论文标题
具有关键指数和混合非局部非线性的Schrödinger方程的归一化解决方案
Normalized Solutions to Schrödinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities
论文作者
论文摘要
我们研究归一化解决方案的存在和不存在$（u_a，λ_a）\ in H^{1}（\ Mathbb {r}^n）\ times \ times \ times \ mathbb {r {r} $与非线性schrödinger方程，具有混合非局部非局部性。 在非局部schrödiger方程的归一化解决方案的背景下，这项研究可以被视为布雷兹斯 - 尼伦贝格问题的对应物，其固定$ l^2 $ -norm $ \ | | u \ | _2 = a> 0 $。 主术语是$ l^2 $ -supergical，即$ p \ in（\ frac {n+α+2} {n} {n}，\ frac {n+α} {n-2} {n-2} $，其中强硬的woodwood-sobolev criticalv criedical $ p = \ frac $ p = \ frac = \ frac {n+α} $ 我们首先证明，如果$ q \ in（\ frac {n+α} {n}，\ frac {n+α+2} {n} {n} {n} {n} {n} {n} {n} {n} {n} {n} {n} {n}）$，则存在两个归一化解决方案，即$μ> 0 $ small，也就是说，一个是负能量，而另一个处于正能水平上。 对于$ q = \ frac {n+α+2} {n} $，我们表明存在$ 0 <μ<\tildeμ$的归一化基态，并且对于$μ> \tildeμ$，$ \tildeμ$没有基态，$μ> \tildeμ$是一个较高的正常数。 如果$ q \ in（\ frac {n+α+2} {n}，\ frac {n+α} {n-2}）$，我们推断出存在任何$μ> 0 $的归一化基态。 我们还获得了$μ<0 $和$ q \ in（\ frac {n+α} {n} {n}，\ frac {n+α+2} {n}] $的情况，我们还获得了一些存在和不存在结果。 此外，我们将标准化基态的渐近行为分析为$μ\ rightarrow 0^{+} $。
We study the existence and nonexistence of normalized solutions $(u_a, λ_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schrödinger equation with mixed nonlocal nonlinearities. This study can be viewed as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions to the nonlocal Schrödiger equation with a fixed $L^2$-norm $\|u\|_2=a>0$. The leading term is $L^2$-supercritical, that is, $p\in (\frac{N+α+2}{N},\frac{N+α}{N-2}]$, where the Hardy-Littlewood-Sobolev critical exponent $p=\frac{N+α}{N-2}$ appears. We first prove that there exist two normalized solutions if $q\in (\frac{N+α}{N},\frac{N+α+2}{N})$ with $μ>0$ small, that is, one is at the negative energy level while the other one is at the positive energy level. For $q=\frac{N+α+2}{N}$, we show that there is a normalized ground state for $0<μ< \tildeμ $ and there exist no ground states for $μ>\tildeμ$, where $\tildeμ$ is a sharp positive constant. If $q\in (\frac{N+α+2}{N},\frac{N+α}{N-2})$, we deduce that there exists a normalized ground state for any $μ>0$. We also obtain some existence and nonexistence results for the case $μ<0$ and $q\in (\frac{N+α}{N},\frac{N+α+2}{N}]$. Besides, we analyze the asymptotic behavior of normalized ground states as $μ\rightarrow 0^{+}$.
