Normalized Solution for p-Kirchhoff Equation with a L2-supercritical Growth

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Zhi-min Ren, Yong-yi Lan
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引用次数: 0

Abstract

In this paper, we investigate the following p-Kirchhoff equation

$$\left\{ {\matrix{{(a + b\,\int_{{\mathbb{R}^N}} {(|\nabla u{|^p} + |u{|^p})dx)\,( - {\Delta _p}u + |u{|^{p - 2}}u) = |u{|^{s - 2}}u + \mu u,\,\,x \in {\mathbb{R}^N},} } \hfill \cr {\int_{{\mathbb{R}^N}} {|u{|^2}dx = \rho ,} } \hfill \cr } } \right.$$

where a > 0, b ≥ 0, ρ > 0 are constants, \({p^ * } = {{Np} \over {N - p}}\) is the critical Sobolev exponent, μ is a Lagrange multiplier, \( - {\Delta _p}u = - {\rm{div}}(|\nabla u{|^{p - 2}}\nabla u)\), \(2 < p < N < 2p,\,\,\,\mu \in \mathbb{R}\) and \(s \in (2{{N + 2} \over N}p - 2,\,\,\,{p^ * })\). We demonstrate that the p-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.

具有 L2 超临界增长的 p-Kirchhoff 方程的归一化解法
在本文中,我们研究了以下 p-Kirchhoff 方程 $$\left\{ {\matrix{{(a + b\,\int_{\mathbb{R}^N}}{(|\nabla u{|^p} + |u{|^p})dx)\,( - {\Delta _p}u + |u{|^{p - 2}}u) = |u{|^{s - 2}}u + \mu u,\,x \ in {\mathbb{R}^N},}}h\fill \cr {\int_{\mathbb{R}^N}}{|u{|^2}dx = \rho ,}}h\fill \cr }}\right.$$ 其中 a > 0, b ≥ 0, ρ > 0 是常数, \({p^ * } = {{Np} \over {N - p}}\) 是临界索波列夫指数, μ 是拉格朗日乘数, \( - {\Delta _p}u = - {\rm{div}}(|\nabla u{|^{p - 2}}\nabla u)\), \(2 <;p < N < 2p,\,\,\mu \in \mathbb{R}\) and\(s \in (2{{N + 2} \over N}p - 2,\,\,\,{p^ * })\).我们利用山口阶梯和一些分析技术证明 p-Kirchhoff 方程有一个归一化解。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
70
审稿时长
3.0 months
期刊介绍: Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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