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

Abstract

In this paper our main goal is to present a new global $L^{\infty }$-estimate for a general class of quasilinear elliptic equations of the form $$-div\mathcal{A}(x,u,\nabla u)=\mathcal{B}(x,u,\nabla u)$$ 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 $L^{\infty }$-estimate of the form $$|u{|}_{\infty ,\hat{I}\text{c◯}}\le C\hat{I}¦(|u{|}_{{\hat{I}}^2,\hat{I}\text{c◯}})$$ where $\hat{I}¦:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a data independent function with ${\lim }_{s\to 0^{+}}\hat{I}¦(s)=0$.

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