In this paper, we study the existence of weak solutions for differential equations of divergence formin 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 : × R × R N → 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.

We prove the existence results in the setting of Orlicz spaces for the following nonlinear elliptic equation:

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