Structure of Positive Radial Solutions to Scalar Field Equations with Harmonic Potential

Abstract We study a nonlinear elliptic equation with a harmonic potential −Δ u+(λ+|x| 2 ) u−|u| p−1 u=0, x∈ R n , (E) which is related to standing waves for a nonlinear Schrodinger equation. The existence of positive solution for (E) in some energy space can be shown by the standard variational method under n ⩾1, λ >− n and 1 p n +2)/( n −2) + . However, the uniqueness is a delicate problem. In this paper, noting that all positive solutions of (E) in the energy space are radially symmetric about the origin, we study the structure of positive radial solutions to (E) and show the uniqueness of positive solution with finite energy when n ⩾3.

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