Existence of solutions for quasilinear elliptic systems in divergence form with variable growth

Yongqiang Fu,Miaomiao Yang

doi:10.1186/1029-242x-2014-23

Abstract

This paper is concerned with the following Dirichlet problem for a quasilinear elliptic system with variable growth: −divσ(x,u(x),Du(x))=f in Ω, u(x)=0 on ∂ Ω, where Ω⊂ R n is a bounded domain. By means of the Young measure and the theory of variable exponent Sobolev spaces, we obtain the existence of solutions in W 0 1 , p ( x ) (Ω, R m ) for each f∈ ( W 0 1 , p ( x ) ( Ω , R m ) ) ∗ .

Highlights

Introduction and main resultIn this paper, we consider the Dirichlet problem for the quasilinear elliptic system– div σ x, u(x), Du(x) = f in, ( . )u(x) = on ∂, where ⊂ Rn (n ≥ ) is a bounded domain
Let Mm×n denote the real vector space of m×n matrices equipped with the inner product M : N = MijNij
Proof In order to prove that G is continuous, it is sufficient to show that G(al) → G(a ) in Rr as al → a in Rr

Summary

For h

([ ]) Let | | < ∞, where | | denotes the Lebesgue measure of , p (x), p (x) ∈ P( ), the necessary and sufficient condition for Lp (x)( ) ⊂ Lp (x)( ) is that p (x) ≤ p (x) for almost every x ∈ , and in this case the embedding is continuous. ([ ]) Let F : × Rm × Mm×n → R be a Carathéodory function and {uk} be a sequence of measurable functions, where uk : → Rm, such that uk → u in measure and Duk generates the Young measure νx. If the sequence {uj} is bounded in Lp(x)( , Rm), there is a Young measure νx generated by {uj} satisfying νx = and the weak L -limit of {uj} is Rm λ dνx(λ).

Then we define

Proof Let us consider the sequence

Then we get