A direct proof of existence of weak solutions to fully anisotropic and inhomogeneous elliptic problems

TL;DR

This paper presents a direct proof establishing the existence and uniqueness of weak solutions for a broad class of nonlinear elliptic equations within fully anisotropic and inhomogeneous Musielak–Orlicz spaces, without imposing common growth conditions.

Contribution

It provides a novel direct proof for weak solutions to complex anisotropic elliptic problems in Musielak–Orlicz spaces, extending previous results to more general growth conditions.

Findings

Existence and uniqueness of weak solutions proved.

Applicable to problems with polynomial, Orlicz, variable exponent, and double phase growth.

No $ abla_2$ or $ abla_2$ conditions required.

Abstract

We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower order terms. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak--Orlicz spaces generated by an $N$-function $M:\Omega\times\mathbb{R}^d\to[0,\infty)$. Neither $\nabla_2$ nor $\Delta_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.