Liouville-type theorem for Kirchhoff equations involving Grushin operators

Abstract

The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations:0.1−M(∫RNω(z)|∇Gu|2dz)divG(ω(z)∇Gu)=f(z)eu,z=(x,y)∈RN=RN1×RN2\\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} $$\\begin{aligned} \\begin{aligned}[b] & -M \\biggl( \\int _{{\\mathbb{R}} ^{N}}\\omega (z) \\vert \\nabla _{G}u \\vert ^{2}\\,dz \\biggr) \\operatorname{div}_{G} \\bigl(\\omega (z) \\nabla _{G}u \\bigr)=f(z)e^{u}, \\\\ &\\quad z=(x,y) \\in R^{N}=R^{N_{1}}\\times R^{N_{2}} \\end{aligned} \\end{aligned}$$ \\end{document} and0.2M(∫RNω(z)|∇Gu|2dz)divG(ω(z)∇Gu)=f(z)u−q,z=(x,y)∈RN=RN1×RN2,\\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} $$\\begin{aligned} \\begin{aligned}[b] & M \\biggl( \\int _{\\mathbb{R}^{N}}\\omega (z) \\vert \\nabla _{G}u \\vert ^{2}\\,dz \\biggr) \\operatorname{div}_{G} \\bigl(\\omega (z) \\nabla _{G}u \\bigr)=f(z)u^{-q}, \\\\ &\\quad z=(x,y) \\in {\\mathbb{R}} ^{N}={\\mathbb{R}} ^{N_{1}}\\times {\\mathbb{R}} ^{N_{2}}, \\end{aligned} \\end{aligned}$$ \\end{document} where M(t)=a+bt^{k}, tgeq 0, with a>0, b, kgeq 0, k=0 if and only if b=0. q>0 and omega (z), f(z)in L^{1}_{mathrm{loc}}({mathbb{R}} ^{N}) are nonnegative functions satisfying omega (z)leq C_{1}|z |_{G}^{theta } and f(z)geq C_{2}|z|_{G}^{d} as |z|_{G} geq R_{0} with d>theta -2, R_{0}, C_{i} (i=1,2) are some positive constants, here alpha geq 0 and |z|_{G}=(|x|^{2(1+ alpha )}+|y|^{2})^{frac{1}{2(1+alpha )}} is the norm corresponding to the Grushin distance. N_{alpha }=N_{1}+(1+alpha )N_{2} is the homogeneous dimension of {mathbb{R}} ^{N}. operatorname{div}_{G} (resp., nabla _{G}) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, θ, d, and N_{alpha }, the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated.