Fractional Kirchhoff problem with critical indefinite nonlinearity

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a class of fractional Kirchhoff equations with critical nonlinearity, employing variational methods and Nehari manifold techniques.

Contribution

It introduces a novel approach combining sublinear and superlinear effects to analyze fractional Kirchhoff problems with indefinite nonlinearities.

Findings

Proved existence of positive solutions under certain conditions.

Established multiplicity results for the problem.

Applied Nehari manifold technique to fractional Kirchhoff equations.

Abstract

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*} M\left(\int_\Omega|(-\Delta)^{\frac{\alpha}{2}}u|^2dx\right)(-\Delta)^{\alpha} u= \lambda f(x)|u|^{q-2}u+|u|^{2^*_\alpha-2}u\;\; \text{in}\; \Omega,\;u=0\;\textrm{in}\;\mathbb R^n\setminus \Omega, \end{equation*} where $\Omega\subset \mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<\alpha<1, \; 2\alpha<n<4\alpha$ and $ \; 1<q<2$. Here $2^*_\alpha={2n}/{(n-2\alpha)}$ is the fractional critical Sobolev exponent, $\lambda$ is a positive parameter and the coefficient $f(x)$ is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results.