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

作者：Dong, Ge; Fang, Xiaochun^*

来源：Boundary Value Problems, 2016, 2016(1): 106.

DOI：10.1186/s13661-016-0612-9

• separable Musielak-Orlicz-Sobolev spaces
• differential equation
• sub-supersolution

In this paper, we study the existence of weak solutions for differential equations of divergence form @@@ -div(a(1)(x,Du)) + a(0)(x,u) = f(x,u,Du), @@@ in Omega 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 x R x R-N -> 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.

• 出版日期2016-5-31
• 单位同济大学