LAUSR

AbstractIn this paper, we studied the following fractional Kirchhoff-type equation: (a+b∫RN|(−△)α2u|2dx)(−△)αu+V(x)u=f(x,u),x∈RN,$$\biggl(a+b \int_{\mathbb{R}^{N}} \bigl\vert (-\triangle)^{\frac{\alpha }{2}}u \bigr\vert ^{2}\,\mathrm{d}x \biggr) (-\triangle)^{\alpha }u+V(x)u=f(x,u), \quad x\in{\mathbb{R}}^{N}, $$ where a, b are positive constants, α∈(0,1)$\alpha\in(0,1)$, N∈(2α,4α)$N\in (2\alpha,4\alpha)$, (−△)α$(-\triangle)^{\alpha}$ is the fractional Laplacian operator, V(x)$V(x)$ and f(x,u)$f(x,u)$ are periodic or asymptotically periodic in x. Under some weaker conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case and asymptotically periodic case, respectively. In particular, our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for α=1$\alpha=1$.