A priori estimates for elliptic problems via Liouville type theorems

Abstract

In this paper we prove a priori estimates for positive solutions of elliptic equations of the \begin{document}$ p $\end{document} -Laplacian type on arbitrary domains of \begin{document}$ \mathbb {R}^N $\end{document} , when a nonlinearity depending on the gradient is considered. Also the case of systems with very general nonlinearities is considered. Our main theorems extend previous results by Polacik, Quitter and Souplet in [ 26 ] in which either the case \begin{document}$ p = 2 $\end{document} with a nonlinearity depending on the gradient or the \begin{document}$ p $\end{document} -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.