论文标题
带有规定质量的Schrödinger方程的新方法：Sobolev亚临界案例和sobolev批判性案例与混合分散
New approaches for Schrödinger equations with prescribed mass: The Sobolev subcritical case and The Sobolev critical case with mixed dispersion
论文作者
论文摘要
在本文中，我们证明了以下schrödinger方程\ begin {equation*}的存在归一化解决方案 \左边\{ \ begin {array} {ll} -Δu-λu= f（u），＆x \ in \ r^n， \ int _ {\ r^n} u^2 \ mathrm {d} x = c \ end {array} \正确的。 带有$ n \ ge3 $，$ c> 0 $，$ c> 0 $，$λ\ in \ r $和$ f \ in \ mathcal {c}（\ r，\ r）$在sobolev subelitical cance中具有弱$ l^2 $ -Subolev Criticalv Criticalv Criticalv Criticalv Criticalv Criticalv Criticalv Criticalv CriticalV $ f（u）=μ| u |^{q-2} u+| u | u |^{2^*-2} u $，$μ> 0 $> 0 $和$ 2 <q <q <2^*= \ f {2n} {n-2} {n-2} $允许$ l^2 $ -L^2 $ -Subcritical，关键，关键或超级临界或超临界或超级临界。我们的方法基于多种多种临界定理，这不仅有助于在Sobolev亚临界情况下削弱以前的$ l^2 $ superitical条件，而且为在$ f（p（ps）上构建有限（ps）序列的替代方案时，当$ f（p（ps）序列）当$ f（f（p f（f（p））涉及拓扑论点的原则，以及所有2美元<q <2^*$的工作。特别是，我们提出了新的策略，以控制Sobolev关键案例中的能量水平，该案例允许以统一的方式对待尺寸$ n = 3 $和$ n \ ge 4 $，并满足了Soave和Jeanjean-le的期望。我们认为，我们的方法和策略可能会得到调整和修改，以在约束环境中攻击更多的变分问题。
In this paper, we prove the existence of normalized solutions for the following Schrödinger equation \begin{equation*} \left\{ \begin{array}{ll} -Δu-λu=f(u), & x\in \R^N, \int_{\R^N}u^2\mathrm{d}x=c \end{array} \right. \end{equation*} with $N\ge3$, $c>0$, $λ\in \R$ and $f\in \mathcal{C}(\R,\R)$ in the Sobolev subcritical case with weaker $L^2$-supercritical conditions and in the Sobolev critical case when $f(u)=μ|u|^{q-2}u+|u|^{2^*-2}u$ with $μ>0$ and $2<q<2^*=\f{2N}{N-2}$ allowing to be $L^2$-subcritical, critical or supercritical. Our approach is based on several new critical point theorems on a manifold, which not only help to weaken the previous $L^2$-supercritical conditions in the Sobolev subcritical case, but present an alternative scheme to construct bounded (PS) sequences on a manifold when $f(u)=μ|u|^{q-2}u+|u|^{2^*-2}u$ technically simpler than the Ghoussoub minimax principle involving topological arguments, as well as working for all $2<q<2^*$. In particular, we propose new strategies to control the energy level in the Sobolev critical case which allow to treat, in a unified way, the dimensions $N=3$ and $N\ge 4$, and fulfill what were expected by Soave and by Jeanjean-Le . We believe that our approaches and strategies may be adapted and modified to attack more variational problems in the constraint contexts.
