Standing waves in nonlinear Schrödinger equations 1
57
3 Existence, uniqueness and properties of solitons
W
e will use the following definition of standing waves for (1.1).
Definition 3.1. A standing wave or soliton of (1.1) is a solution of the form e
iωt
ϕ(x)
with ω ∈ R and ϕ satisfying
(
−∆ϕ + ωϕ −|ϕ|
p−1
ϕ = 0,
ϕ ∈ H
1
(R
N
) \{0} .
(3.1)
Many techniques have been developed to study the existence of solutions to prob-
lems of type (3.1) (see, e.g., [3]). In this section, we look for solutions of (3.1) by using
variational methods (see, e.g., [71] for a general overview).
For the study of solutions of (3.1), we define a functional S : H
1
(R
N
) → R by
setting for v ∈ H
1
(R
N
)
S(v) :=
1
2
k
∇vk
2
L
2
(R
N
)
+
ω
2
kvk
2
L
2
(R
N
)
−
1
p + 1
kvk
p+
1
L
p+1
(R
N
)
.
The functional S is often called action. It is standard that S is of class C
2
(see, for
example, [78]) and for v ∈ H
1
(R
N
) the Fréchet derivative of S at v is given by
S
′
(v) = −∆v + ωv − |v|
p−1
v.
Therefore, ϕ is a solution of (3.1) if and only if ϕ ∈ H
1
(R
N
) \ {0} and S
′
(ϕ) = 0. In
other words, the nontrivial critical points of S are the solutions of (3.1). Therefore, to
prove existence of solutions of (3.1) it is enough to find a nontrivial critical point of S.
This section is divided as follows. First, we prove that if solutions to (3.1) exist
then they are regular, exponentially decaying at infinity and satisfy some functional
identities. Next, we prove the existence of a nontrivial critical point of S. Finally, we
derive various variational characterizations for some special solutions of (3.1).
3.1 Preliminaries
Before studying existence of solutions to (3.1), it is convenient to prove tha
t, if such
solutions exist, they necessarily enjoy the following properties.
Proposition 3.2. Let ω > 0. If ϕ ∈ H
1
(R
N
) satisfies (3.1) then ϕ is regular and
exponentially decaying. More precisely,
(i) ϕ ∈ W
3,r
(R
N
) for all r ∈ [2, +∞), in particular ϕ ∈ C
2
(R
N
);
(ii) there exists ε > 0 such that e
ε|x|
(|ϕ|+ |∇ϕ|) ∈ L
∞
(R
N
).
Sketch of proof. Point (i) follows from the usual elliptic regularity theory by a boot-
strap argument (see [30]). We just indicate how to initiate the bootstrap and refer to
[14] for a detailed proof. Let ϕ be a solution of (3.1). Suppose that ϕ ∈ L
q
(R
N
) for
some q > p. Since |ϕ|
p−1
ϕ ∈ L
q
p
(R
N
) and ϕ satisfies
−∆ϕ + ωϕ = |ϕ|
p−1
ϕ, (3.2)