p-Laplacian方程在非牛顿流体力学、冰川学、燃烧理论、弹性理论、血浆问题、宇宙物理学等领域中有广泛的应用，所以p-Laplacian微分和差分方程有着重要的应用研究价值。目前关于p-Laplacian微分和差分方程同宿解的研究主要采用临界点理论，而临界点理论有一定的局限性。一方面临界点理论应用时需要变分泛函满足紧性条件（PS条件）；另一方面临界点理论虽是研究同宿解存在的有效方法，但不能对其进行定性分析。基于这两方面，本项目主要研究p-Laplacian微分和差分方程的同宿解及其定性分析：.（1）将临界点理论与集中紧性原理相结合的方法推广到p-Laplacian（p(x)-Laplacian）微分和差分方程同宿解问题的研究；.（2）应用变分法与分析技巧研究p-Laplacian微分和差分方程同宿解存在的充要条件，并用分析技巧和数值模拟研究同宿解的存在唯一性。

p-Laplacian equation is widely applied in non-Newtonian fluid mechanics, glaciology, combustion theory, elasticity theory, plasma problem, cosmic physics etc, so p-Laplacian differential and difference equation has important research value. Rencently, critical point theory is the main tool for studying homoclinic solutions of p-Laplacian differential and difference equations, but it has some limitations. The compactness condition (PS condition) of the variational functional is satified; critical point theory is an effective method to study the existence of homoclinic solutions, but it can not obtained the qualitative analysis of homoclinic solutions. Based on the two aspects, we mainly study homoclinic solutions of p-Laplacian differential and difference equations and their qualitative analysis:.(1) We will fisrt employ critical point theory and concentration compabctness principle to study homoclicin solutions of p-Laplacian（p(x)-Laplacian）differential and difference equations；.(2) We will study the necessary and sufficient conditions for the existence of homoclinic solutions of p-Laplacian differential and difference equations by using variational method and analytical technique, then consider the uniqueness by using analytical technique and numerical simulation.