Fractional Kirchhoff problem with critical indefinite nonlinearity

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_Ω|(-Δ)^{\fracα{2}}u|^2dx\right)(-Δ)^α u= λf(x)|u|^{q-2}u+|u|^{2^*_α-2}u\;\; \text{in}\; Ω,\;u=0\;\textrm{in}\;\mathbb R^n\setminus Ω, \end{equation*} where $\Omega \subset {\mathbb{R}}^n$ is a smooth bounded domain, $M(t)=a+\epsilon t,a,\epsilon >0,0<\alpha <1,2\alpha <n<4\alpha$ and $1<q<2$. Here $2_{\alpha }^{\ast }=2n/(n-2\alpha )$ 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.

References in corpus (2)

• Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity
• Multiplicity results of fractional $p$-Laplace equations with sign-changing and singular nonlinearity

Cited by in corpus (1)

• Nehari Manifold for fractional Kirchhoff system with critical nonlinearity