Abstract
We study existence and multiplicity of positive ground states for the scalar curvature equation $$\begin{aligned} \varDelta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n\,, \quad n>2, \end{aligned}$$
Δ
u
+
K
(
|
x
|
)
u
n
+
2
n
-
2
=
0
,
x
∈
R
n
,
n
>
2
,
when the function $$K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+$$
K
:
R
+
→
R
+
is bounded above and below by two positive constants, i.e. $$0<\underline{K} \le K(r) \le \overline{K}$$
0
<
K
̲
≤
K
(
r
)
≤
K
¯
for every $$r > 0$$
r
>
0
, it is decreasing in $$(0,{{{\mathcal {R}}}})$$
(
0
,
R
)
and increasing in $$({{{\mathcal {R}}}},+\infty )$$
(
R
,
+
∞
)
for a certain $${{{\mathcal {R}}}}>0$$
R
>
0
. We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio $$\overline{K}/\underline{K}$$
K
¯
/
K
̲
which guarantees the existence of a large number of ground states with fast decay, i.e. such that $$u(|x|) \sim |x|^{2-n}$$
u
(
|
x
|
)
∼
|
x
|
2
-
n
as $$|x| \rightarrow +\infty $$
|
x
|
→
+
∞
, which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique.
Rational Functions with a Unique Critical Point
