LAUSR

This paper is concerned with the following Kirchhoff-type equation $$\begin{aligned} -\left( a+b\int _{\mathbb {R}^3}|\nabla {u}|^2\mathrm {d}x\right) \triangle u+V(x)u=f(x, u), \quad x\in \mathbb {R}^{3}, \end{aligned}$$-a+b∫R3|∇u|2dx▵u+V(x)u=f(x,u),x∈R3,where $$V\in \mathcal {C}(\mathbb {R}^{3}, (0,\infty ))$$V∈C(R3,(0,∞)), $$f\in \mathcal {C}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})$$f∈C(R3×R,R), V(x) and f(x, t) are periodic or asymptotically periodic in x. Using weaker assumptions $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^3}=\infty $$lim|t|→∞∫0tf(x,s)ds|t|3=∞ uniformly in $$x\in \mathbb {R}^3$$x∈R3 and $$\begin{aligned}&\left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3} \right] \mathrm {sign}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \\&\quad \forall x\in \mathbb {R}^3,\ t>0, \ \tau \ne 0 \end{aligned}$$f(x,τ)τ3-f(x,tτ)(tτ)3sign(1-t)+θ0V(x)|1-t2|(tτ)2≥0,∀x∈R3,t>0,τ≠0with a constant $$\theta _0\in (0,1)$$θ0∈(0,1), instead of the common assumption $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s)\mathrm {d}s}{|t|^4}=\infty $$lim|t|→∞∫0tf(x,s)ds|t|4=∞ uniformly in $$x\in \mathbb {R}^3$$x∈R3 and the usual Nehari-type monotonic condition on $$f(x,t)/|t|^3$$f(x,t)/|t|3, we establish the existence of Nehari-type ground state solutions of the above problem, which generalizes and improves the recent results of Qin et al. (Comput Math Appl 71:1524–1536, 2016) and Zhang and Zhang (J Math Anal Appl 423:1671–1692, 2015). In particular, our results unify asymptotically cubic and super-cubic nonlinearities.