On a class of Kirchhoff problems via local mountain pass

In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: -{ε2 a + εb R3 δu 2 dx ) Δu + V (x)u= f(u) + γu5 in R3 , u ε H1 (R3) , u > 0 in R3 , where ε > 0 is a small parameter, a , b > 0 are constants, γ ε { 0 , 1 }, V is a continuous positive potential with a local minimum, and f is a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in (Adv. Nonlinear Stud. 14 (2014), 483-510; J. Differ. Equ. 252 (2012), 1813-1834; J. Differ. Equ. 253 (2012), 2314-2351)