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.