一类具有快速增权的非线性椭圆方程解的存在性  

On Existence of Solutions for a Class of Nonlinear Elliptic Equation with Fast Increasing Weight

作　　者：郑文静 陈尚杰[1,2] 李麟 ZHENG Wenjing;CHEN Shangjie;LI Lin(School of Mathematics and Statistics,Chongqing Technology and Business University,Chongqing 400067,China;Chongqing Key Laboratory of Social Economy and Applied Statistics,Chongqing 400067,China)

机构地区：[1]重庆工商大学数学与统计学院,重庆400067 [2]经济社会应用统计重庆市重点实验室,重庆400067

出　　处：《西南师范大学学报（自然科学版）》2022年第11期50-56,共7页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：重庆市教育委员会基金项目(KJQN20190081);重庆工商大学基金项目(CTBUZDPTTD201909).

摘　　要：本文研究了全空间上的一类具有快速增权的非线性椭圆方程-a+b∫_(R ^(3)) K(x)|▽u|^(2)d x div(K(x)▽u)=K(x)f(x,u)x∈R^(3)解的存在性问题.其中K(x)=exp|x|24为权函数;非线性项中的函数f(x,u)为连续函数,满足全局次临界条件,且在原点处超线性,在无穷远处超四次增长.在局部AR条件下,证明了该类方程的泛函满足(C)c条件且具有山路几何结构,从而得到了方程非平凡解的存在性.而将局部AR条件替换为全局AR条件时,又得到了该方程基态解(即该方程所有解中能量泛函值最小的解)的存在性.目前关于该方程还没有类似的结果.The existence problem of solutions for a class of nonlinear elliptic equations has been considered in the paper with rapidly increasing weights in the whole space-a+b∫_(R ^(3)) K(x)|▽u|^(2)d x div(K(x)▽u)=K(x)f(x,u)x∈R^(3) where K(x)=exp|x|24 is the weight function,the function f(x,u)in the nonlinear term is a continuous function and satisfies the global subcritical,superlinear at the origin,infinite super-quaternary growth in the distance.Under local AR conditions,it has been proved that the functional of this type of equation satisfies the(C)c conditions and has a mountain pass geometry,so that the existence of non-trivial solutions to the equation is obtained.When the local AR condition is replaced by the global AR condition,the existence of the ground state solution of the equation is obtained,that is,the solution with the smallest energy functional value among all the solutions of the equation.There is no similar result for this equation at present.

关 键 词：变分法 山路定理 (C)c条件 非平凡解 基态解 

分 类 号：O176.3[理学—数学]