The Nehari manifold method for Fractional Kirchhoff problem involving singular and exponential nonlinearity

TL;DR

This paper proves the existence of at least two solutions for a fractional Kirchhoff problem with singular and exponential nonlinearities using Nehari manifold techniques, expanding understanding of such complex nonlinear PDEs.

Contribution

It introduces a novel application of Nehari manifold methods to establish multiple solutions for a fractional Kirchhoff problem with singular and exponential nonlinearities.

Findings

Existence of at least two weak solutions established.

Application of Nehari manifold techniques to fractional Kirchhoff problems.

Results extend the understanding of nonlinear PDEs with singular and exponential terms.

Abstract

In this paper we establish the existence of at least two weak solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearity \begin{equation*} \left\{\begin{split} M\left(\|u\|^{\frac{n}{s}}\right)(-\Delta)^s_{n/s}u & = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\;\text{in}\;\Om, u&>0,\;\text{in}\; \Om, u &= 0,\;\text{in}\; \mb R^n \setminus{\Om}, \end{split} \right. \end{equation*} where $\Om$ is smooth bounded domain in $\mb R^n$, {$n\geq 1$}, $s\in (0,1)$, $\mu>0$ is a real parameter, $\beta <\frac{n}{n-s}$ and $q\in (0,1)$. We have considered the degenerate Kirchhoff case here and used the Nehari manifold techniques to obtain the results.