Uniqueness of prescribed-node radial bound states for a zero-mass double-power equation

Abstract

We study radial bound states of \[ Δu-\abs{u}^{q-1}u+\abs{u}^{p-1}u=0 \quad\text{in }\R^n, \qquad n\ge3,\qquad 1<q<p<\frac{n+2}{n-2}. \] For every integer $k\ge 1$, we prove that there is exactly one radial bound state with $k$ simple zeros and $u(0)>0$. Its absolute maximum is attained at the origin. This confirms the zero-mass, Sobolev-subcritical case of Tang's double-power uniqueness conjecture. The proof adapts Tang's phase-transition analysis and combines it with a finite-level separation argument inspired by Cortázar--García-Huidobro--Yarur. Three main difficulties arise in the zero-mass double-power setting: the sign changes of key comparison quantities in the low-amplitude region, the possible failure of pointwise positivity of an auxiliary quantity when $n=3$ and $p+q<3$, and the absence of exponential decay at infinity. We overcome these difficulties by introducing bridge functionals to propagate the required sign information through low amplitudes, establishing a weighted integral positivity estimate in the exceptional three-dimensional range, and combining a radial-flux tail estimate with finite-level inverse-branch comparison. We also establish the strict ordering of the bound-state shooting heights and show that the shooting variation along the \(k\)-node state has exactly \(k+1\) simple zeros.

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