Concentration of ground state solutions for supercritical zero-mass (N, q)-equations of Choquard reaction

Abstract

We study the following singularly perturbed (N, q)-equation of Choquard type -εNΔNu-εqΔqu=εμ-N(∫RNK(y)F(u(y))|x-y|μdy)K(x)f(u),x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\varepsilon ^N\\Delta _Nu-\\varepsilon ^q\\Delta _qu=\\varepsilon ^{\\mu -N} \\bigg (\\int _{\\mathbb {R}^N}\\frac{K(y)F(u(y))}{|x-y|^{\\mu }}dy\\bigg )K(x)f(u),~x\\in \\mathbb {R}^N, \\end{aligned}$$\\end{document}where Δru=div(|∇u|r-2∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _ru = \ ext {div}(|\ abla u|^{r-2}\ abla u)$$\\end{document} denotes the usual r-Laplacian operator with r∈{q,N}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\in \\{q,N\\}$$\\end{document} and 1<q<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<q<N$$\\end{document}, ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document} is a sufficiently small parameter, K∈C0(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\in C^0(\\mathbb {R}^N)$$\\end{document} satisfies some technical assumptions, 0<μ<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\mu <N$$\\end{document} and F is the primitive of f that fulfills a supercritical exponential growth in the Trudinger–Moser sense. Due to the new version of Trudinger–Moser type inequality introduced in Shen and Rădulescu (Zero-mass (N, q)-Laplacian equation with Stein-Weiss convolution part 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}: supercritical exponential case. submitted), we aim to derive the existence and concentration of ground state solutions for the given equation using variational method, where the concentrating phenomenon appears at the maximum point set of K as ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+$$\\end{document}.