Opuscula Math. 39, no. 2 (2019), 195-206

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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  • Chiara Lini
  • Università degli Studi di Perugia, Dipartimento di Matematica e Informatica, Via Vanvitelli1 - 06123 Perugia, Italy
  • Communicated by Dušan Repovš.
  • Received: 2018-09-19.
  • Accepted: 2018-11-03.
  • Published online: 2018-12-07.
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Cite this article as:
Roberta Filippucci, Chiara Lini, Existence results and a priori estimates for solutions of quasilinear problems with gradient terms, Opuscula Math. 39, no. 2 (2019), 195-206, https://doi.org/10.7494/OpMath.2019.39.2.195

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