Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities

We study the following nonlinear Kirchhoff equation on the whole space−(1+b∫R3(|∇u|2+V(x)u2)dx)[Δu+V(x)u]=f(x,u) in R3, where b>0b>0 is a constant and the nonlinear term f(x,u)f(x,u) is only locally defined for |u||u| small and satisfies some mild conditions. It is proved that the above problem admits infinity many weak solutions provided that some additional assumptions are satisfied. Moreover, we show that these solutions tend to zero in L∞(R3)L∞(R3). Our method relies upon a variant of the symmetric mountain pass lemma due to Kajikiya and Moser iteration method.