Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

Abstract

<p style='text-indent:20px;'>In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u>0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu>0 $\end{document}</tex-math></inline-formula> is a real parameter, <inline-formula><tex-math id="M6">\begin{document}$ \beta <{n/(n-s)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q\in (0,1) $\end{document}</tex-math></inline-formula>.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.</p>