双幂非线性定态薛定谔方程正解的唯一性：固定频率情形与固定质量情形

中文摘要 AI 辅助

对于问题 \begin{align*} -\Delta u + \lambda u = u^p + u^q, \quad u \in H^1(\mathbb{R}^N), ~~ N \ge 2, \end{align*} 其中 $\lambda > 0$ 且 $1 < q < p < 2^* - 1$，正径向解的存在性早已为人所知。对于纯幂非线性，众所周知该解是唯一的。相比之下，双幂情形的唯一性问题要复杂得多，且依赖于维数和指数。已知在三维情形下，双幂非线性方程的唯一性结果一般不成立，并且自 H. Berestycki 和 P.-L. Lions 的工作 [2]（Arch. Ration. Mech. Anal. 1983）以来，在 $N \ge 4$ 的整个次临界范围内证明正径向解的唯一性一直是一个长期未解决的问题。在本文中，我们给出了当 $N \ge 5$ 时对所有 $\lambda > 0$ 和所有 $1 < q < p < 2^* - 1$ 的唯一性结果。在维数 $N = 2,3,4$ 时，在 $p$ 和 $q$ 的附加条件下，我们获得了对所有 $\lambda > 0$ 的唯一性结果。我们的结果与 J. Dávila、M. del Pino 和 I. Guerra [6]（Proc. London Math. Soc. 2013）的结果形成鲜明对比，后者在 $N=3$ 时获得了一些非唯一性结果。同时，我们对其数值模拟的启示给出了肯定的回答和严格的证明，澄清了 [6] 留下的开放问题。最后，我们完全解决了上述方程在质量次临界和质量临界区域中对于所有 $N \ge 2$ 的 $L^2$ 质量约束下的唯一性问题。

英文摘要

The existence of a positive radial solution to the problem \begin{align*} -Δu + λu = u^p + u^q, \quad u \in H^1(\mathbb{R}^N), ~~ N \ge 2, \end{align*} where $λ> 0$ and $1 < q < p < 2^* - 1$, has been known for a long time. For the pure-power nonlinearity, it is well known that this solution is unique. In contrast, the uniqueness problem for the double-power case is substantially more delicate and depends on the dimension and the exponents. It has been known that the uniqueness result is in general not true for the equation with double-power nonlinearities in dimension three and it is a long-standing open question to prove the uniqueness of positive radial solutions throughout the full subcritical range for $N \ge 4$ since the work of H. Berestycki, P.-L. Lions [2] (Arch. Ration. Mech. Anal. 1983). In this paper we provide the uniqueness results for all $λ> 0$ and all $1 < q < p < 2^* - 1$ when $N \ge 5$. In dimensions $N = 2,3,4$, with additional conditions on $p$ and $q$, we obtain the uniqueness results for all $λ> 0$. Our results are in sharp contrast to those of J. Dávila, M. del Pino and I. Guerra [6] (Proc. London Math. Soc. 2013), where some non-uniqueness results were obtained for $N=3$. At the same time, we give a positive answer and rigorous proof to the implication of their numerical simulation, clarify the open issues left by [6]. Finally, we completely resolve the uniqueness problem of the above equation with $L^2$-mass constraints in the mass-subcritical and mass-critical regimes for all $N \ge 2$.

发表机构

• Tsinghua University（清华大学）

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