A priori estimates for elliptic problems via Liouville type theorems

Laura Baldelli; Roberta Filippucci

doi:10.3934/dcdss.2020148

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2020, Volume 13, Issue 7: 1883-1898. Doi: 10.3934/dcdss.2020148

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A priori estimates for elliptic problems via Liouville type theorems

Laura Baldelli^1, and
Roberta Filippucci^2, ,

1.
Department of Mathematics, University of Firenze, viale Morgagni 40-44, 50134 Firenze, Italy
2.
Department of Mathematics, University of Perugia, via Vanvitelli 1, 06123 Perugia, Italy

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

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday, with deep gratitude, esteem and affection

Received: November 2018

Revised: November 2018

Published: April 2020

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Abstract

In this paper we prove a priori estimates for positive solutions of elliptic equations of the $ p $-Laplacian type on arbitrary domains of $ \mathbb {R}^N $, when a nonlinearity depending on the gradient is considered. Also the case of systems with very general nonlinearities is considered. Our main theorems extend previous results by Polacik, Quitter and Souplet in [26] in which either the case $ p = 2 $ with a nonlinearity depending on the gradient or the $ p $-Laplacian case with a nonlinearity not depending on the gradient is treated. The technique is based on the use of a method developed in [26] whose main tools are rescaling arguments combined with a key "doubling" property, which is different from the celebrated blow up technique due to Gidas and Spruck in [16]. A discussion on the sharpness of the main result in the scalar case is presented.

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Mathematics Subject Classification: Primary: 35J92, 35J70; Secondary: 35J47.

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