Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 25, pp. 1-29.

Title: Fractional Kirchhoff Hardy problems with weighted Choquard and singular nonlinearity
       
Authors: Sarika Goyal (Bennett Univ., Greater Noida, Uttar Pradesh, India)
         Tarun Sharma (Bennett Univ., Greater Noida, Uttar Pradesh, India)

Abstract: 
In this article, we study the existence and multiplicity of solutions to the
fractional Kirchhoff Hardy problem involving weighted Choquard and singular nonlinearity
$$
\displaylines{
M(\|u\|^2)(-\Delta)^su - \gamma\frac{u}{|x|^{2s}}
 = \lambda \frac{l(x)}{ u^q} + \frac{1}{|x|^{\alpha}}
\Big({\int_{\Omega}\frac{r(y)|u(y)|^p}{|y|^{\alpha}|x-y|^\mu}\,dy}\Big)r(x)|u|^{p-2}u \quad
\text{in } \Omega,
\cr
 u>0 \text{ in } \Omega, \quad u = 0  \text{ in } \mathbb{R}^N\setminus\Omega,
}$$
where $\Omega\subseteq \mathbb{R}^N$ is an open bounded domain with smooth boundary
containing 0 in its interior, $N>2s$ with $s\in(0,1)$, $0<q<1$, $0<\mu<N$, $\gamma$
and $\lambda$ are positive parameters, $\theta\in [1, p)$ with
$1 < p < 2^*_{\mu,s,\alpha}$, where $2^*_{\mu,s,\alpha}$ is the upper critical exponent
in the sense of weighted Hardy-Littlewood-Sobolev inequality.
Moreover $M$ models a Kirchhoff coefficient, l is a positive weight and r is a
sign-changing function. Under the suitable assumption on l and r, we established
the existence of two positive solutions to the above problem by Nehari-manifold and
fibering map analysis with respect to the parameters.The results obtained
here are new even for s=1.

Submitted December 30, 2021. Published March 25, 2022.
Math Subject Classifications: 35A15, 35J75, 36B38.
Key Words: Fractional Kirchhoff Hardy operator; singular nonlinearity;
\hfill\break\indent weighted Choquard type nonlinearity;
 Nehari-manifold; fibering map.