Weak and entropy solutions to nonlinear Neumann boundary-value problems with variable exponents

Abstract

In this article, we study the following nonlinear Neumann boundary-value problem − div a(x, ∇u) + |u| ^p(x)−2 u = f in Ω, on ∂Ω, where Ω is a smooth bounded open domain in ℝ ^N , is the outer unit normal derivative on ∂Ω, div a(x, ∇u) a p(x)-Laplace type operator. We prove the existence and uniqueness of a weak solution for f ∈ L ^{(p ₋)′}(Ω), the existence and uniqueness of an entropy solution for L ¹-data f independent of u and the existence of weak solutions for f dependent on u. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

Keywords:

• generalized Lebesgue and Sobolev spaces
• weak solution
• entropy solution
• p(x)-Laplace operator

AMS Subject Classifications::

• 35J20
• 35J25
• 35D30
• 35B38
• 35J60

Acknowledgement

The authors want to express their deepest thanks to the editor and anonymous referees for comments and suggestions on this article.