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

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 $M:\mathrm{\Omega }\times {\mathbb{R}}^d\to [0,\mathrm{\infty })$. Neither ${\mathrm{\nabla }}_2$ nor ${\mathrm{\Delta }}_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.