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we with compact derivative, whose Gateauxderivative is given by??(u)(v)

we obtain the compact embedding X ,? C0(¯?) forN2< p?. Therefore, there existsa positive constant c > 0 such that?u?C0(¯?)? c?u?a(1.4)for each u ? X,16.We need the following fundamental lemma in the proof of our results.Lemma 1.6. For each u ? X let ?(u) =??(|?u|p(x)+a(x)|u|p(x))dx, then we have(i) ?u?a< 1(= 1;> 1) ? ?(u) < 1(= 1;> 1),(ii) If ?u?a< 1 ? ?u?p+a ? ?(u) ? ?u?p?a ,(iii) If ?u?a> 1 ? ?u?p?a ? ?(u) ? ?u?p+a .The proof of this lemma is similar to the proof in 13. Here and in the sequel,we suppose the following condition on the Carathéodory function f : ?? × R ? R:(F) |f(x,s)| ? ?(x) + b|s|?(x)?1for all (x,s) ? ?? × R, where ? ? L?(x)?(x)?1(??),b ? 0 is a constant and ? ? C(??) such that1 < ??:= infx?¯??(x) ? ?(x) ? ?+:= supx?¯??(x) < p?.(1.5)Moreover, setF(x,t) :=?t0f(x,?)d?,for all (x,t) ? ?? × R.2Multiple SolutionsIn this section we present the existence of one non-trivial solution for the problem(1.1), then we study the existence of multiple solutions for (1.1) .Definition 2.1. We say that a function u ? X is a weak solution of problem (1.1)if??|?u|p(x)?2?u?v dx +??a(x)|u|p(x)?2uv dx ? ????f(x,u)v d? = 0,for all v ? X.5Theorem 2.2. Suppose f : ??×R ? R is a Carathéodory function satisfying (F).Assume that there exist d ? 1 and k ? c with dp+?a?1

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