| Abstract: |
| In this talk, we are concerned with the existence and concentration behavior of ground state solutions for the following Kirchhoff type equations involving the 1-Laplacian operator
$$\left(a+b\left(\displaystyle\int_{\mathbb R^N}\epsilon|Du|+\int_{\mathbb R^N}V(x)|u|\right)^{\alpha-1}\right)\left(-\epsilon\Delta_{1}u+V(x)\displaystyle\frac{u}{|u|}\right) = f(u)$$
in $\mathbb R^N$, where $a, b>0$, $N\geq2$, $\epsilon>0$ is a small parameter, $\alpha\in(1,\frac{N}{N-1})$, and the operator $\Delta_1$ is the well known 1-Laplacian operator. Under suitable conditions on $V$ and $f$, using non-smooth critical point theory, Lions' Concentration-Compactness Principle and some ingenious analyses, we first prove the existence of ground state solutions $u_\epsilon$ for a small parameter $\epsilon > 0$ in $BV(\mathbb R^N)$, the space of functions of bounded variation, which is the natural functional setting for the 1-Laplacian. Subsequently, we demonstrate that as $\epsilon \to 0$, this family of solutions concentrates around a global minimum of the potential $V$. This work is jointly with Jiaqian Zhang and Jianjun Zhang. |
|