Nontrivial solutions for Kirchhoff-type problems with a parameter

The paper deals with the following Kirchhoff-type problems:{−(a+b∫RN|∇u|2dx)△u+λV(x)u=f(x,u) in RN,u∈H1(RN), where constants a,b>0a,b>0, λ≥1λ≥1 is a parameter, and f(x,u)f(x,u) is either sublinear or superlinear at infinity. Under relaxed assumptions on V(x)V(x), we establish the existence and multiplicity results without the compactness of embedding of the working space.