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Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves

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  • For $N\geq3$ and $p>1$, we consider the nonlinear Schrödinger equation

    $i\partial_{t}w+\Delta_{x}w+V(x) |w| ^{p-1}w=0\text{ where }w=w(t,x):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{C}$

    with a potential $V$ that decays at infinity like $| x|^{-b}$ for some $b\in (0,2)$. A standing wave is a solution of the form

    $w(t,x)=e^{i\lambda t}u(x)\text{ where }\lambda>0\text{ and }u:\mathbb{R}^{N}\rightarrow\mathbb{R}.$

    For $ 1 < p < 1+(4-2b)/(N-2)$, we establish the existence of a $C^1$-branch of standing waves parametrized by frequencies $\lambda $ in a right neighbourhood of $0$. We also prove that these standing waves are orbitally stable if $ 1 < p < 1+(4-2b)/N$ and unstable if $1+(4-2b)/N < p < 1+(4-2b)/(N-2)$.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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