Existence of positive solutions to Kirchhoff equations with vanishing potentials and general nonlinearity

We study the existence of positive solutions to the following Kirchhoff type equation with vanishing potential and general nonlinearity: $$\begin{aligned} \left\{ \begin{aligned}&-(\varepsilon ^2a+\varepsilon b{\int _{\mathbb {R}^3}}{|\nabla v|}^{2})\Delta v+V(x)v=f(v), ~~~~x\in \mathbb {R}^3, \\&v>0,~~~v\in H^{1}(\mathbb {R}^3), \end{aligned} \right. \end{aligned}$$-(ε2a+εb∫R3|∇v|2)Δv+V(x)v=f(v),x∈R3,v>0,v∈H1(R3),where $$\varepsilon >0$$ε>0 is a small parameter, $$a,b>0$$a,b>0 are constants and the potential V can vanish, i.e., the zero set of V, $$\mathcal {Z}:=\{x\in \mathbb {R}^3|V(x)=0\}$$Z:={x∈R3|V(x)=0} is non-empty. In our case, the method of Nehari manifold does not work any more. We first make a truncation of the nonlinearity and prove the existence of solutions for the equation with truncated nonlinearity, then by elliptic estimates, we prove that the solution of truncated equation is just the solution of our original problem for sufficiently small $$\varepsilon >0$$ε>0.