COMBINED EFFECTS OF SINGULAR AND HARDY NONLINEARITIES IN FRACTIONAL KIRCHHOFF CHOQUARD EQUATION

The aim of this paper is to investigate the existence and the multiplicity of solutions to the singular Kirchhoff non-local problem with Hardy and Choquard nonlinearities. The problem is defined as follows:

$ \begin{equation*} \left\{ \begin{array}{ll} &\quad M\Big( \int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dx dy\Big) -\Delta^s_p u-\alpha \frac{|u|^{p-2}u}{|x|^{sp}}\\ &=\lambda f(x) u^{-\gamma}+ g(x) \Big( \int_{\Omega}\frac{u^{p_{\mu, s}^*}(y)}{|x-y|^\mu}dy\Big)u^{p_{\mu, s}^*-1} \; \text{in}\; \Omega, \\ &u>0, \;\;\;\;\quad \text{in }\Omega, \\ &u=0, \;\;\;\;\quad \text{in }\mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{equation*} $

where, $ \Omega\subset \mathbb{R}^N $ is a bounded domain, $ s\in (0, 1) $, $ N>sp $, $ \gamma\in (0, 1), $ $ \alpha, $ $ \lambda $ are two positive real parameters $ 0<\mu<N $, $ p^*_{s} = \frac{Np}{N - sp} $ is the fractional critical Sobolev exponent, while $ p_{\mu, s} = \frac{(Np - \mu)}{(N-sp)} $ and $ p_{\mu, s}^*=\Big(\frac{p}{2}\Big).\Big(\frac{2N-\mu}{N-sp}\Big) $ denote the critical and upper critical exponent in the sense of Hardy Littlewood Sobolev inequality respectively, $ M(t) = a + b{t}^{\theta - 1}, \;\text{with } a>0, b>0 $ and $ \theta\in \Big(1, \min\{ 2p_{\mu, s}^*/p, p_{\mu, s}^*\}\Big). $ Furthermore, $ f $ is a non-negative weight and $ g $ is a sign-changing weight. The novelty in this work lies in the combination of a fractional framework and a singular term with the Hardy and Choquard nonlinearities. To establish the existence of at least two positive solutions for the problem, the Nehari manifold approach is employed.