In this paper, we consider the strong instability of standing waves for the nonlinear Schrödinger–Poisson equation$$\begin{aligned} i\partial _t\psi +\Delta \psi -(|x|^{-1}*|\psi |^2)\psi +|\psi |^{p}\psi =0~~~~ (t,x)\in [0,T^*)\times {\mathbb {R}}^3. \end{aligned}$$In the \(L^2\)-critical case, i.e., \(p=\frac{4}{3}\), we prove that the standing waves are strongly unstable by blow-up. This result is a complement to the result of Kikuchi (Adv Nonlinear Stud 7:403–437, 2007) and Bellazzini et al. (Proc Lond Math Soc 107:303–339, 2013), where the instability of standing waves were studied in the \(L^2\)-supercritical case, i.e., \(\frac{4}{3}< p<4\).

中文翻译：

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非线性Schrödinger–Poisson方程在$$ L ^ 2 $$ L 2-临界情况下驻波的不稳定性

在本文中，我们考虑了非线性Schrödinger-Poisson方程$$ \ begin {aligned} i \ partial _t \ psi + \ Delta \ psi-（| x | ^ {-1} * | \ psi | ^ 2）\ psi + | \ psi | ^ {p} \ psi = 0 ~~~~（t，x）\ in [0，T ^ *）\次{\ mathbb {R}} ^ 3。\ end {aligned} $$在\（L ^ 2 \）临界情况下，即\（p = \ frac {4} {3} \），我们证明了驻波由于爆炸而非常不稳定。此结果是Kikuchi（Adv Nonlinear Stud 7：403–437，2007）和Bellazzini等人的结果的补充。（Proc Lond Math Soc 107：303–339，2013），其中在\（L ^ 2 \）-超临界情况下，即\（\ frac {4} {3} <p < 4 \）。