Existence of multiple solutions for fractional p-Kirchhoff equations with concave-convex nonlinearities

Abstract

In this paper, we investigate the existence of multiple solutions for Kirchhoff-type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions as follows: {M(∫R2n|u(x)−u(y)|p|x−y|n+spdxdy)(−△)psu=λ|u|q−2u+αα+β|u|α−2u|v|β,in Ω,M(∫R2n|v(x)−v(y)|p|x−y|n+spdxdy)(−△)psv=μ|v|q−2v+βα+β|v|β−2v|u|α,in Ω,u=v=0,in Rn∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} M(\\int_{\\mathbb{R}^{2n}}\\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\\,dx\\, dy)(-\\triangle )_{p}^{s}u=\\lambda|u|^{q-2}u+\\frac{\\alpha}{\\alpha+\\beta}|u|^{\\alpha -2}u|v|^{\\beta}, & \\mbox{in }\\Omega, \\\\ M(\\int_{\\mathbb{R}^{2n}}\\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\\,dx\\, dy)(-\\triangle )_{p}^{s}v=\\mu|v|^{q-2}v+\\frac{\\beta}{\\alpha+\\beta}|v|^{\\beta -2}v|u|^{\\alpha}, & \\mbox{in }\\Omega, \\\\ u=v=0, & \\mbox{in }\\mathbb{R}^{n}\\setminus\\Omega, \\end{cases} $$\\end{document} where Ω is a smooth bounded set in mathbb{R}^{n}, n>ps with sin(0,1) fixed, {lambda,mu}>0 are two parameters, 1< q< p< p(tau+1)<alpha+beta<p^{*}, p^{*}=frac{np}{n-sp}, M is a continuous function, given by M(h)=k+lh^{tau}, k>0, l,taugeq0, and (-triangle)_{p}^{s} is the fractional p-Laplacian operator. We will prove that the problem has at least two solutions by using the Nehari manifold method and fibering maps.