本项目拟采用变分方法研究修正的p-Laplacian方程的驻波解的存在性和多解性。该类问题属非局部问题，起源于一个有弹性细绳的自由振动模型，也常用于种群动力学系统中模拟动物种群的密度。不仅如此，其退化的局部形式亦出现于等离子物理和凝聚态理论的研究中。本项目首先将研究其变分结构。而由于其具有修正项和拟线性的原因，该类问题虽然在形式上有变分泛函，但我们在构造变分泛函时用的工作空间甚至不是线性空间，仅只是完备度量空间，传统的极大极小方法和临界点理论不适用于其研究。因此，我们将应用近年发展起来的完备度量空间上连续泛函的非光滑临界点理论研究其非平凡（或非半平凡）弱解和多胞解的存在性和多解性，获得修正的p-Laplacian方程的驻波解的存在性和多解性的一些充分条件，并对解做数值模拟，说明这些条件的合理性和广泛性。

This program shall aim to study the existence and multiplicity of standing waves for modified p-Laplacian equations via variational methods. This is a class of nonlocal problems proposed as a model for free vibrations of elastic strings and can model the population density. Moreover, its degenerative local form appeared in the study of plasma physis and condensed matter theory. Firstly, we shall study the variational structures for the problems. Generally, because of the reason of modified term and nonlocality, the problems are in the form of the variational functionals.Unfortunately, our working spaces are not vector spaces, only completed metric spaces. Therefore, classical minimax method and critical point theory may not be applied. From above all, we will study the existence and multiplicity of nontrivial(or nonsemi-trivial) weak solutions and multibump solutions by nonsmooth critical point theory for continuous functional in completed metric spaces. The numerical simulation of solutions may be presented to illustrate the feasibility and effectiveness of our results.