| Volume 15, Number 1, 2025, Pages - DOI：10.11948/JAAC-2023-0429 |
| Combined effects of singular and Hardy nonlinearities in fractional Kirchhoff Choquard equation |
| Kamel Saoudi,Rana Alkhal,Mouna Kratou |
| Keywords:Kirchhoff problem, Choquard term, fractional Sobolev spaces, Hardy potential, singularities, Nehari manifolds. |
| Abstract: |
| The aim of this paper is to investigate the existence and the multiplicity of solutions to the
singular Kirchhoff nonlocal problem with Hardy and Choquard nonlinearities
\begin{equation*}
\left\{
\begin{array}{ll}
M\Big(\displaystyle \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(\displaystyle\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 $0<\mu0, b>0$ and $\theta\in \Big(1, \min\{ 2p_{\mu,s}^*/p, p_{\mu,s}^*\}\Big),$ $f$ is a non-negative weight and $g$ is a
sign-changing weight. The novelty in the current work is the combination of fractional framework and singular term with the Hardy and Choquard nonlinearities. In order to provide the existence of at least two positive solutions to the above problem, we use Nehari manifold approach. |
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