Normalized solutions to HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

IF 1.4 3区 数学 Q1 MATHEMATICS
Ziheng Zhang, Jianlun Liu, Hong-Rui Sun
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引用次数: 0

Abstract

This paper is concerned with the following HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

$$\begin{aligned} {\left\{ \begin{array}{ll} -{\Delta }u-\mu (I_\alpha *[h|u|^p])h|u|^{p-2}u-(I_\alpha *|u|^{2^*_\alpha })|u|^{2^*_\alpha -2}u=\lambda u\ \ \text{ in }\ \mathbb {R}^N, \\ \int _{\mathbb {R}^N} u^2 dx = c, \end{array}\right. } \end{aligned}$$

where \(\mu ,c>0\), \(N \ge 3\), \(0<\alpha <N\), \(2_\alpha :=\frac{N+\alpha }{N}<p<2^*_\alpha :=\frac{N+\alpha }{N-2}\), \(\lambda \in \mathbb {R}\) is a Lagrange multiplier, \(I_\alpha \) is the Riesz potential and \(h:\mathbb {R}^N\rightarrow (0,\infty )\) is a continuous function. Under a class of reasonable assumptions on h, we prove the existence of normalized solutions to the above problem for the case \(\frac{N+\alpha +2}{N}\le p<\frac{N+\alpha }{N-2}\) and discuss its asymptotical behaviors as \(\mu \rightarrow 0^+\) and \(c\rightarrow 0^+\) respectively. When \(\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}\), we obtain the existence of one local minimizer after considering a suitable minimization problem.

具有非自主非局部扰动的 HLS 上临界聚焦 Choquard 方程的归一化解
本文关注的是以下具有非自主非局部扰动的 HLS 上临界聚焦 Choquard 方程 $$\begin{aligned} {\left\{ \begin{array}{ll} - {\Delta }u-\mu (I_\alpha *[h|u|^p])h|u|^{p-2}u-(I_\alpha *|u|^{p]){\Delta }u-\mu (I_\alpha *[h|u|^p])h|u|^{p-2}u-(I_\alpha *|u|^{2^*_\alpha })|u|^{2^*_\alpha -2}u=\lambda u\ \text{ in }\mathbb {R}^N、\\ u^2 dx = c, end{array}\right.}\end{aligned}$where \(\mu ,c>0\),\(N \ge 3\), \(0<\alpha <N\),\(2_\alpha :=\frac{N+\alpha }{N}<p<2^*_\alpha :=frac{N+\alpha }{N-2}\),\(\lambda \in \mathbb {R}\)是拉格朗日乘数,\(I_\alpha \)是里兹势,\(h:\mathbb {R}^N\rightarrow (0,\infty )\) 是连续函数。在关于 h 的一类合理假设下,我们证明了在 \(\frac{N+\alpha +2}{N}le p<\frac{N+\alpha }{N-2}\) 的情况下上述问题的归一化解的存在,并讨论了它分别作为 \(\mu \rightarrow 0^+\) 和\(c\rightarrow 0^+\) 的渐近行为。当 \(\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}\) 时,在考虑一个合适的最小化问题后,我们得到了一个局部最小化的存在。
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来源期刊
Analysis and Mathematical Physics
Analysis and Mathematical Physics MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.70
自引率
0.00%
发文量
122
期刊介绍: Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.
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