In this paper we study the existence of a renormalized solution for the nonlinear p(x)--elliptic problem in the Weighted--Variable--Exponent Soblev spaces, of the form: - div (a(x, u, ∇ u)) + H(x, u, ∇ u) = f ∈ Ω, where the right-hand side f belong to L1(Ω) and H(x, s, ξ) is the nonlinear term satisfying some growth condition, but no sign condition on s.
In this article, we study the existence of a renormalized solution for the nonlinear p(x)-parabolic problem associated to the equation:The main contribution of our work is to prove the existence of a renormalized solution in the Sobolev space with variable exponent. The critical growth condition on H(x, t, u, ∇u) is with respect to∇u, no growth with respect to u and no sign condition or the coercivity condition.
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