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Comparison and Positive Solutions for Problems with the (p, q)-Laplacian and a Convection Term

Published online by Cambridge University Press:  16 April 2014

Luiz F. O. Faria ,
Olímpio H. Miyagaki  and
Dumitru Motreanu
Luiz F. O. Faria
Affiliation:
Departamento de Matemática – ICE, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330, Minas Gerais, Brazil
Olímpio H. Miyagaki
Affiliation:
Departamento de Matemática – ICE, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330, Minas Gerais, Brazil
Dumitru Motreanu
Affiliation:
Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France, (xlink:href="motreanu@univ-perp.fr">motreanu@univ-perp.fr)
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Abstract

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p, q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.

Keywords

quasi-linear elliptic equation(p, q)-Laplacianconvection termpositive solutionapproximationcomparisonPrimary 35J60Secondary 35J92
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 3 , October 2014 , pp. 687 - 698
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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References

1.Benci, V., Fortunato, D. and Pisani, L., Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 1 (1998), 315–344.Google Scholar
2.Brézis, H., Functional analysis, Sobolev spaces and partial differential equations (Springer, 2011).Google Scholar
3.Cherfils, L. and Il'yasov, Y., On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Commun. Pure Appl. Analysis 4 (2005), 9–22.CrossRefGoogle Scholar
4.Correa, F. J. S. A. and Nascimento, R. G., On the existence of solutions of a nonlocal elliptic equation with a p-Kirchhoff-type term, Int. J. Math. Math. Sci. 2008 (2008), 364085.Google Scholar
5.Covei, D. P., Existence and asymptotic behavior of positive solution to a quasilinear elliptic problem in ℝN, Nonlin. Analysis 4 (2008), 2615–2622.Google Scholar
6.Díaz, J. I. and Saa, J. E., Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris I 305 (1978), 521–524.Google Scholar
7.Figueiredo, G. M., Existence of positive solutions for a class of p&q elliptic problems with critical growth on ℝN, J. Math. Analysis Applic. 378 (2011), 507–518.CrossRefGoogle Scholar
8.Ladyzhenskaya, O. A. and Ural'tseva, N. N., Linear and quasilinear elliptic equations (Academic Press, 1968).Google Scholar
9.Lieberman, G. M., Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Analysis 12 (1988), 1203–1219.CrossRefGoogle Scholar
10.Lieberman, G. M., The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. PDEs 16 (1991), 311–361.Google Scholar
11.Mawhin, J. and Willem, M., Critical point theory and Hamiltonian systems (Springer, 1989).Google Scholar
12.Miyajima, S., Motreanu, D. and Tanaka, M., Multiple existence results of solutions for the Neumann problems via super- and sub-solutions, J. Funct. Analysis 262 (2012), 1921–1953.CrossRefGoogle Scholar
13.Motreanu, D., Motreanu, V. V. and Papageorgiou, N. S., Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Annali Scuola Norm. Sup. Pisa V 10 (2011), 1–27.Google Scholar
14.Pucci, P. and Serrin, J., The maximum principle (Birkhäuser, Basel, 2007).CrossRefGoogle Scholar
15.Ruiz, D., A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Diff. Eqns 199 (2004), 96–114.CrossRefGoogle Scholar
16.Sidiropoulos, N. E., Existence of solutions to indefinite quasilinear elliptic problems of p-q-Laplacian type, Electron. J. Diff. Eqns 2010 (2010), no. 162.Google Scholar
17.Zou, H. H., A priori estimates and existence for quasi-linear elliptic equations, Calc. Var. PDEs 33 (2008), 417–437.Google Scholar