Abstract
The paper introduces a new type of nonlinear elliptic Dirichlet problem driven by the (p, q)-Laplacian where the reaction term is in the convection form (meaning that it exhibits dependence on the solution and its gradient) composed with a (possibly nonlinear) general map called intrinsic operator on the Sobolev space. Under verifiable hypotheses, we establish the existence of at least one (weak) solution. A second main result deals with the uniqueness of solution. Finally, a third result provides the existence and uniqueness of solution to a problem of this type involving a translation viewed as an intrinsic operator. Examples show the applicability of these results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H. Amann, M. Crandall, On some existence theorems for semi-linear elliptic equations. Indiana Univ. Math. J. 27(5), 779–790 (1978)
D. Averna, D. Motreanu, E. Tornatore, Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence. Appl. Math. Lett. 61, 102–107 (2016)
H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18(1), 115–175 (1968)
S. Carl, V.K. Le, D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer Monographs in Mathematics (Springer, New York, 2007)
D. De Figueiredo, M. Girardi, M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques. Differ. Integral Equ. 17(1–2), 119–126 (2004)
F. Faraci, D. Motreanu, D. Puglisi, Positive solutions of quasi-linear elliptic equations with dependence on the gradient. Calc. Var. Partial Differ. Equ. 54(1), 525–538 (2015)
L.F.O. Faria, O.H. Miyagaki, D. Motreanu, Comparison and positive solutions for problems with (p, q)-Laplacian and convection term. Proc. Edinb. Math. Soc. (2) 57(3), 687–698 (2014)
L.F.O. Faria, O.H. Miyagaki, D. Motreanu, M. Tanaka, Existence results for nonlinear elliptic equations with Leray-Lions operator and dependence on the gradient. Nonlinear Anal. 96, 154–166 (2014)
L.F.O. Faria, O.H. Miyagaki, M. Tanaka, Existence of a positive solution for problems with (p, q)-Laplacian and convection term in \(\mathbb {R}^n\). Bound. Value Probl. 2016, Paper No. 158, 20 pp.
V.V. Motreanu, Uniqueness results for a Dirichlet problem with variable exponent. Commun. Pure Appl. Anal. 9(5), 1399–1410 (2010)
D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differ. Equ. 199(1), 96–114 (2004)
E. Zeidler, Nonlinear Functional Analysis and Its Applications, vols. II A/B (Springer, Berlin, 1990)
H.H. Zou, A priori estimates and existence for quasi-linear elliptic equations. Calc. Var. Partial Differ. Equ. 33(4), 417–437 (2008)
Acknowledgements
The authors thank Dr. Lucas Fresse for significant suggestions improving the work. The first author is partially supported by CNPq/BRAZIL PROC. 400633/2017-5.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Motreanu, D., Motreanu, V.V. (2019). (p, q)–Laplacian Equations with Convection Term and an Intrinsic Operator. In: Andrica, D., Rassias, T. (eds) Differential and Integral Inequalities. Springer Optimization and Its Applications, vol 151. Springer, Cham. https://doi.org/10.1007/978-3-030-27407-8_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-27407-8_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-27406-1
Online ISBN: 978-3-030-27407-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)