In this paper we prove a priori estimates for positive solutions of elliptic equations of the p-Laplacian type on arbitrary domains of 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 Duke Math. J in 2007 in which either the case p=2 with a non linearity 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 the paper by Polacik, Quitter and Souplet 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 1981. A discussion on the sharpness of the main result in the scalar case is presented.
A priori estimates for elliptic problems via Liouville type theorems
BALDELLI, LAURA;Filippucci, Roberta
2020
Abstract
In this paper we prove a priori estimates for positive solutions of elliptic equations of the p-Laplacian type on arbitrary domains of 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 Duke Math. J in 2007 in which either the case p=2 with a non linearity 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 the paper by Polacik, Quitter and Souplet 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 1981. A discussion on the sharpness of the main result in the scalar case is presented.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
