A new variational principle, convexity, and supercritical Neumann problems
Authors: Cowan Craig , Moameni Abbas 
Published: April 2017

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column minus normal upper Delta u plus u equals a left-parenthesis x right-parenthesis f left-parenthesis u right-parenthesis 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u greater-than 0 2nd Column a m p semicolon in normal upper Omega comma 3rd Row 1st Column StartFraction partial-differential u Over partial-differential nu EndFraction equals 0 2nd Column a m p semicolon on partial-differential normal upper Omega period EndLayout EndLayout">

(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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis u right-parenthesis equals integral Underscript normal upper Omega Endscripts left-brace a left-parenthesis x right-parenthesis upper F Superscript asterisk Baseline left-parenthesis StartFraction minus normal upper Delta u plus u Over a left-parenthesis x right-parenthesis EndFraction right-parenthesis minus a left-parenthesis x right-parenthesis upper F left-parenthesis u right-parenthesis right-brace d x">

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.