Uniqueness of radial bound states for an equation with an odd cubic-like nonlinearity

Abstract

We investigate radial bound states with prescribed nodes for the odd cubic equation \[ Δu+u(\abs{u}-b)(c-\abs{u})=0 \quad\text{in }\R^n,\qquad n\ge2,\quad 0<b<c. \] Writing $a=b/c$, we prove that a radial bound state with exactly $k\ge1$ simple zeros and positive central value exists if and only if $0<a<1/2$. For each $k$, this solution is unique for all $0<a<1/2$ in dimension two, for $a\ge(7-3\sqrt5)/2$ in dimension three, and for $a\ge(43-5\sqrt{33})/32$ in dimension four. For one node, we identify exact ranges obtained from a tangent comparison and a quotient inequality in every dimension $n\ge3$; the threshold is dimension-independent for $3\le n\le23$ and is characterized implicitly for $n\ge24$. These results address the natural odd counterpart of the cubic nonlinearity highlighted in Tang's prescribed-node uniqueness conjecture [M. Tang, \emph{Invent. Math.}, 2026]. The main challenges are the second positive zero of the cubic source, the lack of a global comparison inequality, and supercritical cubic growth in higher dimensions. We overcome them by an amplitude confinement argument, controlled extensions preserving sharp comparison estimates, and a subcritical truncation for existence. A complementary small-$a$ uniqueness range in dimensions three to five follows from a recent massive double-power result.