On the existence and multiplicity of solutions for a class of sub-Laplacian problems involving critical Sobolev–Hardy exponents on Carnot groups

ABSTRACT

In this work, we study the following sub-elliptic equations on Carnot group $\mathbb{G}$ with Hardy-type singularity and critical Sobolev–Hardy exponents $-{\mathrm{\Delta }}_{\mathbb{G}}u=\lambda {\psi }^{\alpha }\frac{|u{|}^{2^{\ast }(\alpha )-2}u}{d(z{)}^{\alpha }}+\beta f(z)|u{|}^{p-2}u,z\in \mathbb{G},$where $-{\mathrm{\Delta }}_{\mathbb{G}}$ stands for the sub-Laplacian operator on Carnot group $\mathbb{G}$, $0<\alpha \le 2$, and $2^{\ast }(\alpha )=\frac{2(Q-\alpha )}{Q-2}$ is the critical Sobolev–Hardy exponent, Q is the homogeneous dimension with respect to the dilation ${\delta }_{\gamma }$ naturally associated with $-{\mathrm{\Delta }}_{\mathbb{G}}$, d is the natural gauge associated with the fundamental solution of $-{\mathrm{\Delta }}_{\mathbb{G}}$ on $\mathbb{G}$, $\psi =|{\mathrm{\nabla }}_{\mathbb{G}}d|$ and ${\mathrm{\nabla }}_{\mathbb{G}}$ is the horizontal gradient associated with ${\mathrm{\Delta }}_{\mathbb{G}}$. Through variational methods combined with the theory of genus, we prove that our problems admit infinitely many solutions.

KEYWORDS:

• Subelliptic problem
• critical Sobolev–Hardy exponents
• Hardy-type potential
• infinitely many solutions
• Carnot groups

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35H20
• 35J70
• 35B40

Disclosure statement

No potential conflict of interest was reported by the author.