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

Abstract

Abstract In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential ( 1 + b [ u ] α 2 ) ( ( - Δ x ) α u - Δ y u ) + V ( x , y ) u = f ( x , y , u ) , ( x , y ) ∈ ℝ N = ℝ n × ℝ m , \left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m}, where [ u ] α = ( ∫ ℝ N ( | ( - Δ x ) α 2 u | 2 + | ∇ y u | 2 ) d x d y ) 1 2 {\left[ u \right]_\alpha } = {\left( {\int {_{{\mathbb{R}^N}}\left( {{{\left| {{{\left( { - {\Delta _x}} \right)}^{{\alpha \over 2}}}u} \right|}^2} + {{\left| {{\nabla _y}u} \right|}^2}} \right)dxdy} } \right)^{{1 \over 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 ub . In addition, a convergence property of ub as b ↘ 0 is established.