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

Abstract

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 $p=2+\frac{8}{N}$. In this paper, a perturbation functional is addedto the Kirchhoff-type constrained variational 问题 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 条件, we also prove that the minimizer is the ground state 解 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 constrainedcritical exponent.