Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation

Abstract

In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential$$\left(1+b{\left[u\right]}_{\alpha }^2\right)\left({\left(-{\mathrm{\Delta }}_x\right)}^{\alpha }u-{\mathrm{\Delta }}_{y}u\right)+V\left(x,y\right)u=f\left(x,y,u\right),\left(x,y\right)\in {ℝ}^N={ℝ}^n\times {ℝ}^m,$$where ${\left[u\right]}_{\alpha }={\left(\int {}_{{ℝ}^N}\left({\left|{\left(-{\mathrm{\Delta }}_x\right)}^{\frac{\alpha }{2}}u\right|}^2+{\left|{\nabla }_{y}u\right|}^2\right)dxdy\right)}^{\frac{1}{2}}$. Based on variational approach and a variant of the quantitative strain lemma, for each b > 0, we show the existence of a least energy nodal solution u_b. In addition, a convergence property of u_b as b ↘ 0 is established.