In this paper, we study the uniqueness of nodal radial solutions to nonlinear elliptic equations in the unit ball in R 3 . Under suitable conditions, we prove that, for any given positive integer k, the problem we considered has at most one solution possessing exactly k−1 nodes. Together with the results presented by Nagasaki [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (2): 211–232, 1989] and Tanaka [Proc. Roy. Soc. Edinburgh Sect. A. 138 (6): 1331–1343, 2008], we can prove that more types of nonlinear elliptic equations have the uniqueness of nodal radial solutions.

Abstract: In this paper, the nonlinear Schrödinger-type equation −(∇ + iA) ^2 u + u + λ[I_α*(K|u|^2)]Ku=af(|u|)u/|u| in R ^3 is considered in the presence of magnetic field, where A ∈ C ^1 (R ^3 ,R^ 3 ), α ∈ (0,3), I_α denotes the Riesz potential, K ∈ L^ p (R ^3 ,(0,∞)) for some p ∈ (6/(1 + α),∞], a ∈ L^ q (R 3 ,[0,∞)) \ {0} for some q ∈ (3/2,∞], and f ∈ C(R,[0,∞)) is assumed to be asymptotically linear at infinity. Under suitable assumptions regarding A, K, a, and f, variational methods are used to establish the existence of ground-state solutions of the above equation for sufficiently small values of the parameter λ.