本项目旨在应用变分方法研究一类拟线性变分数阶Laplacian系统的可解性。该类问题属变分数阶非局部问题，起源于一个由温度变化引起的流体力学模型，也常用于介质力学和人口动力学的研究中。不仅如此，其退化的局部形式亦出现于博弈论的研究中。由于变分数阶、拟线性和非局部性的原因，该类系统虽然在形式上有变分泛函，但在构造变分泛函时用的工作空间不是线性空间，仅只是完备度量空间，而且所构造的泛函可能是既无上界也无下界的强不定泛函，通常的极大极小方法和古典的Morse指标理论不适用于该系统的研究。因此，本项目将从如下四方面进行研究：.⑴应用Gelerkin逼近法建立度量空间上的强不定Morse指标理论；.⑵利用度量空间上的强不定Morse指标理论研究其弱解、多胞解和节点解的存在性和多解性；.⑶利用对偶方法将该系统转化为其相应的积分系统进行研究；.⑷利用极大极小方法和移动平面法研究其正解的存在性和多重性。

This project shall aim to study the solvability for a class of quasilinear variable-order fractional laplacian systems by variational methods. This is a class of nonlocal problems proposed as a fluid mechanics model caused by temperature changes and can model the medium mechanics and population dynamics. Moreover, its degenerative local form appeared in the study of game theory. Generally, because of the reason of variable-order fractional derivative,quasi-linearity and nonlocality,the problems are in the form of the variational functionals.Unfortunately, our working spaces are not vector spaces, only completed metric spaces and the functional may be a strongly infinite functional with neither upper nor lower bounds. Therefore, classical minimax method and morse index theory may not be applied. From above all, we will study the project as follows:.⑴We will use the Gelerkin approximation method to establish the strongly infinite morse index theory in the metric space；.⑵We will study the existence and multiplicity of weak solutions , multibump solutions and nodal solutions by strongly infinite morse index theory in the metric space；.⑶We will study the problems by changing them into their corresponding integral systems via a dual approach；.⑷We will use the minimax method and the moving plane method to study the existence and multiplicity of positive solutions for this system.