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

In this paper our main goal is to present a new global $$L^\infty $$ L ∞ -estimate for a general class of quasilinear elliptic equations of the form $$\begin{aligned} -\text{ div }\,{{\mathcal {A}}}(x,u,\nabla u)={{\mathcal {B}}}(x,u,\nabla u) \end{aligned}$$ - div A ( x , u , ∇ u ) = B ( x , u , ∇ u ) under minimal structure conditions on the functions $${{\mathcal {A}}}$$ A and $${{\mathcal {B}}},$$ B , and in arbitrary domains of $${{{\mathbb {R}}}}^N.$$ R N . The main focus and the novelty of the paper is to prove $$L^\infty $$ L ∞ -estimate of the form $$\begin{aligned} |u|_{\infty , \Omega }\le C \Phi (|u|_{\beta ,\Omega }) \end{aligned}$$ | u | ∞ , Ω ≤ C Φ ( | u | β , Ω ) where the constant C encodes the contribution of the data, and $$\Phi : {{{\mathbb {R}}}}^+\rightarrow {{{\mathbb {R}}}}^+$$ Φ : R + → R + is a data independent, continuous, and nondecreasing function with $$\lim _{s\rightarrow 0^+}\Phi (s)=0.$$ lim s → 0 + Φ ( s ) = 0 .