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

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 -functionNeither ∇ 2 nor ∆ 2 conditions are imposed on M . Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.