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 \(a, b>0\) , \(N\ge 2\) , \(\epsilon >0\) 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 \(\epsilon \rightarrow 0\) , this family of solutions concentrates around a global minimum of the potential V. Our results generalize and improve the ones in Alves and Pimenta (Calc. Var. 56:143, 2017) and some other related literatures even when \(\epsilon =1\) , and also extend the classical theory of Kirchhoff equations to the challenging context of the 1-Laplacian operator.
