Positive and negative solutions for the nonlinear fractional Kirchhoff equation in ${\mathbb {R}}^{N}$

Abstract

This paper is concerned with the following nonlinear fractional Kirchhoff equation

$$\begin{aligned} (a+\lambda \int _{{\mathbb {R}}^{N}}|(-\varDelta )^{\frac{s}{2}}u|^{2}dx)(-\varDelta )^{s}u+V(x)u=f(x,u)+ w(x)|u|^{q-2}u,\ \ \ x\in {\mathbb {R}}^{N}, \end{aligned}$$

where $N>2s,\ a>0, \lambda \ge 0$ is a parameter, $(-\varDelta )^{s}$ denotes the fractional Laplacian operator of order $s\in (0, 1),\ 2_{s}^{\star }=\frac{2N}{N-2s},\ V$ and f are continuous, and $w(x)\in L^{\frac{2_{s}^{\star }}{2_{s}^{\star }-q}}({\mathbb {R}}^{N}, {\mathbb {R}}^{+})$ with $1<q<2$. By using variational methods, Pohozaev identity for the fractional Laplacian and iterative technique, two positive solutions and two negative solutions are obtained when the nonlinearity f does not satisfy the usual Ambrosetti–Rabinowitz condition.