论文标题
具有混合分数拉普拉斯的非线性schrödinger方程的归一化解决方案的存在和动力学
Existence and dynamics of normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
论文作者
论文摘要
在本文中，我们关注方程解决方案的存在和动态，其混合分数laplacians $$（-Δ）^{s_1} u +（ - δ） $ n \ geq 1 $，$ 0 <s_2 <s_1 <1 $，$ 2+ \ frac {4s_1} {n} {n} \ leq p <\ infty $如果$ n \ leq 2s_1 $，$ 2+ \ frac {4s_1} {4s_1} {n} {n} {n} \ leq p <$ n} > 2S_1 $，$λ\ in \ r $出现在Lagrange乘法器中是未知的。分数laplacian $（ - δ）^s $的特征为$ \ MATHCAL {f}（（ - δ）^{s} u）（ξ）= |ξ|^{2S} \ Mathcal {f}（f}（f}（ξ）（U）（U）（U）（U）$ n $ n $ deN $ n $ n $，for n $ n $ n $ n $，首先，我们确定基态解决方案的存在和界面解决方案的多样性。然后，我们研究了相关时间依赖方程的库奇问题解决方案的动力学。此外，我们建立了基态解决方案的轨道不稳定。
In this paper, we are concerned with the existence and dynamics of solutions to the equation with mixed fractional Laplacians $$ (-Δ)^{s_1} u +(-Δ)^{s_2} u + λu=|u|^{p-2} u $$ under the constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<s_2<s_1<1$, $2+ \frac {4s_1}{N} \leq p< \infty $ if $N \leq 2s_1$, $2+ \frac {4s_1}{N} \leq p<\frac{2N}{N-2s_1}$ if $N >2s_1$, $λ\in \R$ appearing as Lagrange multiplier is unknown. The fractional Laplacian $(-Δ)^s$ is characterized as $\mathcal{F}((-Δ)^{s}u)(ξ)=|ξ|^{2s} \mathcal{F}(u)(ξ)$ for $ξ\in \R^N$, where $\mathcal{F}$ denotes the Fourier transform. First we establish the existence of ground state solutions and the multiplicity of bound state solutions. Then we study dynamics of solutions to the Cauchy problem for the associated time-dependent equation. Moreover, we establish orbital instability of ground state solutions.
