Differential equations of divergence form in separable Musielak-Orlicz-Sobolev spaces

Abstract

In this paper, we study the existence of weak solutions for differential equations of divergence form $$-\operatorname {div}\bigl(a_{1}(x,Du)\bigr)+a_{0}(x,u)=f(x,u,Du), $$ in Ω coupled with a Dirichlet or Neumann boundary condition in separable Musielak-Orlicz-Sobolev spaces where $a_{1}$ satisfies the growth condition, the coercive condition, and the monotone condition, and $a_{0}$ satisfies the growth condition without any coercive condition or monotone condition. The right-hand side $f:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb {R}$ is a Carathéodory function satisfying a growth condition dependent on the solution u and its gradient Du. We prove the existence of weak solutions by using a linear functional analysis method. Some sufficient conditions guarantee the existence enclosure of weak solutions between sub- and supersolutions. Our method does not require any reflexivity of the Musielak-Orlicz-Sobolev spaces.