Concentration profile, energy, and weak limits of radial solutions to semilinear elliptic equations with Trudinger-Moser critical nonlinearities.
Concentration profile, energy, and weak limits of radial solutions to semilinear elliptic equations with Trudinger-Moser critical nonlinearities.
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具有 Trudinger-Moser 临界非线性的半线性椭圆方程的径向解的浓度分布、能量和弱极限。
DOI：
10.1007/s00526-021-01951-5
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发表时间：
2021
影响因子：
2.1
通讯作者：
Daisuke Naimen
中科院分区：
文献类型：
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作者：
及川一誠;Issei Oikawa;Daisuke Naimen
我们研究了下一个特鲁丁格-莫泽临界方程，-Δu=λueu 2+α|u|βin B，u=0在∂B上，其中α>；0α>；0，(λ，β)∈(0，∞)*(0，2)(λ，β)∈(0，∞)×(0，2))和B⊂R^2 B⊂R2是以原点为中心的单位球。我们对能量有界的径向解序列的渐近行为进行了分类。通过Blow-up分析和Scaling技巧，我们导出了极限轮廓、能量和几个集中解的渐近公式以及弱极限的精确信息。特别地，我们得到了关于弱极限在集中点的幅度的一个新的必要条件。这证明了格罗西等人(数学·安即将出现)2020年在径向情况下的猜想。此外，在β≤1β≤1的情况下，我们证明了任何序列至多带有一个气泡。这为证明(λ，β)(λ，β)在适当的范围内不存在低能节点径向解提供了新的证明。最后，我们讨论了我们的分类结果的几个对应项。特别地，我们证明了带有多个气泡且弱收敛于一个变号解的解序列的存在性。
We investigate the next Trudinger–Moser critical equations,-Δ u= λ ueu 2+ α| u| β in B, u= 0 on∂ B, where α> 0 α> 0,(λ, β) ∈ (0, ∞) * (0, 2)(λ, β)∈(0,∞)×(0, 2) and B ⊂ R^ 2 B⊂ R 2 is the unit ball centered at the origin. We classify the asymptotic behavior of energy bounded sequences of radial solutions. Via the blow–up analysis and a scaling technique, we deduce the limit profile, energy, and several asymptotic formulas of concentrating solutions together with precise information of the weak limit. In particular, we obtain a new necessary condition on the amplitude of the weak limit at the concentration point. This gives a proof of the conjecture by Grossi et al.(Math Ann, to appear) in 2020 in the radial case. Moreover, in the case of β ≤ 1 β≤ 1, we show that any sequence carries at most one bubble. This allows a new proof of the nonexistence of low energy nodal radial solutions for (λ, β)(λ, β) in a suitable range. Lastly, we discuss several counterparts of our classification result. Especially, we prove the existence of a sequence of solutions which carries multiple bubbles and weakly converges to a sign-changing solution.