Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

Abstract

We investigate the following problem -div(v(x)|∇u|m-2∇u)+V(x)|u|m-2u=|x|-θ∗|u|b|x|α|u|b-2|x|αu+λ|x|-γ∗|u|c|x|β|u|c-2|x|βuinRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\mathrm{div}(v(x)|\ abla u|^{m-2}\ abla u)+V(x)|u|^{m-2}u= \\left( |x|^{-\ heta }*\\frac{|u|^{b}}{|x|^{\\alpha }}\\right) \\frac{|u|^{b-2}}{|x|^{\\alpha }}u+\\lambda \\left( |x|^{-\\gamma }*\\frac{|u|^{c}}{|x|^{\\beta }}\\right) \\frac{|u|^{c-2}}{|x|^{\\beta }}u \\quad \ ext { in }{\\mathbb {R}}^{N}, \\end{aligned}$$\\end{document}where b,c,α,β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b, c, \\alpha , \\beta >0$$\\end{document}, θ,γ∈(0,N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta ,\\gamma \\in (0,N)$$\\end{document}, N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 3$$\\end{document}, 2≤m<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le m< \\infty$$\\end{document} and λ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\in {\\mathbb {R}}$$\\end{document}. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.