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 Ð ℝ3 j∇uj 2 dx)Δu + V(x)u = Q(x)f(uÞ, in ℝ3, where a, b > 0, f ∈ C1ð ℝ3, ℝÞ, and V, Q ∈ C1(ℝ3, ℝ+Þ. VðxÞ and Q(x) 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.