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
$1+(4-2b)/N < p < 1+(4-2b)/(N-2)$.
{{journal.titleEn}} 
DownLoad: