Liouville type theorems for nonlinear elliptic equations involving operator in divergence form

Abstract

The aim of this paper is to study the properties of the solutions of \documentclass[12pt]{minimal}\begin{document}${\rm div} (\mathcal {A}(x, \nabla u))\break +f_{1}(u)-f_{2}(u)=0$\end{document} div (A(x,∇u))+f1(u)−f2(u)=0 in all \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^{N}.$\end{document}RN. We obtain Liouville type boundedness for the solutions. We show that \documentclass[12pt]{minimal}\begin{document}$|u|\le (\frac{\alpha }{\beta })^{\frac{1}{m-q+1}}$\end{document}|u|≤(αβ)1m−q+1 on \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^{N},$\end{document}RN, under the assumptions f1(u) ⩽ αuq–1 and f2(u) ⩾ βum, for some 0 < α ⩽ β and m > q−1 > 0. If u does not change the sign, we prove that u is constant.