Sub-elliptic problems with multiple critical Sobolev-Hardy exponents on Carnot groups

Abstract

In this paper we first prove the attainability of the following best Sobolev-Hardy constant on Carnot group

$$\begin{aligned} S_{\mu ,\alpha }:= \inf \limits _{u \in S^{1} ({\mathbb {G}})\setminus \{0\}} \frac{ \int _{{\mathbb {G}}} |\nabla _{{\mathbb {G}}}u|^2 dz- \mu \int _{{\mathbb {G}}} \frac{\psi ^{2}(z)|u|^2}{d(z)^{2}}dz}{(\int _{{\mathbb {G}}} \frac{\psi (z)^{\alpha }|u|^{2^*(\alpha )}}{d(z)^{\alpha }}dz)^\frac{2}{2^*(\alpha )}}, \end{aligned}$$

where ${\mathbb {G}}$ is a Carnot group, $\nabla _{{\mathbb {G}}}u$ is the horizontal gradient associated with $\varDelta _{{\mathbb {G}}}$, d is the natural gauge associated with the fundamental solution of $-\varDelta _{{\mathbb {G}}}$ on ${\mathbb {G}}$, $\psi$ is the geometrical function defined as $\psi =|\nabla _{{\mathbb {G}}}d|$, $0\le \alpha <2$, $0\le \mu <\mu _{G}:=(\frac{Q-2}{2})^{2}$, $2^*(\alpha )=\frac{2(Q-\alpha )}{Q-2}$ is the critical Sobolev-Hardy exponent and Q is the homogeneous dimension of ${\mathbb {G}}$. The attainability of the best constant $S_{\mu ,\alpha }$ allows us to establish the existence of nontrivial weak solution to the following sub-elliptic equation and system involving the Hardy-type potentials and multiple critical nonlinearities

$$\begin{aligned} -\varDelta _{{\mathbb {G}}}u- \mu \frac{\psi (z)^{2}u}{d(z)^{2}}= {\frac{\psi (z)^{\alpha }|u|^{2^*(\alpha )-2}u}{d(z)^{\alpha }}}+ {\frac{\psi (z)^{\beta }|u|^{2^*(\beta )-2}u}{d(z)^{\beta }}}\quad \text {in}\,\,{\mathbb {G}}, \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{aligned}&-\varDelta _{{\mathbb {G}}}u- \mu \frac{\psi (z)^{2}u}{d(z)^{2}}= {\frac{\psi (z)^{\alpha }|u|^{2^*(\alpha )-2}u}{d(z)^{\alpha }}}+ \frac{\lambda \, \eta }{\eta + \theta } {\frac{\psi (z)^{\alpha }|u|^{\eta -2}u|v|^{\theta }}{d(z)^{\alpha }}} \quad&\text {in}\,\,\,{\mathbb {G}},\\&-\varDelta _{{\mathbb {G}}}v- \mu \frac{\psi (z)^{2}v}{d(z)^{2}}= {\frac{\psi (z)^{\alpha }|v|^{2^*(\alpha )-2}v}{d(z)^{\alpha }}}+ \frac{\lambda \, \theta }{\eta + \theta } {\frac{\psi (z)^{\alpha }|u|^{\eta }|v|^{\theta -2}v}{d(z)^{\alpha }}} \quad&\text {in}\,\,\,{\mathbb {G}}, \end{aligned} \right. \end{aligned}$$

where $\varDelta _{{\mathbb {G}}}$ is the sub-Laplacian operator on Carnot group ${\mathbb {G}}$, $0\le \mu <\mu _{G}$, $0\le \alpha ,\beta <2$, $\eta$, $\theta >1$ with $\eta +\theta =2^*(\alpha )$ and $\lambda >0$ is a parameter. The results are obtained by variational methods and local compactness of Palais-Smale sequences.