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 Caratheodory 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.

Highlights

  • Let ⊂ RN be a bounded domain with Lipschitz boundary

  • Le [ ] established a subsupersolution method for variational inequalities with Leray-Lions operators in Sobolev spaces with variable exponents

  • We refer to some results of sub-supersolution methods for variational inequalities and the existence of solutions for differential equations studied in variable exponent Sobolev or Orlicz-Sobolev spaces

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Summary

Introduction

Let ⊂ RN be a bounded domain with Lipschitz boundary. Le [ ] established a subsupersolution method for variational inequalities with Leray-Lions operators in Sobolev spaces with variable exponents. Orlicz spaces in [ ] and study the existence of solutions for the following nonlinear problem:. ) with Dirichlet boundary or Neumann boundary condition in separable Musielak-Orlicz-Sobolev spaces. ) of can be omitted because of the reflexivity of the Musielak-Orlicz-Sobolev spaces in [ ]. We refer to some results of sub-supersolution methods for variational inequalities and the existence of solutions for differential equations studied in variable exponent Sobolev or Orlicz-Sobolev spaces (see, e.g., [ – ]). A real function defined on × R+, where R+ = [ , +∞), will be said a generalized N function (i.e. a Musielak-Orlicz function), denoted by ∈ N( ), if it satisfies the following conditions:.

Define s
Kn and h
It implies that l
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