Opuscula Math. 39, no. 2 (2019), 195-206
https://doi.org/10.7494/OpMath.2019.39.2.195

Opuscula Mathematica

# Existence results and a priori estimates for solutions of quasilinear problems with gradient terms

Roberta Filippucci
Chiara Lini

Abstract. In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in $$\mathbb{R}^N$$ with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of $$u$$ and by removing the growth condition for the nonlinearity $$f$$ at $$u=0$$.

Keywords: existence result, quasilinear problems, a priori estimates.

Mathematics Subject Classification: 35J92, 35J70.

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1. C. Azizieh, P. Clément, A priori estimates and continuation methods for positive solutions of $$p$$-Laplace equations, J. Diff. Eq. 179 (2002), 213-245.
2. J.P. Bartier, Global behavior of solutions of a reaction-diffusion equation with gradient absorption in unbounded domains, Asymptot. Anal. 46 (2006), 325-347.
3. H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), 601-614.
4. P. Clément, R. Manásevich, E. Mitidieri, Positive solutions for a quasilinear system via blow-up, Comm. Partial Differential Equations 18 (1993), 2071-2106.
5. L. Damascelli, F. Pascella, Monotonicity and symmetry of solutions of $$p$$-Laplace equations, $$1\lt p\leq 2$$, via the moving plane method, Ann. Scuola Norm. Pisa, Cl. Sci. 26 (1998), 689-707.
6. E. Di Benedetto, $$C^{1,\alpha}$$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850.
7. D. De Figueiredo, P.L. Lions, R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), 41-63.
8. D. De Figueiredo, J. Yang, A priori bounds for positive solutions of a non-variational elliptic system, Comm. Partial Differential Equations 26 (2001), 2305-2321.
9. L. Dupaigne, M. Ghergu, V. Rădulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl. 87 (2007), 563-581.
10. R. Filippucci, C. Lini, Existence of solutions for quasilinear Dirichlet problems with gradient terms, Discrete Contin. Dyn. Syst. Ser. S, Special Issue on the occasion of the 60th birthday of Vicentiu D. Radulescu, 12 (2019), 267-286.
11. L. Gasinski, N.S. Papageorgiou, Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differential Equations 263 (2017), 1451-1476.
12. M. Ghergu, V. Rădulescu, Nonradial blow-up solutions of sublinear elliptic equations with gradient terms, Comm. Pure Appl. An. 3 (2004), 465-474.
13. M. Ghergu, V. Rădulescu, On a class of sublinear elliptic problems with convection term, J. Math. Anal. Appl. 311 (2005), 635-646.
14. M. Ghergu, V. Rădulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, vol. 37, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2008.
15. B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525-598.
16. B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883-901.
17. M.A. Krasnoselskii, Fixed point of cone-compressing or cone-extending operators, Soviet. Math. Dokl. 1 (1960), 1285-1288.
18. G.M. Lieberman, Boundary regulary for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219.
19. S. Lorca, P. Ubilla, A priori estimate for a quasilinear problem depending on the gradient, J. Math. Anal. Appl. 367 (2010), 60-74.
20. E. Mitidieri, S.I. Pohozaev, The absence of global positive solutions to quasilinear elliptic inequalities, Dokl. Math. 57 (1998), 250-253.
21. D. Motreanu, M. Tanaka, Existence of positive solutions for nonlinear elliptic equations with convection terms, Nonlinear Anal. 152 (2017) 38.
22. P. Polacik, P. Quitter, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J. 139 (2007), 1203-1219.
23. P. Pucci, J. Serrin, The strong maximum principle, Progress in nonlinear differential equations and their applications, vol. 73, Birkhäuser, 2007.
24. P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202.
25. V. Rădulescu, Bifurcation and asymptotics for elliptic problems with singular nonlinearity. Elliptic and parabolic problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel 63 (2005), 389-401.
26. V. Rădulescu, M. Xiang, B. Zhang, Existence of solutions for a bi-nonlocal fractional p-Kirchhoff type problem, Comput. Math. Appl. 71 (2016), 255-266.
27. D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Differential Equations 199 (2004), 96-114.
28. J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247-302.
29. J. Serrin, H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), 79-142.
30. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150.
31. N. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747.
32. J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202.
33. X.-J. Wang, Y.-B. Deng, Existence of multiple solutions to nonlinear elliptic equations of nondivergence form, J. Math. Anal. Appl. 189 (1995), 617-630.
• Chiara Lini
• Università degli Studi di Perugia, Dipartimento di Matematica e Informatica, Via Vanvitelli1 - 06123 Perugia, Italy
• Communicated by Dušan Repovš.
• Accepted: 2018-11-03.
• Published online: 2018-12-07.