一类带有梯度项非线性椭圆问题的解存在性和多重性  

Existence and multiplicity of solutions to a class of nonlinear elliptic equation with gradient terms

作　　者：熊慧如 方飞 XIONG Huiru;FANG Fei(School of Mathematics and Statistics,Beijing Technology and Business University,Beijing 102488,China)

机构地区：[1]北京工商大学数学与统计学院

出　　处：《厦门大学学报（自然科学版）》2024年第6期1099-1105,共7页Journal of Xiamen University：Natural Science

摘　　要：[目的]研究如下的带梯度项的非线性椭圆方程的非平凡解的存在性和多重性,Lu-Δu-12(x·▽u)=f(x,u),其中f∈C(R^(N)×R,R),N≥3.[方法]主要使用变分方法.环绕定理是变分法中的一个重要结果,它给出了泛函的(PS)序列存在的一个充分条件,并且在假定了(PS)紧致条件后就可以得到一个临界点.喷泉定理被用来证明非线性椭圆方程无穷多解的存在性.[结果]得到三个结果.结果1:通过假设非线性项是次临界的,在原点附近是次线性的,且满足(AR)条件,利用环绕定理证明了该方程一个非平凡解的存在性.结果2:在结果1的假设基础上,还假设了非线性项满足对称条件,然后利用喷泉定理证明了该方程无穷多解的存在性.结果3:通过假设非线性项是次临界的,在原点附近满足局部环绕条件,证明了该方程一个平凡解的存在性.[结论]首次利用环绕定理证明了非线性项满足局部环绕条件时非平凡解的存在性,且首次利用喷泉定理得到了该方程无穷多解的存在性.[Objective]The main purpose of this article is to investigate the existence and multiplicity of nontrivial solutions for nonlinear elliptic equations with gradient terms,Lu-Δu-12(x·▽u)=f(x,u),Where f∈C(R^(N)×R,R),N≥3.This type of elliptic equation has wide applications in fields such as mathematics,physics,engineering,and economics.By analyzing the existence and multiplicity of solutions to such equations,we can further deepen our understanding of the essence of such equations.[Methods]To achieve our research objectives,we use the variational methods to prove our conclusions.The basic idea of the variational methods is to reduce the problem of solving nonlinear partial differential equations and operator equations to the problem of finding critical points(especially extreme points)of a generalized function in a certain function space.Variational methods have wide applications in partial differential equations,integral equations,and many mathematical and physical problems.The variational methods began with Johann Bernoulli's public proposal of the brachistochrone curve problem in June 1696.The 18th century was the pioneering period of the variational methods,which established the Euler equation that extreme values should satisfy and solved a large number of specific problems based on it.In the 19th century,the variational methods was widely applied to mathematics and physics,and sufficient conditions for extreme value functions were established.At the beginning of the 20th century,three of the 23 famous mathematical problems mentioned by Hilbert in his speech at the International Congress of Mathematicians in Paris were related to variational methods.The idea of variational methods runs through Courant and Hilbert's book“Mathematical and Physical Methods”.The minimax method,represented by the Mountain Pass Theorem established by Ambresetti and Rabinowitz in the 1970s,is an important part of modern variational theory,and Morse's extensive variational methods(Morse Theory)is a hallmark of the developmen

关 键 词：梯度项 解的存在性 环绕定理 喷泉定理 

分 类 号：O175.25[理学—数学]