一类变指数椭圆方程的可解性研究

This project mainly studies the solvability of a class of variable exponent elliptic equation when the nonlinear term satisfies the coupling of different local growth conditions, such as local "sub-linear", "super-linear", critical exponent, supercritical exponent and so on. We consider the solvability and multiplicity of these equations by improving some traditional theoretical methods and developing new variational techniques, and enrich and perfect the well known results of elliptic equation with variable exponent. Our main contents are as follows: 1)If the nonlinear term satisfies the coupling of local "sub-linear" and local "super-linear", we study the existence and multiplicity of nontrivial solutions by using interval divided thoughts and the functional disturbance; 2)If the nonlinear term involves local critical exponent or local supercritical exponent, we give some conditions of the solvability of these equations by applying the principle of concentration, Emden-Fowler transform and Lyapunov-Schmidt.. These equations of the project have extensive practical background. By proposing and solving these problems, it will help to expand applications of this theory in electrorheological fluid, elastic mechanics and image processing and so on.

本项目主要研究非线性项满足“次线性”、“超线性”、临界和超临界指数等各种局部增长条件相互耦合情形的变指数椭圆方程的可解性。通过改进传统的理论方法，发展新的变分技巧，来研究该类方程的可解性和多解性，进而丰富和完善已有变指数椭圆方程的研究结果。主要内容包括：1) 当非线性项满足局部“次线性”与局部“超线性”增长条件时，用区间划分思想和泛函扰动等变分方法来研究该类方程非平凡解的存在性和多重性；2) 当非线性项满足局部临界指数或局部超临界指数增长条件时，用集中紧性原理、Emden-Fowler变换与Lyapunov-Schmidt约化等手段来研究该类方程的可解性条件。. 本项目研究的方程具有广泛的实际背景，这些问题的提出和解决，将有助于拓展相关理论在电流变流体、弹性力学和图像处理等领域的应用。

在本项目中，我们对变指数非线性项涉及局部“次线性”与局部“超线性”增长的半线性椭圆方程、p(x)-Laplace方程和p(x)-Kirchhoff方程非平凡解的可解性和多解性进行了研究. 对非线性项涉及局部“次线性”增长、局部“超线性”增长、临界或超临界指数增长的半线性椭圆方程和涉及凹凸非线性项的p(x)-Laplace方程，我们通过修正非线线性项建立扰动方程，利用变分技巧和估计方法，分别获得了这类方程正解、变号解和径向对称解的存在性和多重性. 对带有局部消失位势项的p(x)-Laplace方程，利用Clark定理和对称山路引理，获得了无穷多低能量解和高能量解的存在性. 此外，部分结果也被推广到Kirchhoff方程，并讨论了一类涉及负模增长的新的p(x)-Kirchhoff方程，一定程度上丰富和完善了已有变指数椭圆方程的一些研究成果.