含约束临界Kirchhoff型约束变分问题极小解的存在性及其极限行为

Existence and limit behavior of minimizers of Kirchhoff-type constrained variational problems with constrained critical exponent

非线性Kirchhoff型约束变分问题当非线性项只含一个幂次项且指数为约束临界p=2+8/N时,由现有文献可知该问题不存在极小解.本文考虑了含低阶扰动项和约束临界指数项的Kirchhoff型约束变分问题,利用伸缩技巧、集中紧原理和Pohozaev恒等式,得到了扰动项指数和系数对该变分问题极小解存在性的影响,并证明该极小解是相对应的Kirchhoff方程的基态解.进一步,本文通过精细的能量估计,探讨扰动项指数趋于约束临界指数时极小能量和极小解的极限行为.

There are no minimizers for nonlinear Kirchhoff-type constrained variational problems when the nonlinear term only includes an exponential term and the exponent is the constrained critical exponent p=2+8/N.In this paper,a perturbation functional is added to the Kirchhoff-type constrained variational problem with constrained critical exponent.Then,for this functional,a complete classification with respect to the exponent and the coefficient in the perturbation term for its normalized critical points is obtained by using the scaling technique,the concentration-compactness principle and the Pohozaev identity.Under some conditions,we also prove that the minimizer is the ground state solution of the corresponding Kirchhoff equation.Furthermore,by using the energy estimation,the limit behaviors of the minimum energy and the minimizer are discussed when the perturbation term exponent tends to the constrained critical exponent.