Differential Equations Seminar: On existence and concentration phenomena for Kirchhoff type problems

This talk focuses on the existence, asymptotic behaviour and local uniqueness of solutions for Kirchhoff type equations and constrained minimizers of Kirchhoff type energy functionals. Four main results are presented.
First, the existence of multi-peak solutions for a singularly perturbed Kirchhoff equation is established via variational method. Second, for a Kirchhoff equation with a radially symmetric potential in the whole space, the existence of infinitely many non-radial solutions is proved by means of finite dimensional reduction method. Third, the concentration behavior and local uniqueness of constrained minimizers for a Kirchhoff energy functional are investigated. Fourth, the existence and asymptotic behaviour of constrained minimizers are studied for a Kirchhoff energy functional with a degenerate potential. These results characterize the solution structure and concentration phenomena for Kirchhoff elliptic problems under nonlocal effects. These are joint works with Lu Lu, Wei Shuai and Chun-Lei Tang.