A mass supercritical problem revisited
In any dimension N ≥ 1 and for given mass m > 0 , we revisit the nonlinear scalar field equation with an L 2 constraint: - Δ u = f ( u ) - μ u in R N , ‖ u ‖ L 2 ( R N ) 2 = m , u ∈ H 1 ( R N ) , ( P m ) where μ ∈ R will arise as a Lagrange multiplier. Assuming only that the nonlinearity f is con...
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| Published in: | Calculus of variations and partial differential equations Vol. 59; no. 5 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.10.2020
Springer Nature B.V Springer Verlag |
| Subjects: | |
| ISSN: | 0944-2669, 1432-0835 |
| Online Access: | Get full text |
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| Summary: | In any dimension
N
≥
1
and for given mass
m
>
0
, we revisit the nonlinear scalar field equation with an
L
2
constraint:
-
Δ
u
=
f
(
u
)
-
μ
u
in
R
N
,
‖
u
‖
L
2
(
R
N
)
2
=
m
,
u
∈
H
1
(
R
N
)
,
(
P
m
)
where
μ
∈
R
will arise as a Lagrange multiplier. Assuming only that the nonlinearity
f
is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states to
(
P
m
)
and reveal the basic behavior of the ground state energy
E
m
as
m
>
0
varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other
L
2
constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any
N
≥
2
and establish the existence and multiplicity of nonradial sign-changing solutions when
N
≥
4
. Finally we propose two open problems. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0944-2669 1432-0835 |
| DOI: | 10.1007/s00526-020-01828-z |