Concentrating solutions for a fractional Kirchhoff equation with critical growth

Article type: Research Article

Authors: Ambrosio, Vincenzo^{; *}

Affiliations: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy. E-mail: v.ambrosio@univpm.it

Correspondence: [*] Corresponding author. E-mail: v.ambrosio@univpm.it.

Abstract: In this paper we consider the following class of fractional Kirchhoff equations with critical growth: (ε2sa+ε4s−3b∫R3|(−Δ)s2u|2dx)(−Δ)su+V(x)u=f(u)+|u|2s∗−2uin R3,u∈Hs(R3),u>0in R3, where ε>0 is a small parameter, a,b>0 are constants, s∈(34,1), 2s∗=63−2s is the fractional critical exponent, (−Δ)s is the fractional Laplacian operator, V is a positive continuous potential and f is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions uε which concentrates around a local minimum of V as ε→0.

Keywords: Fractional Kirchhoff equation, variational methods, critical growth

DOI: 10.3233/ASY-191543

Journal: Asymptotic Analysis, vol. 116, no. 3-4, pp. 249-278, 2020