上同调指标与具临界非线性项的拟线性椭圆方程

Quasilinear equations not only have physical background, but also are of great significance in mathematics. The spectra for quasilinear operators can not be described completely, and it is difficult to apply the critical point theories which are based on the eigenvalues and used to the semilinear equations successfully to quasilinear equations. They need further developing and innovating. So using the methods of defining eigenvalues with cohological index, combining the critical point theorems and the theories for the elliptic equations, this project is to get the existence and multiplicity of solutions for the quasilinear elliptic equations involving p-Laplacian、N-Laplacian、p(x)-Laplacian and fractional p-Laplacian with critical nonlinearities in bounded domains and whole spaces respectively; and to investigate the relationship of solutions between the fractional p Laplacian and p Laplacian equations.

拟线性椭圆方程既具有强烈的物理背景,在数学理论上也有重要意义. 到目前为止拟线性算子的谱还没有得到完全的描述,基于特征值的临界点定理对半线性椭圆方程成果丰富, 但对拟线性椭圆方程却无法应用, 急需人们进一步发展和创新. 本项目将利用上同调指标定义特征值的方法,结合临界点定理及偏微分方程基本理论, 致力于研究有界区域及全空间中具临界非线性项的拟线性椭圆方程包括p-Laplacian、N-Laplacian、p(x)-Laplacian及分数阶p-Laplacian方程解的存在性与多重性,同时探求分数阶p-Laplacian方程解与对应的p-Laplacian方程解的关系.

拟线性椭圆方程既具有强烈的物理背景, 在数学理论上也有重要意义. 到目前为止拟线性算子的谱还没有得到完全的描述, 基于特征值的临界点定理对半线性椭圆方程成果丰富, 但对拟线性椭圆方程却无法应用. 本项目基于上同调指标及伪指标, 建立了两个新的不需依赖线性子空间的临界点定理, 推广了著名的环绕定理(P.H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 5(1978) 215-223.)及Bartolo 等人的抽象的临界点定理(P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal.7(1983), 981-1012. ), 并将其应用于包括p-Laplacian、N-Laplacian、分数阶p-Laplacian等具临界非线性项的拟线性椭圆方程中, 得到了解的存在性与多重性.