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In this paper, by using systemtric mountain pass theorem and Pohozaev type indentity, we study the existence of nontrivial solutions and the necessary conditions for the existence of solutions about no homogeneous quasilinear equation in exterior unit ball. The mondel of equation iswhere （?） ,1 < p, q < N.Ωis the exterior unit ball.Let C0∞（Ω） denote the function space consisting of radially C∞smooth functions on Ωwith copact supporting set. Dr1,p（Ω） be the completion of C0,r∞（Ω） with respect to the normalDefine（?） is measurable, （?）},（?） is measurable, （?）},where q > 1, s≥1. we defineIt is a Banach space under the normAccording to the relationship among a, b, p, q and N, we define index s* and obtain the result of compact embedding and existence of nontrivial solutions. The main result of this paper is following theorems.Theorem 1.1 Let 1 < p, q < N, If s≥s*, then In particularly, if s > s*, the embedding is compact embedding.Theoreml.2 Let 1 < p, q < N, If s > max{p, q, s*}, then the equation （P） have infinitely many radial solutions.Theorem 1.3 If the equation （P） have a nontrivial solution , then a, b, p, q and N must satisfyor
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