Abstract
In this article, we study the following degenerate unilateral problems: $$ -\mbox{ div} (a(x,\nabla u))+H(x,u,\nabla u)=f,$$ which is subject to the Weighted Sobolev spaces with variable exponent $W^{1,p(x)}_{0}(\Omega,\omega)$, where $\omega$ is a weight function on $\Omega$, ($\omega$ is a measurable, a.e. strictly positive function on $\Omega$ and satisfying some integrability conditions). The function $H(x,s,\xi)$ is a nonlinear term satisfying some growth condition but no sign condition and the right hand side $f\in L^1(\Omega)$.
Highlights
We make the following assumptions on a, H and f : The function a : Ω × RN → RN is a Caratheodory function satisfying the following assumptions:
In various applications, we can meet boundary value obstacle problems like problem (1.1) for elliptic equations whose ellipticity is ”disturbed” in the sense that some degeneration or singularity appears
For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces
Summary
It is worth pointing out that the conditions (H1) and (H2) are essential Without it the space W 1,p(x)(Ω, ω) is not necessarily a Banach space even though p(x) is a constant. Extracting a subsequence, still denoted by un, we can write un ⇀ u in W01,p(x)(Ω, ω) which implies un → u a.e. in Ω, and since Dn → 0 a.e. in Ω, there exists a subset B of Ω, of zero measure, such that for x ∈ Ω \ B, |u(x)| < ∞, |∇u(x)| < ∞, k(x) < ∞, un(x) → u(x), Dn(x) → 0. Since the sequence a(x, ∇un) is bounded in (Lp′(x)(Ω, ω∗))N and a(x, ∇un) → a(x, ∇u) a.e. in Ω, by Lemma 3.1, we can establish that a(x, ∇un) ⇀ a(x, ∇u) in (Lp′(x)(Ω, ω∗))N a.e. in Ω.
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