Existence and multiplicity results for fractional $p$-Kirchhoff equation with sign changing nonlinearities

Abstract

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-Δ)^{s}_p u &=λf(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} Ω,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus Ω, \end{array} \end{equation*} where $(-\Delta {)}_p^s$ is the fractional $p$-Laplace operator, $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary, $f\in L^{\frac{r}{r-q}}(\Omega )$ and $g\in L^{\infty }(\Omega )$ are sign changing, $M$ is continuous function, $ps<n<2ps$ and $1<q<p<r\le p_s^{\ast }=\frac{np}{n-ps}$.

Advances in pure and applied Mathematics 2016

References in corpus (1)

• A note on repelling periodic points for meromorphic functions with bounded set of singular values