Abstract
In this paper, we study the existence of entropy solutions for some nonlinear $p(x)-$elliptic equation of the type $$Au - \mbox{div }\phi(u) + H(x,u,\nabla u) = \mu,$$ where $A$ is an operator of Leray-Lions type acting from $W_{0}^{1,p(x)}(\Omega)$ into its dual, the strongly nonlinear term $H$ is assumed only to satisfy some nonstandard growth condition with respect to $|\nabla u|,$ here $\>\phi(\cdot)\in C^{0}(I\!\!R,I\!\!R^{N})\>$ and $\mu$ belongs to ${\mathcal{M}}_{0}^{b}(\Omega)$.
Highlights
Variable exponent Sobolev spaces have attracted an increasing attention of many researchers, the impulse for this mainly comes from their applications in various fields, as in image processing and electro-rheological fluids
This paper is organized as follows: We introduce in the section 2 some assumptions on a(x, s, ξ) and H(x, s, ξ) and some important lemmas useful to prove our main result
The aim of this paper is to study the existence of solutions for the strongly nonlinear p(x)−elliptic problem: Au − div φ(u) + H(x, u, ∇u) = f − div F in Ω, u= 0 on ∂Ω, (2.7)
Summary
The Sobolev space with variable exponent W 1,p(·)(Ω) is defined by:. Where a : Ω × IR × IRN → IR is a Caratheodory function satisfying the following conditions:. A function u is called an entropy solution of the strongly nonlinear p(x)-elliptic problem (2.6) if u ∈ T01,p(x)(Ω), H(x, u, ∇u) ∈ L1(Ω) and a(x, u, ∇u) · ∇Tk(u − v) dx + φ(u) · ∇Tk(u − v) dx. Assuming that (2.3) − (2.6) hold, let f ∈ L1(Ω) and φ(·) ∈ C0(IR, IRN ), the problem (2.7) has at least one entropy solution. Proof of Theorem 3.2 Step 1: Approximate problems Let (fn)n∈IN be a sequence of smooth functions such that fn → f in L1(Ω) and |fn| ≤ |f |.
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