This paper deals with the existence solution for the following type of boun- dary value problems: { Δ ( jxjp(x) jΔujp(x)2 Δu ) = jujq(x)2 u; in Ω; u = Δu = 0; on @Ω; where Ω is a smooth bounded domain in ℜN. It is established for a negative , there exists at least one weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces and a variant of the Mountain Pass theorem.
