关键词: Fractional Kirchhoff equation, singular problems, variational and topological methods

Abstract: This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional $p$-Laplacian operator. More precisely, we study the following nonlocal problem: \begin{align*} \begin{cases} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|x|^{\alpha_1p}|y|^{\alpha_2p}|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\right) \mathcal{L}^{s}_pu= |x|^{\beta} f(u)\,\, \ &{\rm in}\ \Omega,\\ u=0\ \ \ \ &{\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*} where $\mathcal{L}^{s}_p$ is the generalized fractional $p$-Laplacian operator, $N\geq1$, $s\in(0,1)$, $\alpha_1,\alpha_2,\beta\in\mathbb{R}$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, and $M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0$, $f:\Omega\rightarrow\mathbb{R}$ are continuous functions. Firstly, we introduce a variational framework for the above problem. Then, the existence of least energy solutions is obtained by using variational methods, provided that the nonlinear term $f$ has $(\theta p-1)$-sublinear growth at infinity. Moreover, the existence of infinitely many solutions is obtained by using Krasnoselskii's genus theory. Finally, we obtain the existence and multiplicity of solutions if $f$ has $(\theta p-1)$-superlinear growth at infinity. The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.

Key words: Fractional Kirchhoff equation, singular problems, variational and topological methods