论文标题
Trudinger-Moser临界非线性的径向溶液对半线性椭圆方程的浓度曲线，能量和弱极限
Concentration profile, energy, and weak limits of radial solutions to semilinear elliptic equations with Trudinger-Moser critical nonlinearities
论文作者
论文摘要
我们研究了下一个Trudinger -Moser批判方程，\ [\ [\ begin {case}-ΔU=λU^{u^2+α| $（λ，β）\ in（0，\ infty）\ times（0,2）$和$ b \ subset \ mathbb {r}^2 $是以原点为中心的单位球。我们对径向溶液的能量有限序列的渐近行为进行了分类。通过打击分析和缩放技术，我们推断了浓缩溶液的极限曲线，能量和几种渐近公式，以及弱极限的精确信息。特别是，我们获得了浓度点弱极限幅度的新必要条件。这可以证明2020年在径向情况下，格罗西·摩西尼（Grossi-Mancini-Naimen-Pistoia）的猜想。此外，在$β\ le1 $的情况下，我们表明任何序列最多都带有一个气泡。这允许在合适范围内使用$（λ，β）$的低能节点径向溶液不存在新的证明。最后，我们讨论了分类结果的几个同行。尤其是，我们证明了一系列解决方案的存在，这些溶液携带多个气泡并弱收敛到签名溶液。
We investigate the next Trudinger-Moser critical equations, \[ \begin{cases} -Δu=λue^{u^2+α|u|^β}&\text{ in }B,\\ u=0&\text{ on }\partial B, \end{cases} \] where $α>0$, $(λ,β)\in(0,\infty)\times(0,2)$ and $B\subset \mathbb{R}^2$ 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-Mancini-Naimen-Pistoia in 2020 in the radial case. Moreover, in the case of $β\le1$, 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 $(λ,β)$ 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.
