Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities

IF 1.3 2区 数学 Q1 MATHEMATICS Mathematische Annalen Pub Date : 2024-09-12 DOI:10.1007/s00208-024-02982-x
Sitong Chen, Xianhua Tang
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引用次数: 0

Abstract

This paper focuses on the existence of normalized solutions for the following Kirchhoff equation:

$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2\textrm{d}x\right) \Delta u+\lambda u=u^5+\mu |u|^{q-2}u, & x\in {\mathbb {R}}^3, \\ \int _{{\mathbb {R}}^3}u^2\textrm{d}x=c, \\ \end{array} \right. \end{aligned}$$

where \(a,b,c>0\), \(\mu \in {\mathbb {R}}\) and \(2<q<6\), \(\lambda \in {\mathbb {R}}\) will arise as a Lagrange multiplier that is not a priori given. By using new analytical techniques, the paper establishes several existence results for the case \(\mu >0\):

  1. (1)

    The existence of two solutions, one being a local minimizer and the other of mountain-pass type, under explicit conditions on c when \(2<q<\frac{10}{3}\).

  2. (2)

    The existence of a mountain-pass type solution under explicit conditions on c when \(\frac{10}{3}\le q<\frac{14}{3}\).

  3. (3)

    The existence of a ground state solution for all \(c>0\) when \(\frac{14}{3}\le q<6\).

Furthermore, the paper presents the first non-existence result for the case \(\mu \le 0\) and \(2<q<6\). In particular, refined estimates of energy levels are proposed, suggesting a new threshold of compactness in the \(L^2\)-constraint. This study addresses an open problem for \(2<q<\frac{10}{3}\) and fills a gap in the case \(\frac{10}{3}\le q<\frac{14}{3}\). We believe that our approach can be applied to a broader range of nonlinear terms with Sobolev critical growth, and the underlying ideas have potential for future development and applicability.

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具有索波列夫临界指数和混合非线性的基尔霍夫方程的归一化解
本文重点研究以下基尔霍夫方程的归一化解的存在性: $$\begin{aligned}\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2\textrm{d}x\right) \Delta u+\lambda u=u^5+\mu |u|^{q-2}u, &;x\in {\mathbb {R}}^3, \int _{\mathbb {R}}^3}u^2\textrm{d}x=c, \\end{array}.\对\end{aligned}$where (a,b,c>0\), (\mu \in {\mathbb {R}}\) and (2<q<6\), (\lambda \in {\mathbb {R}}\) will arise as a Lagrange multiplier that is not a priori given.通过使用新的分析技术,本文为 \(\mu >0\) 的情况建立了几个存在性结果:(1)当 \(2<q<\frac{10}{3}\)时,在 c 的显式条件下存在两个解,一个是局部最小解,另一个是山路类型的解。(2)当\(\frac{10}{3}le q<\frac{14}{3}\) 时,在c的显式条件下存在一个山越式解。 (3)当\(\frac{14}{3}le q<6\) 时,对于所有\(c>0\)存在一个基态解。此外,本文首次提出了\(\mu \le 0\) and\(2<q<6\) 情况下的不存在结果。特别是提出了对能级的精细估计,暗示了在\(L^2\)约束下紧凑性的新阈值。这项研究解决了 \(2<q<\frac{10}{3}\) 的一个未决问题,并填补了 \(\frac{10}{3}le q<\frac{14}{3}\) 情况下的一个空白。我们相信,我们的方法可以应用于范围更广的具有索博列夫临界增长的非线性项,其基本思想具有未来发展和应用的潜力。
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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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