Weak solutions for anisotropic nonlinear elliptic problem with variable exponent and measure data.

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Résumé

Let Ω⊂ℝN(N≥3) be a bounded smooth domain and μ be a bounded Radon measure.
In this paper, the authors study the following anisotropic nonlinear boundary value problem:
−∑i=1N∂∂xiai(x,∂u∂xi)=μ in Ω,u|∂Ω=0,
where ai(⋅,⋅):Ω×ℝ→ℝ is a Carathéodory function (i=1,2,…,N) and there exists C1>0 such that
|ai(x,ξ)|≤C1(1+|ξ|pi(x)−1) for all ξ∈ℝ and a.e. x∈Ω,i=1,2,…,N.
Under some further conditions on ai(x,ξ) and pi(x), the existence of a weak solution for the above nonlinear elliptic problem is proved in an anisotropic variable exponent Sobolev space.

Mots-clés

weak solution; elliptic equation; variable exponent; anisotropic Sobolev spaces; Marcinkiewicz spaces; Radon measure