Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 10, pp. 1-19.

Title: Multiplicity results of nonlocal singular PDEs with critical Sobolev-Hardy exponent
       
Authors: Adel Daoues   (Ecole Superieure des Sciences et de Tech., Sousse, Tunisia)
         Amani Hammami (Ecole Superieure des Sciences et de Tech., Sousse, Tunisia)
         Kamel Saoudi  (Imam Abdulrahman Bin Faisal Univ., Saudi Arabia)
         
Abstract:
 In this article we study a nonlocal equation involving singular and critical
 Hardy-Sobolev non-linearities,
$$\displaylines{
 (-\Delta_p)^su-\mu \frac{|u|^{p-2}u}{|x|^{sp}}
 =\lambda u^{-\alpha}+\frac{|u|^{p_s^*(t)-2}u}{|x|^t}, \quad\text{in }\Omega,\cr
 u>0,\quad\text{in }\Omega,\cr
 u=0,\quad\text{in }\mathbb{R}^N\setminus\Omega,
}$$
where $\Omega \subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary
and $(-\Delta_p)^s$ is the fractional p-Laplacian operator.
We combine some variational techniques with a perturbation method to show the existence
of multiple solutions.


Submitted November 22, 2022. Published January 26, 2023.
Math Subject Classifications: 35R11, 35J75, 35J60, 46E35.
Key Words: Nonlocal elliptic problem; singular non-linearity; variational method;
    Sobolev and Hardy non-linearities; perturbation method; multiple positive solutions.