分数阶Laplacian方程若干问题的研究

Fractional Laplacian operator is an important extension of the Laplacian operator. Fractional Laplacian equations are introduced by many practical fields. The research has important values in theory and application, in recent years which gradually forms a hot research hotspot in the world. Rich research achievements about sign-changing solutions and singular solution and jumping nonlinear problem solution of semilinear elliptic equation and quasilinear elliptic equation are obtained, but there are few corresponding results on the fractional Laplacian equation results. By using variational method and topological degree, all kinds of analysis techniques combined with properties of Laplacian operator fractional order, the existence of sign-changing solution and existence and multiplicity of singular solutions for nonlinear fractional Laplacian equation with variational structure boundary value problems, and existence and multiplicity of solutions of the fractional Laplacian problem with jumping nonlinear term, are going to be studied respectively in this programme. We shall establish some theory results, which promote and enrich nonlinear fractional Laplacian equation theory research. We try hard to achieve a breakthrough on the way to solve the problem at the same time.

分数阶Laplacian算子是Laplacian 算子的一个重要推广。分数阶Laplacian方程在许多实际领域被提出，其研究具有重要的理论和应用价值, 近年来在国际上逐步形成一个研究热点。半线性椭圆方程及拟线性椭圆方程的变号解、奇异解及跳跃非线性问题的解的研究有着丰富的研究成果，但分数阶Laplacian方程的变号解、奇异解及跳跃非线性问题的解的研究结果却很少。 本项目拟结合分数阶Laplacian算子的各种性质，运用变分法与拓扑度方法，及各种分析技巧，分别研究若干具有变分结构的非线性分数阶Laplacian方程边值问题变号解、奇异解的存在性与多重性 , 及具有跳跃非线性项的分数阶Laplacian问题解的存在性与多重性，建立若干理论结果，促进和丰富非线性分数阶Laplacian方程的理论研究。同时在解决问题的方法上力争有所突破和创新。

分数阶Laplacian方程有重要的理论和应用意义，本项目系统地研究了几类分数阶Laplacian方程问题，我们解决问题的方法有创新，发展和拓广了非线性分数阶Laplacian方程的研究。分数阶p&q Laplacian问题奇异正解的存在性和多重性的研究是个难点，我们克服了这个困难，运用变分方法，Nehari流形方法及巧妙运用一些分析技巧，解决了这个问题。研究了分数阶Laplacian算子的Fucik谱的性质，利用其性质，获得了非局部椭圆算子方程组多重解的存在性以及带Hardy位势的拟线性椭圆方程组弱解的存在性。利用最小化及变形引理，研究了分数阶非线性方程变号解的存在性, 获得了带变号加权函数的分数阶Kirchhoff方程解与变号解的存在性，及非线性分数阶Schrodinger–Poisson方程组解的存在性定理。本项目中我们还研究了若干非线性分数阶方程边值问题解的存在性与多重性。