Infinitely many nodal solutions for a modified Kirchhoff type equation

Abstract

In this paper, we consider the modified Kirchhoff type equation, that is, the Kirchhoff type equation with a quasilinear term (1) where a, b>0, and is a radial potential function and bounded below by a positive number. The appearance of nonlocal term and quasilinear term makes the variational functional of (1) totally different from the classical Schrödinger equation. By introducing the Miranda theorem, via a construction and gluing method, for any given integer , we prove that Equation (1) admits a radial nodal solution having exactly k nodes. Moreover, the energy of is monotonically increasing in k and for any sequence , and up to a subsequence, converges strongly to some as , which is a nodal solution with exactly k nodes to the local quasilinear Schrödinger equation (2) These results improve and generalize the previous results in the literature from the local quasilinear Schrödinger equation to the nonlocal quasilinear Kirchhoff equation.