$L^\infty$-bounds of solutions for a class of strongly nonlinear elliptic equations in Musielak spaces

Abstract

In this paper we establish the existence of bounded solutions to a strongly nonlinear elliptic problem of the form $$ -\mathop{\rm div}\mathcal{A}(x,u,{\nabla}u)+g(x,u,\nabla u)= f \quad\text{in }{\Omega}, $$ with $u\in W^1_0L_\varphi({\Omega})\cap L^{\infty}(\Omega)$, where $$ \mathcal{A}(x,s,\xi)\cdot\xi\geq \overline{\varphi}_{x}^{-1} (\varphi(x,h(|s|)))\varphi(x,|\xi|), $$ $h\colon {\mathbb{R}^+} \to \mathopen ]0,1] $ is a continuous decreasing function with unbounded primitive and $g$ is a non-linearity satisfying $|g(x,s,\xi)|\leq\beta(s)\varphi(x,|\xi|)$. We assume the $\Delta_{2}$-condition on the Musielak function $\varphi$.