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

The aim of this paper is to study the properties of the solutions of $\Delta _{p}u+f_{1}(u)-f_{2}(u)=0$ in all $\mathbb{R}^{N}.$ We obtain Liouville type boundedness for the solutions. We show that $\vert u\vert\leq (\frac {\alpha }{\beta })^{\frac {1}{m-q+1}}$ on $\mathbb{R}^{N},$ under the assumptions $f_{1}(u)\leq \alpha u^{p-1}$ and $f_{2}(u)\geq \beta u^{m},$ for some $0<\alpha \leq \beta$ and $m>q-1\geq p-1>0.$ If $u$ does not change sign, we prove that $u$ is constant.