Infinitely many nodal solutions for a modified Kirchhoff type equation

paper, we consider the modified Kirchhoff type equation, that is, the Kirchhoff type equation with a quasilinear term {- ( a + b integral (R3) |del u|(2)dx) Delta u + V(|x|)u - 1/2 Delta(|u|(2))u = |u|(p-2)u in R-3, u -> 0 as |x|->infinity, where a, b > 0, p epsilon(4, 22*) and V(|x|) is a radial potential function and bounded below by a positive number. The appearance of nonlocal term b integral (R3) |del u|(2)dx Delta u and quasilinear term 1/2 Delta(|u|(2))u makes the variational functional of (1) totally different from the classical Schrodinger equation. By introducing the Miranda theorem, via a construction and gluing method, for any given integer k >= 1, we prove that Equation (1) admits a radial nodal solution U-b (k) having exactly k nodes. Moreover, the energy of U-b (k) is monotonically increasing in k and for any sequence {b(n)}, and up to a subsequence, U-bn (k) converges strongly to some U-0 (k) not equal 0 as b(n)-> 0(1), which is a nodal solution with exactly k nodes to the local quasilinear Schrodinger equation {-a Delta u + V(|x|)u - 1/2 Delta(|u|(2))u = |u|(p-2)u in R-3, u -> 0 as |x|->infinity. These results improve and generalize the previous results in the literature from the local quasilinear Schrodinger equation to the nonlocal quasilinear Kirchhoff equation.