论文标题
Sobolev CriticalSchrödinger方程的多个归一化解决方案
Multiple normalized solutions for a Sobolev critical Schrödinger equation
论文作者
论文摘要
我们研究了常规波的存在，规定的$ l^2 $ norm（弥撒），用于具有混合功率非线性的非线性schrödinger方程 $$ i \ partial_t ϕ+ δϕ+ μϕ | ϕ |^{q -2}+ ϕ | ϕ | ϕ |^{2^* - 2} = 0，\ quad（t，x）\ in r \ times r^n，in $$ 其中$ n \ geq 3 $，$ ϕ：r \ times r^n \ to c $，$μ> 0 $，$ 2 <q <q <2 + 4/n $和$ 2^* = 2n/（n-2）$是关键的sobolev endent。已经证明，对于小质量，基态存在，并且对应于相关能量功能的局部最小值。还确定，尽管非线性是sobolev批判性的，但基础状态的集合是轨道稳定的。在这里，我们证明，当$ n \ geq 4 $ $时，也存在不是基态，并且位于能量功能的山水水平上。在有限的时间内，这些解决方案是不稳定的。我们的研究是由N. Soave提出的一个问题引起的。
We study the existence of standing waves, of prescribed $L^2$-norm (the mass), for the nonlinear Schrödinger equation with mixed power nonlinearities $$ i \partial_t ϕ+ Δϕ+ μϕ|ϕ|^{q-2} + ϕ|ϕ|^{2^* - 2} = 0, \quad (t, x) \in R \times R^N, $$ where $N \geq 3$, $ϕ: R \times R^N \to C$, $μ> 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. It was already proved that, for small mass, ground states exist and correspond to local minima of the associated Energy functional. It was also established that despite the nonlinearity is Sobolev critical, the set of ground states is orbitally stable. Here we prove that, when $N \geq 4$, there also exist standing waves which are not ground states and are located at a mountain-pass level of the Energy functional. These solutions are unstable by blow-up in finite time. Our study is motivated by a question raised by N. Soave.
