Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction

Abstract

In this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity ε2αa+ε4α-3b∫R3|(-Δ)α2u|2dx(-Δ)αu+V(x)u=H(u-β)f(u)inR3,u∈Hα(R3),u>0inR3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\left( \\varepsilon ^{2\\alpha }a+\\varepsilon ^{4\\alpha -3}b\\int _{{\\mathbb {R}}^3}|(-\\Delta )^{\\frac{\\alpha }{2}} u|^2{{\\mathrm{d}}}x\\right) (-\\Delta )^\\alpha {u}+V(x)u = H(u-\\beta )f(u) &{} \\quad \ ext{ in }\\,\\,{\\mathbb {R}}^3, \\\\ u\\in H^\\alpha ({\\mathbb {R}}^3),\\quad u>0 &{} \\quad \ ext{ in }\\,\\, {\\mathbb {R}}^3, \\end{array} \\right. \\end{aligned}$$\\end{document}where varepsilon ,beta >0 are small parameters, alpha in (frac{3}{4},1) and a, b are positive constants, (-Delta )^{alpha } is the fractional Laplacian operator, H is the Heaviside function, V is a positive continuous potential, and f is a superlinear continuous function with subcritical growth. By using minimax theorems together with the non-smooth theory, we obtain existence and concentration properties of positive solutions to this non-local system.