Existence of solutions for quasilinear Dirichlet problems with gradient terms

Roberta Filippucci; Chiara Lini

doi:10.3934/dcdss.2019019

Article Contents

2019, Volume 12, Issue 2: 267-286. Doi: 10.3934/dcdss.2019019

This issue Previous Article Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence Next Article Robin problems for the p-Laplacian with gradient dependence

Existence of solutions for quasilinear Dirichlet problems with gradient terms

Roberta Filippucci^, and
Chiara Lini

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

* Corresponding author: Roberta Filippucci
* Corresponding author: Roberta Filippucci

Received: April 30, 2017

Revised: November 30, 2017

Published: April 2019

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35J92, 35J70; Secondary: 35J25.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	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.
	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.
	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.
	P. Clement , R. 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.
	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.
	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.
	D. De Figueiredo , P. 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.
	L. Dupaigne , M. 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.
	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.
	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.
	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.
	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.
	M. A. Krasnoselskii , Fixed point of cone-compressing or cone-extending operators, Soviet. Math. Dokl., 1 (1960) , 1285-1288.
	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.
	E. Mitidieri and S. I. Pohozaev , The absence of global positive solutions to quasilinear elliptic inequalities, Dokl. Math., 57 (1998) , 250--253.
	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.
	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.
	V. Rădulescu , M. 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.
	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.
	J. Serrin , Local behavior of solutions of quasilinear equations, Acta Math., 111 (1964) , 247-302. doi: 10.1007/BF02391014.
	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.
	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.
	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.
	J. L. Vázquez , A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984) , 191-202. doi: 10.1007/BF01449041.