On existence and concentration of positive solutions for a fractional Kirchhoff equation with critical exponential growth

Abstract

In this paper we consider the following fractional Kirchhoff type problem 0 & {\\rm in}\\ \\mathbb{R}, \\end{array}\\right. \\end{align*} $$]]> { ( ϵ a + ϵ b 2 π ∫ R 2 | u ( x ) − u ( y ) | 2 | x − y | 2 d x d y ) ( − Δ ) 1 2 u + V ( x ) u = f ( u ) i n R , u ∈ H 1 / 2 ( R ) , u > 0 i n R , where ϵ is a positive parameter and a,b>0 are constants; V is a positive continuous potential and f is a nonlinear term with critical exponential growth in the Trudinger–Moser sense. We obtain the existence and concentration of positive solutions by variational methods.