| Abstract: |
| In this talk we discuss the problem of existence and multiplicity of radial ground states with fast decay (GS for short)
for
$$ \Delta u + [1+\epsilon k(|x|)] u^{\frac{n+2}{n-2}}=0$$
where $x \in \mathbb{R}^n$, $n \ge 3$, $k \in C^1$, $k(|x|) \in [0,1]$, $\epsilon>0$ small.
Nowadays several different conditions sufficient for the existence of GS are available in literature. Further, if $k$
has a unique critical point and it is a maximum the GS is unique, see \textbf{[KYY]}. On the other side
if the unique critical point is a minimum (and some other conditions are fulfilled)
a large number of GS are found, if $\epsilon>0$ is small enough, see \textbf{[CL]}.
A similar result was obtained in \textbf{[FF]} replacing $[1+\epsilon k(|x|)]$ by a slowly varying function $k(|x|^{\epsilon})$.
Our purpose is to give a constructive argument which enable us to reprove the result in \textbf{[CL]}
but giving an estimate on how small $\epsilon$ should be. In fact
$\epsilon$ need not to be very small, e.g. we have at least $k$ GS for $\epsilon |
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