Global $L^{\mathrm{\infty }}$-estimate for general quasilinear elliptic equations in arbitrary domains of ${\mathbb{R}}^N$

In this paper our main goal is to present a new global $L^{\mathrm{\infty }}$-estimate for a general class of quasilinear elliptic equations of the form $$-div\mathcal{A}(x,u,\mathrm{\nabla }u)=\mathcal{B}(x,u,\mathrm{\nabla }u)$$ under minimal structure conditions on the functions $\mathcal{A}$ and $\mathcal{B}$, and in arbitrary domains of ${\mathbb{R}}^N$. The main focus and the novelty of the paper is to prove $L^{\mathrm{\infty }}$-estimate of the form $$\mathrm{\mid }u{\mathrm{\mid }}_{\mathrm{\infty },\mathrm{\Omega }}\le C\mathrm{\Phi }(\mathrm{\mid }u{\mathrm{\mid }}_{\beta ,\mathrm{\Omega }})$$ where $\mathrm{\Phi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a data independent function with ${\lim }_{s\to 0^{+}}\mathrm{\Phi }(s)=0$.