Fractional Kirchhoff Hardy problems with weighted Choquard and singular nonlinearity

Abstract

In this article, we study the existence and multiplicity of solutions to the fractional Kirchhoff Hardy problem involving weighted Choquard and singular nonlinearity $$\displaylines{ M(\|u\|^2)(-\Delta)^su - \gamma\frac{u}{|x|^{2s}} = \lambda \frac{l(x)}{ u^q} + \frac{1}{|x|^{\alpha}} \Big({\int_{\Omega}\frac{r(y)|u(y)|^p}{|y|^{\alpha}|x-y|^\mu}\,dy}\Big)r(x)|u|^{p-2}u \quad \hbox{in } \Omega, \cr u>0 \hbox{ in } \Omega, \quad u = 0 \hbox{ in } \mathbb{R}^N\backslash\Omega, }$$ 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.