Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions

Abstract

This paper is dedicated to studying the following fractional Kirchhoff-type equation $$\begin{aligned} \left( a+b\int _{\mathbb {R}^N}|(-\triangle )^{\alpha /2}{u}|^2\mathrm {d}x\right) (-\triangle )^{\alpha }u+V(|x|)u =f(|x|, u), \ \ \ \ x\in {\mathbb {R}}^{N}, \end{aligned}$$where \(a, b>0\), either \(N=2\) and \(\alpha \in (1/2,1)\) or \(N=3\) and \(\alpha \in (3/4,1)\) holds, \(V\in \mathcal {C}(\mathbb {R}^{N}, [0,\infty ))\) and \(f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})\). By combining the constraint variational method with some new inequalities, we prove that the above problem possesses a radial sign-changing solution \(u_b\) for \(b\ge 0\) without the usual Nehari-type monotonicity condition on f, and its energy is strictly larger than twice that of the ground state radial solutions of Nehari-type. Moreover, we establish the convergence property of \(u_b\) as \(b\searrow 0\). In particular, our results unify both asymptotically cubic and super-cubic cases, which improve and complement the existing ones in the literature.