Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in R N $\mathbb{R}^{N}$

Abstract

Abstract In this paper, we studied the following fractional Kirchhoff-type equation: ( a + b ∫ R N | ( − △ ) α 2 u | 2 d x ) ( − △ ) α u + V ( x ) u = f ( x , u ) , x ∈ R N , $$\biggl(a+b \int_{\mathbb{R}^{N}} \bigl\vert (-\triangle)^{\frac{\alpha }{2}}u \bigr\vert ^{2}\,\mathrm{d}x \biggr) (-\triangle)^{\alpha }u+V(x)u=f(x,u), \quad x\in{\mathbb{R}}^{N}, $$ where a, b are positive constants, α ∈ ( 0 , 1 ) $\alpha\in(0,1)$ , N ∈ ( 2 α , 4 α ) $N\in (2\alpha,4\alpha)$ , ( − △ ) 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 $\alpha=1$ .

Keywords

• fractional Kirchhoff-type problem
• Nehari-type ground state solution
• asymptotically periodic
• asymptotically cubic and super-cubic growth