Infinitely many nodal solutions with a prescribed number of nodes for the Kirchhoff type equations

Abstract

In this paper, we investigate the existence of multiple radial sign-changing solutions with the nodal characterization for a class of Kirchhoff type problems{−(a+b|∇u|L22)Δu+V(|x|)u=K(|x|)f(u)in RN,u∈H1(RN), where N=1,2,3,a,b>0, V,K are radial and bounded away from below by positive numbers. Under some weak assumptions on f∈C0(R;R), by taking advantage of the Gersgorin disc's theorem and Miranda theorem, we develop some new analytic techniques and prove that this problem admits infinitely many nodal solutions {Ukb} having a prescribed number of nodes k, whose energy is strictly increasing in k. Moreover, the asymptotic behaviors of Ukb as b→0+ are established. These results improve and generalize the previous results in the literature.