Liouville type theorems for nonlinear elliptic equations on the whole space $\mathbb{R}^{N}$

Abstract

The aim of this paper is to study the properties of the solutions of Δ p u + f 1 (u) − f 2 (u) = 0 in all R N . We obtain Liouville type boundedness for the solutions. We show that |u| ≤ ( α β ) 1 m−q+1 on R N , under the assumptions f 1 (u) ≤ αu p−1 and f 2 (u) ≥ βu m , for some 0 < α ≤ β and m > q− 1 ≥ p−1 > 0. If u does not change sign, we prove that u is constant.