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 integral 3 del u 2dx Delta u+V x u=Q x f u ,in3, where a, b>0, f is an element of C1 3, , and V,Q is an element of C1 3,+. V x 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.