Abstract:
In this paper, we study the subelliptic equation with weight
$$
-\Delta_\lambda
u=
\bigg(\frac{1}{|x|_\lambda^{Q-\alpha}}*|x|^\beta_\lambda|u|^p\bigg)|x|^\beta_\lambda|u|^{p
-2}u , \qquad x\in\mathbb R^N,
$$
where
$\alpha>0,$$ \beta\geq 0$,
$p>2$,
and
$\Delta_\lambda$
is a subelliptic operator
of the form
$$
\Delta_\lambda=\sum_{i=1}^N \partial_{x_i}(\lambda_i^2\partial_{x_i}).
$$
Here $Q$ is the homogeneous dimension on $\mathbb R^N$
associated with the operator $\Delta_\lambda$, and
the $\lambda_i$, $i=1,\dots,N$, satisfy some general
hypotheses. Our purpose is to establish the nonexistence
of nontrivial stable solutions for
$$
\max(Q-4-2\beta,0)<\alpha<Q.
$$
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