In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: $$\begin{aligned} \textstyle\begin{cases} (-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{\mu }{u^{\gamma }}+f & \text{in } \Omega, \\ u>0 & \text{in } \Omega, \\ u=0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ). \end{cases}\displaystyle \end{aligned}$$ Here $0 < s<1$ , $\lambda >0$ , $\gamma >0$ , and $\Omega \subset \mathbb{R}^{N}$ ( $N > 2s$ ) is a bounded smooth domain such that $0 \in \Omega $ . Moreover, $0 \leq \mu,f \in L^{1}(\Omega )$ . For $0< \lambda \leq \Lambda _{N,s}$ , $\Lambda _{N,s}$ being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of growth of solutions. Moreover, we mention a nonexistence and a complete blowup result for the case $\lambda > \Lambda _{N,s}$ . Besides, we consider the parabolic equivalence of the above problem in the case $\mu \equiv 1$ and some suitable $f(x,t)$ , that is, $$\begin{aligned} \textstyle\begin{cases} u_{t}+(-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{1}{u^{\gamma }}+f(x,t) & \text{in } \Omega \times (0,T), \\ u>0 & \text{in } \Omega \times (0,T), \\ u =0 & \text{in } (\mathbb{R}^{N} \setminus \Omega ) \times (0,T), \\ u(x,0)=u_{0} & \text{in } \mathbb{R}^{N}, \end{cases}\displaystyle \end{aligned}$$ where $u_{0} \in X_{0}^{s}(\Omega )$ satisfies an appropriate cone condition. In the case $0<\gamma \leq 1$ or $\gamma >1$ with $2s(\gamma -1)<(\gamma +1)$ , we show the existence of a unique solution for any $0< \lambda < \Lambda _{N,s}$ and prove a stabilization result for certain range of λ.

中文翻译：

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受 Hardy 势及其抛物线等价影响的非局部 Lazer-McKenna 型问题

在本文中，我们研究了哈代势对以下涉及奇异非线性的分数问题的解是否存在的影响： $$\begin{aligned} \textstyle\begin{cases} (-\Delta )^{s } u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{\mu }{u^{\gamma }}+f & \text{in } \Omega, \\ u >0 & \text{in } \Omega, \\ u=0 & \text{in } (\mathbb{R}^{N} \setminus \Omega )。\end{cases}\displaystyle \end{aligned}$$ 这里 $0 < s<1$ ， $\lambda >0$ ， $\gamma >0$ 和 $\Omega \subset \mathbb{R}^{N }$ ( $N > 2s$ ) 是一个有界平滑域，使得 $0 \in \Omega $ 。此外， $0 \leq \mu,f \in L^{1}(\Omega )$ 。对于 $0< \lambda \leq \Lambda _{N,s}$ ， $\Lambda _{N,s}$ 是分数 Hardy 不等式中的最佳常数，我们找到了关于数据 μ 和 f 问题的正弱解存在的充分必要条件。此外，对于 f 的规则数据，在适当的假设下，我们获得一些存在性和唯一性结果并计算解的增长率。此外，我们提到 $\lambda > \Lambda _{N,s}$ 的情况不存在和完全爆炸结果。此外，我们在 $\mu \equiv 1$ 和一些合适的 $f(x,t)$ 的情况下考虑上述问题的抛物线等价，即 $$\begin{aligned} \textstyle\begin{cases} u_{t}+(-\Delta )^{s} u = \lambda \frac{u}{ \vert x \vert ^{2s}} + \frac{1}{u^{\gamma }}+f (x,t) & \text{in } \Omega \times (0,T), \\ u>0 & \text{in } \Omega \times (0,T), \\ u =0 & \text {in } (\mathbb{R}^{N} \setminus \Omega ) \times (0,T), \\ u(x,0)=u_{0} & \text{in } \mathbb{R}^{N}, \end{cases}\displaystyle \end{aligned}$$ where $u_{0} \in X_{0}^{s}(\Omega )$满足适当的圆锥条件。在 $0<\gamma \leq 1$ 或 $\gamma >1$ 和 $2s(\gamma -1)<(\gamma +1)$ 的情况下，我们证明了任何 $0< \lambda 的唯一解的存在< \Lambda _{N,s}$ 并证明一定范围的 λ 的稳定结果。