一类广义Kirchhoff方程基态变号解的存在性  

Existence of Ground State Sign-changing Solutions for a Class of Generalized Kirchhoff Equations

作　　者：黄婷 晏颖 商彦英[1] HUANG Ting;YAN Ying;SHANG Yanying(School of Mathematics and Statistics,Southwest University,Chongqing 400715,China)

机构地区：[1]西南大学数学与统计学院,重庆400715

出　　处：《西南师范大学学报（自然科学版）》2023年第1期18-25,共8页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11471267)。

摘　　要：研究了一类广义Kirchhoff方程-a+b∫R^(3)|u|2 d x△u+V(x)u=g(u)其中a,b>0是常数.由于在方程中出现了非局部项b∫R^(3)|u|2 d x△u,所以,方程的变分泛函与b=0时方程的变分泛函具有不同的性质.与相关文献相比,g不需要满足单调性条件,并且非线性项g包含g(t)=|t|^(p-2) t(2<p≤4)这种情况,V也不需要满足强制性条件.首先引入辅助算子,构造伪梯度向量场,证明了下降流不变集的存在性.其次,由于4超线性AR条件不成立,所以引入了一种非局部扰动方法,即增加了一个高阶项β|u|^(r-2)u和另一个非局部扰动.对于扰动问题,通过改进的AR条件和下降流不变集下的极大极小参数得到了扰动问题的变号解,进而得到了原方程的变号解.最后,证明了该变号解是原方程的基态变号解.Study a class of generalized Kirchhoff equations-a+b∫R^(3)|u|2 d x△u+V(x)u=g(u)where a,b>0 are constants.Since there are nonlocal terms b∫R^(3)|u|2 d x△u in the equation,the variational generalization of the equation has different properties from the case b=0.In contrast to the related literature,g does not need to satisfy the monotonicity condition and the nonlinear term g contains g(t)=|t|^(p-2) t(2<p≤4)this case,V also does not need to satisfy the mandatory condition.Firstly,an auxiliary operator is introduced to construct a pseudo-gradient vector field to prove the existence of a descending flow invariant set.Second,since the 4-hyperlinear AR condition does not hold,it is necessary to introduce a nonlocal perturbation method by adding a higher-order termβ|u|r-2 u and another nonlocal perturbation.For the perturbed problem,the variational solution of the perturbed problem is obtained by the improved AR condition and the extremely minimal parameter under the descending flow invariant set,which in turn leads to the variational solution of the original problem.Finally,it is proved that the variational solution is the base-state variational solution of the original problem.

关 键 词：KIRCHHOFF方程 变号解 非局部扰动方法 下降流不变集 

分 类 号：O176.3[理学—数学]