^{1, α}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.

[7]

D. De FigueiredoP. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. 

[8]

L. DupaigneM. Ghergu and V. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl., 87 (2007), 563-581.  doi: 10.1016/j.matpur.2007.03.002.

[9]

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.

[10]

M. Ghergu and V. Rădulescu, On a class of sublinear elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646.  doi: 10.1016/j.jmaa.2005.03.012.

[11]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math, 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[12]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[13]

M. A. Krasnoselskii, Fixed point of cone-compressing or cone-extending operators, Soviet. Math. Dokl., 1 (1960), 1285-1288. 

[14]

G. M. Lieberman, Boundary regulary for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[15]

E. Mitidieri and S. I. Pohozaev, The absence of global positive solutions to quasilinear elliptic inequalities, Dokl. Math., 57 (1998), 250--253. 

[16]

D. Motreanu and M. Tanaka, Existence of positive solutions for nonlinear elliptic equations with convection terms, Nonlinear Anal., 152 (2017), 38-59.  doi: 10.1016/j.na.2016.12.011.

[17]

P. Pucci and J. Serrin, The Strong Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhauser Publ., Switzerland, 2007, X, 234 pages.

[18]

V. RădulescuM. Xiang and B. Zhang, Existence of solutions for a bi-nonlocal fractional p-Kirchhoff type problem, Comput. Math. Appl., 71 (2016), 255-266.  doi: 10.1016/j.camwa.2015.11.017.

[19]

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.

[20]

J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.

[21]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.

[22]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.

[23]

N. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.

[24]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.

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