Solutions for fractional p(x,⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p(x,\\cdot )$\\end{document}-Kirchhoff-type equations in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{N}$\\end{document}

Abstract

In this paper, we discuss the fractional p(x,⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p(x,\\cdot )$\\end{document}-Kirchhoff-type equations M(∫RN×RN1p(x,y)|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dxdy)(−Δp(x,.))su+|u|p¯(x)−2u=f(x,u).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ M\\left (\\int _{\\mathbb{R}^{N}\ imes \\mathbb{R}^{N}} \\frac{1}{p(x,y)} \\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}dxdy\\right )(-\\Delta _{p(x,.)})^{s} u+|u|^{\\bar{p}(x)-2}u=f(x,u).$$\\end{document} We weaken the conditions on the nonlinear term f and get the existence and multiplicity of solutions via variational methods, which improves some previous results.