Proof of Proposition 2.2

We next prove Proposition 2.2. Note that we write H¹ for H¹(R³, C), and similarly for L²_V and L⁴. A sequence {(uⁿ₁, uⁿ₀, uⁿ₋₁)} in (H¹)³is said to be (weakly) convergent in H¹if it is (weakly) convergent in (H¹)³ = H¹⊕ H¹⊕ H¹ = H¹(R³, R³), which is equivalent to say {uⁿ_j} is (weakly) convergent in H¹ for each j = 1, 0, −1. The same convention applies to (weak) convergence in L²_V and in L⁴.

We’ll use without proof the following facts.

1. B is a reflexive Banach space, in which weak convergence is equivalent to weak convergence in H¹, in L²_V, and in L⁴separately.

2. Since B+is a convex and closed subset of B, B+ is a weakly closed subset of B (Mazur’s theorem).

3. H¹, L²_V, and L⁴are uniformly convex.

Remark 8.2. For our purpose, we can in fact “define” weak convergence in B to be weak convergence in H¹, in L²_V and in L⁴, without knowing that this definition is really equivalent to weak convergence in the Banach space B. Some arguments should then be modified, for example the reason B+is weakly closed in B. This, though works, is of course very unsatisfactory.

Remark 8.3. Although these facts are well-known, some of them usually do not ap-pear in standard courses. Indeed, I myself got the answer of the first claim from mathoverflow.net(Thanks Dr. William B. Johnson), and the uniform convexity of gen-eral Sobolev spaces (but not only Lebesgue spaces) was found on a page of

math.stackexchange.com(asked by Tom´as and answered by martini).

We’ll also need the following observation.

Lemma 8.13. For β1 < 0 (and |β1| < β0 by the assumption (A2)), we have for every u ∈ A

E₀[u] + E₁[u] = (β₀+ β₁) Z

|u|⁴ − β₁ Z

u²₀− 2u₁u−1

2

.

In particular,

E₀[u] + E₁[u] ≥ (β₀+ β₁) Z

|u|⁴.

Proof. The assertion is a direct consequence of the following identity:

|u|⁴−2u²₀(u₁+ u₋₁)²+ (u²₁− u²₋₁)² = u²₀− 2u₁u₋₁
2

. (8.7)

Remark 8.4. Identity (8.7) is also used in the proof of Theorem 3.1 (equation 3.5).

For convenience we restate Proposition 2.2 below.

Proposition 8.14. Let {uⁿ} be a sequence in B⁺. Suppose N [uⁿ] → 1, M[uⁿ] → M , and E [uⁿ] is uniformly bounded in n, then {uⁿ} has a subsequence {u^n(k)}^∞_k=1 converging weakly to someu^∞ ∈ A, which satisfies E[u^∞] ≤ lim inf_k→∞E[u^n(k)]. If we assume further thatE[uⁿ] → E_g, thenu^∞ ∈ G, and u^n(k)→ u^∞in the norm of B.

Proof. We first remark that with the norm defined by (2.1), B is a reflexive Banach space, in which weak convergence is equivalent to weak convergence in H¹, in L²_V, and in L⁴separately. We omit the verifications of these standard facts. Moreover, since

B⁺ is a convex and closed subset of B, B+ is a weakly closed subset of B (Mazur’s theorem).

Note that the uniform boundedness of E [uⁿ] implies {uⁿ} is a bounded sequence in B. This is obvious if β1 > 0, and is also true for β₁ < 0 by Lemma 8.13. Thus,

First, by the weak lower semi-continuity of a norm, we have Z that (i) (A1) implies (u^n(k)_j )²is very small outside a large enough bounded set, and that (ii) on any bounded set, u^n(k)_j → u^∞_j in L² by compact embedding H¹ ,→ L². As to (i), note that since {uⁿ} is a bounded sequence, we have in particularR V |u^n(k)|² ≤ C for some C > 0 independent of k. By the assumption (A1), for any ε > 0, there exists R_ε> 0 such that V (x) ≥ C/ε for |x| ≥ R_ε. Thus we have

From this fact and the strong convergence mentioned in (ii), we obtain lim sup

Since ε > 0 is arbitrary, (8.9) and (8.10) implies (8.8), and hence u^∞ ∈ A.

Next, the assertion E [u^∞] ≤ lim inf_kE[u^n(k)] follows a general weak lower semi-continuity theorem. See e.g. Theorem 1.6 of [20]. Indeed, by that theorem we have

Z

for every continuous function f : R³ → [0, ∞). As a consequence, we have E_kin[u^∞] ≤ lim inf

Note carefully that (8.11) requires that f be nonnegative, and hence the assertion of the weak lower semi-continuity of E0 + E₁ in (8.12) also uses Lemma 8.13. It is then clear that the limit inferiors in (8.12) must all be limits provided E [uⁿ] tends to the ground-state energy E_g. Otherwise we get the contradiction

E_g = lim

Since H¹ is uniformly convex, (8.13) together with the fact u^n(k)* u^∞weakly in H¹ imply u^n(k) → u^∞strongly in H¹. Similarly we can prove u^n(k) → u^∞in L²_V and in L⁴, and hence u^n(k) → u^∞in the norm of B.

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