Affiliations: [a] Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy. E-mail: v.ambrosio@univpm.it | [b] Faculty of Education, and Faculty of Mathematics and Physics & Institute of Mathematics, Physics and Mechanics, University of Ljubljana, SI-1000 Ljubljana, Slovenia. E-mail: dusan.repovs@guest.arnes.si
Abstract: In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems:
−(ε2a+εb∫R3|∇u|2dx)Δu+V(x)u=f(u)+γu5in R3,u∈H1(R3),u>0in R3,
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).
