Utilizing a new variational principle that allows us to deal with problems beyond the usual locally compact structure, we study problems with a supercritical nonlinearity of the type
(1)
{
−
Δ
u
+
u
=
a
(
x
)
f
(
u
)
a
m
p
;
in
Ω
,
u
>
0
a
m
p
;
in
Ω
,
∂
u
∂
ν
=
0
a
m
p
;
on
∂
Ω
.
\begin{equation}\tag {1} \begin {cases} -\Delta u + u = a(x) f(u) & \text {in $\Omega $}, \\ u>0 & \text {in $\Omega $}, \\ \frac {\partial u}{\partial \nu } = 0 & \text {on $\partial \Omega $}. \end{cases} \end{equation}
To be more precise,
Ω
\Omega
is a bounded domain in
R
N
\mathbb {R}^N
which satisfies certain symmetry assumptions,
Ω
\Omega
is a domain of “
m
m
revolution" (
1
≤
m
>
N
1\leq m>N
and the case of
m
=
1
m=1
corresponds to radial domains), and
a
>
0
a > 0
satisfies compatible symmetry assumptions along with monotonicity conditions. We find positive nontrivial solutions of (1) in the case of suitable supercritical nonlinearities
f
f
by finding critical points of
I
I
where
\[
I
(
u
)
=
∫
Ω
{
a
(
x
)
F
∗
(
−
Δ
u
+
u
a
(
x
)
)
−
a
(
x
)
F
(
u
)
}
d
x
I(u)=\int _\Omega \left \{ a(x) F^* \left ( \frac {-\Delta u + u}{a(x)} \right ) - a(x) F(u) \right \} dx
\]
over the closed convex cone
K
m
K_m
of nonnegative, symmetric, and monotonic functions in
H
1
(
Ω
)
H^1(\Omega )
where
F
′
=
f
F’=f
and where
F
∗
F^*
is the Fenchel dual of
F
F
. We mention two important comments: First, there is a hidden symmetry in the functional
I
I
due to the presence of a convex function and its Fenchel dual that makes it ideal to deal with supercritical problems lacking the necessary compactness requirement. Second, the energy
I
I
is not at all related to the classical Euler–Lagrange energy associated with (1). After we have proven the existence of critical points
u
u
of
I
I
on
K
m
K_m
, we then unitize a new abstract variational approach to show that these critical points in fact satisfy
−
Δ
u
+
u
=
a
(
x
)
f
(
u
)
-\Delta u + u = a(x) f(u)
.
In the particular case of
f
(
u
)
=
|
u
|
p
−
2
u
f(u)=|u|^{p-2} u
we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.