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.

在本文中，我们研究了具有奇异项 $$ \ begin {eqnarray} \ left \ {\ begin {array} {@ {} ll @ {}}的非局部 $ p $ -Laplace方程的非负解的存在性和唯一性} -B \ Bigl（\ frac {1} {p} \ int _ {\ unicode [STIX] {x1D6FA}} | \\ unicode [STIX] {x1D6FB} u | ^ {p} \ text {d} x \ Bigr ）\ unicode [STIX] {x1D6E5} _ {p} u = \ frac {h（x）} {u ^ {\ unicode [STIX] {x1D6FE}}}} + k（x）u ^ {q}，\ quad ＆x \ in \ unicode [STIX] {x1D6FA}，\\ u> 0，\ quad＆x \ in \ unicode [STIX] {x1D6FA}，\\ u = 0，\ quad＆x \ in \ unicode [STIX ] {x2202} \ unicode [STIX] {x1D6FA}，\ end {array} \ right。\ end {eqnarray} $$ 其中 $ \ unicode [STIX] {x1D6FA} \ subset \ mathbb {R} ^ {N}（ N \ geqslant 1）$ 是具有光滑边界 $ \ unicode [STIX] {x2202} \ unicode [STIX] {x1D6FA} $ 的有界域， $ h \ in L ^ {1}（\ unicode [STIX] {x1D6FA}）$ ， $ h> 0 $ 在 $ \ unicode [STIX] {x1D6FA} $ ， $ k \ in L ^ {\ infty}中几乎到处（\ unicode [STIX] {x1D6FA}）$ 是非负函数， $ B：[0，+ \ infty）\ rightarrow [m，+ \ infty）$ 对于某些正常数 $ m $ ， $ p是连续的> 1 $ ， $ 0 \ leqslant q \ leqslant p-1 $ 和 $ \ unicode [STIX] {x1D6FE}> 1 $ 。 将给出问题对$（h（x），\ unicode [STIX] {x1D6FE}）$ 的“兼容条件”，以允许至少一个解决方案。更确切地说，表明只有当存在一个问题时，该问题才能接受至少一个解决方案。 $ u_ {0} \ in W_ {0} ^ {1，p}（\ unicode [STIX] {x1D6FA}）$$ 这样使得 $ \ int _ {\ unicode [STIX] {x1D6FA}} h（x）u_ { 0} ^ {1- \ unicode [STIX] {x1D6FE}} \ text {d} x <\ infty $ 。当 $ k（x）\ equiv 0 $时 ，弱解是唯一的。