Existence, Nonexistence, and Multiple Results for the Fractional p-Kirchhoff-type Equation in ${\mathbb{R}^N}$

Abstract

In this paper, we investigate the fractional p-Kirchhoff-type equation

$$\begin{array}{ll} M\left(\int\int\limits_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\right)(-\Delta)_p^su+V(x)|u|^{p-2}u+b(x)|u|^{q-2}u\\ \quad = \lambda a(x)|u|^{m-2}u, \; x\in\mathbb{R}^N,\end{array}$$

where ${\lambda}$ is a real parameter, ${(-\Delta)_p^s }$ is the fractional p-Laplacian operator with ${0 < s < 1 < p}$ and ${ps < N}$, ${V(x), a(x), b(x): \mathbb{R}^N\to (0,\infty)}$ are three positive weights, and M is a continuous and positive function. The case ${1 < q < m < p_s^*}$ is considered. Using variational methods, we prove the existence, nonexistence, and multiplicity of solutions for the above equation depending on ${\lambda, m,q}$ and according to the weight functions a(x) and b(x). Our results extend the previous works of Pucci et al. [23] and of Xiang et al. [29].