Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schrödinger equations

TL;DR

This work analyzes positive radial solutions of the double-power stationary Schrödinger equation in three dimensions, revealing a non-uniqueness phenomenon for small frequencies by proving the existence of exactly two positive solutions: the ground state and a second constrained-minimizer whose $L^{\infty}$ norm blows up as $\omega\to0$. The authors develop a comprehensive blow-up analysis, establishing that the finite-$L^{\infty}$ branch converges to the ground-state profile $U^{\dagger}$ under a suitable rescaling, while the large-$L^{\infty}$ branch concentrates to the Aubin–Talenti bubble $W$ through a detailed convergence in $\dot{H}^1$ and $L^q$. They derive sharp asymptotics relating the parameters $\alpha_{\omega}$ and $\beta_{\omega}$ across regimes $2<p<3$, $p=2$, and $1<p<2$, including universal limits and a singular-limit case for $1<p<2$, where the blow-up profile converges to a singular solution $U_{\theta_{0},\infty}$. The paper also proves uniqueness and non-degeneracy (and Morse-index information) for the large solutions, employing resolvent expansions (Jensen–Kato) and Lyapunov–Schmidt reductions to control the delicate limiting behavior. Collectively, these results provide a precise bifurcation picture and stability implications for standing-wave dynamics in the corresponding nonlinear Schrödinger flow.

Abstract

In this paper, we investigate positive radial solutions to double-power nonlinear stationary Schrodinger equations in three space dimensions. It is now known that the non-uniqueness of H^{1}-positive solutions can occur in three dimensions when the frequency is sufficiently small. Under suitable conditions, in addition to the ground state solution (whose L^{\infty} norm vanishes as the frequency tends to zero), there exists another positive solution that minimizes a different constrained variational problem, with an L^{\infty} norm diverging as the frequency tends to zero (see Theorem 1.4). We classify all positive solutions with small frequency into two categories: the ground state and the Aubin-Talenti type solution. As a consequence, we establish the multiplicity of positive solutions. Finally, we also examine the non-degeneracy and Morse index of each positive solution.

Theorems & Definitions (214)

• Remark 1.1
• Theorem 1.1: MR4445670MR4638619MR3166237
• Remark 1.2
• Theorem 1.2: Theorem 1.1 and Proposition 1.5 of Soave MR4096725
• Theorem 1.3: MR4476243MR4096725MR4433054
• Remark 1.3
• Theorem 1.4: Theorem 1.2 (3) of Wei and Wu MR4638619
• Remark 1.4
• Theorem 2.1
• Theorem 2.2