In this paper, we study both the existence and uniqueness of nonnegative solutions for the nonlocal
$p$
-Laplace equation with singular term
$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}-B\Bigl(\frac{1}{p}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{p}\text{d}x\Bigr)\unicode[STIX]{x1D6E5}_{p}u=\frac{h(x)}{u^{\unicode[STIX]{x1D6FE}}}+k(x)u^{q},\quad & x\in \unicode[STIX]{x1D6FA},\\ u>0,\quad & x\in \unicode[STIX]{x1D6FA},\\ u=0,\quad & x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}(N\geqslant 1)$
is a bounded domain with smooth boundary
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$
,
$h\in L^{1}(\unicode[STIX]{x1D6FA})$
,
$h>0$
almost everywhere in
$\unicode[STIX]{x1D6FA}$
,
$k\in L^{\infty }(\unicode[STIX]{x1D6FA})$
is a non-negative function,
$B:[0,+\infty )\rightarrow [m,+\infty )$
is continuous for some positive constant
$m$
,
$p>1$
,
$0\leqslant q\leqslant p-1$
, and
$\unicode[STIX]{x1D6FE}>1$
. A “compatibility condition” on the couple
$(h(x),\unicode[STIX]{x1D6FE})$
will be given for the problem to admit at least one solution. To be a little more precise, it is shown that the problem admits at least one solution if and only if there exists a
$u_{0}\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$
such that
$\int _{\unicode[STIX]{x1D6FA}}h(x)u_{0}^{1-\unicode[STIX]{x1D6FE}}\text{d}x<\infty$
. When
$k(x)\equiv 0$
, the weak solution is unique.