| Abstract: |
| In this talk, I will introduce some recent progresses on the nonlinear Choquard equation with the lower critical exponent
$$
-\Delta u + Vu=\bigl(I_\alpha \ast |u|^{\frac{\alpha}{N}+1}\bigr)|u|^{\frac{\alpha}{N}-1}u +f(x,u)\qquad \text{ in }\mathbb{R}^N
$$
where $N\geq 1$, $\alpha\in(0,N)$ is the order of the Riesz potential $I_\alpha$. The exponent $\frac{\alpha}{N}+1$ is critical with respect to the Hardy--Littlewood--Sobolev inequality.
Our study contains two existence results on the ground state solutions for Choquard equations with the lower critical exponent, one is
the case that potential $V$ is a confining potential, the other is when the local perturbation $f$ satisfies some suitable assumptions.
Our approaches are based on variational methods. These are joint works with Prof. Jean Van Schaftingen. |
|