中文摘要：

利用基于临界点理论的变分方法和Ekeland变分原理，研究含凹凸非线性的参数型p（ x）-Laplace方程的Dirichlet问题的正解的存在性。在该方程中，超线性项不需要满足Ambrosetti-Rabinowitz条件，对于取值较小的参数，证明了所研究的问题至少有2个非平凡的光滑正解。

英文摘要：

Using variational methods based on the critical point theory and the Ekeland variational principle,a non-linear parametric Dirichlet problem,driven by the anisotropic p（ x）-Laplacian with the combined effects of concave and convex terms,is studied. In this problem,the superlinear nonlinearity does not need to satisfy the Ambrosetti-Rabinowitz condition. It is shown that for small values of the parameter,the problem has at least two nontrivial smooth positive solutions.