Abstract
In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: − ( ε 2 a + ε b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( u ) + γ u 5 in R 3 , u ∈ H 1 ( R 3 ) , u > 0 in R 3 , where ε > 0 is a small parameter, a , b > 0 are constants, γ ∈ { 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).
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