An Analysis of Quasilinear Elliptic Systems with $L^{\infty}$-Type Data

Abstract

The present study establishes the existence and uniqueness of a solution of weak energy for a boundary value problem within a smooth, bounded, open domain $\Omega$ in $\mathbb{R}^{n}$ where $ n \geq 3 $. The problem is defined by the following equation: $\begin{cases} {-div} \left[a(z,\upsilon,D\upsilon)\right]+\vert \upsilon\vert^{p{(z)}-2}\upsilon =f & \text { in } \Omega, \\ ~~ \upsilon =0 & \text { on } \partial \Omega, \end{cases}$ where the function $f$ is constrained to lie within the space $L^{\infty}\left(\Omega ; \mathbb{R}^{m}\right)$. The proof of existence relies on the utilization of the concept of Young measures.