Existence and nonexistence of solutions for a class of Kirchhoff type equation involving fractional p-Laplacian

Abstract

In this paper, we consider the following Kirchhoff type equation involving the fractional p-Laplacian: $$\begin{aligned} \begin{aligned} M \left( \iint _{\mathbb {R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \mathrm {d}x\mathrm {d}y \right) (-\Delta )^{s}_{p} u +\lambda V(x)|u|^{p-2}u =K(x)f(u), \;x\in \mathbb {R}^{N}, \end{aligned} \end{aligned}$$ where $$\lambda $$ is a real parameter, $$\left( -\Delta \right) ^{s}_{p}$$ is the fractional p-Laplacian operator, with $$0<s<1<p<\infty $$ and $$sp<N$$ . Functions M, V and K satisfy some suitable conditions. For f is superlinear at infinity, we establish the existence of multiple solutions and infinitely many solutions to above equation, which extend the main result in Pucci et al. (Calc Var Partial Differ Equations 54:2785–2806, 2015). For f is asymptotically linear at infinity, we first study the influence of function K on the existence and nonexistence of solutions for the above equation, which complement the main result in Jia and Luo (J Math Anal Appl 467:893–915, 2018).