The existence of least energy sign-changing solution for Kirchhoff-type problem with potential vanishing at infinity
Summary: In this paper, we study the Kirchhoff-type equation: \[ -\Big(a+b \int_{\mathbb{R}^3} |\nabla u|^2 d x\Big)\varDelta u+V(x)u=Q(x) f(u), \text{ in } \mathbb{R}^3, \] where \(a, b>0\), \(f\in C^1(\mathbb{R}^3, \mathbb{R})\), and \(V, Q \in C^1(\mathbb{R}^3, \mathbb{R}^+)\). \(V(x)\) 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.
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