The Existence of Least Energy Sign-Changing Solution for Kirchhoff-Type Problem with Potential Vanishing at Infinity

In this paper, we study the Kirchhoff-type equation: −a+b∫ℝ3 ∇u2dxΔu+Vxu=Qxfu,in ℝ3, where a, b>0, f∈C1ℝ3,ℝ, and V,Q∈C1ℝ3,ℝ+. Vx and Qx are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution u to the above equation. Moreover, we obtain that the sign-changing solution u has exactly two nodal domains. Our results can be seen as an improvement of the previous literature