NORMALIZED SOLUTIONS TO SCHRÖDINGER EQUATIONS IN THE STRONGLY SUBLINEAR REGIME

. We look for solutions to the Schrödinger equation − ∆ u + λu = g ( u ) in R N coupled with the mass constraint R R N | u | 2 dx = ρ 2 , with N ≥ 2 . The behaviour of g at the origin is allowed to be strongly sublinear, i


Introduction
We look for solutions (u, λ) ∈ H 1 (R N ) × R to the following equation (1.1) −∆u + λu = g(u) in R N , N ≥ 2 paired with the constraint (1.2) where ρ > 0 is fixed and g satisfies suitable assumptions (see (g0)-(g4) below).Equations like (1.1) arise when one looks for standing-wave solutions to the evolution equation (1.3) i∂ t Ψ + ∆Ψ + g(Ψ) = 0, i.e., Ψ(t, x) = e iλt u(x).The condition (1.2), instead, is motivated by the property that, formally, the quantity is conserved in time.Moreover, the L 2 -norm squared of a solution to (1.3) has a precise physical meaning in the contexts which this equation is derived from: it represents, e.g., the power supply in nonlinear optics and the total number of atoms in the Bose-Einstein condensation.The model for g that inspires this paper is (1.4) g(s) = s ln s 2 , which was introduced in [4] to obtain the separability of non-interacting subsystems; observe that, in this case, which is known in the literature as the strongly sublinear (or infinite-mass) regime.After that, both (1.1) -possibly with an external potential -and (1.3), in both cases with g as in (1.4), raised great 1 interest: we mention [13,31,[33][34][35] for the former context and [8,9,11,14] (see also [10,Section 9.3]) for the latter one.The common feature of most of the articles above is that the authors can work in the subspace of H 1 (R N ) where the energy functional (see (1.7) below) is of class C 1 (we mention that, instead, the approach of [13] is based on critical point theory for lower semi-continuous functionals, while that of [14] on a truncation argument).Of course, this strategy is impossible with generic nonlinearities as in this paper; speaking of which, generic nonlinear terms were recently considered in [15,24].Concerning normalized solutions, i.e., in the presence of (1.2), to the best of our knowledge, they are only treated in two very recent pieces of work: in [30], with (1.5) g(s) = αs ln s 2 + µ|s| p−2 s, where α, µ ∈ R and 2 < p ≤ 2 * , and in [37], which mainly concerns systems of two equations.It is worth mentioning that, when g is as in (1.4), if (v, 0) solves (1.1) and v = 0, then (u, λ) given by is a solution to (1.1)-(1.2).Nonetheless, the approach of [30,37] still makes use of H.In particular, to recover compactness, the authors consider the subspace of radially symmetric functions as in [9,11], which embeds compactly into L q (R N ) for every q ∈ [2, 2 * ); this is different from the classical result concerning the subspace of radial functions in H 1 (R N ), which embeds compactly into L q (R N ) only for q ∈ (2, 2 * ).The additional compact embedding into L 2 (R N ) proves to be extremely useful owing to the constraint (1.2); in particular, this is what allows the authors to work with every ρ > 0 when 2 < p < 2 + 4/N , where p is the same as in (1.5).Once again, here we cannot utilize the compact embedding into L 2 (R N ) because g need not be given by (1.5).
The function g in (1.1) satisfies the following assumptions.Define G(s) = (g0) g : R → R is continuous and g(0) = 0. (g1) lim s→0 G + (s)/|s| 2 = 0 and lim sup s→0 g + (s)/s < ∞, where g The assumptions (g0), (g2), and (g4) are classical in the context of elliptic problems (cf.[2,3]), while (g1) is a weaker version of what is normally assumed in the presence of L 2 -constraints, i.e., lim s→0 g(s)/s = 0; (g3) indicates that the growth of G + is mass-subcritical at infinity.With these assumptions, we can deal with nonlinearities that are much more general than (1.5); for example, we can consider functions g that behave like −|s| ω−1 s or |s| ω−1 s ln s 2 for |s| small, with 0 < ω < 1.We can also cover the case when g goes to 0 more slowly than any power, e.g., when g behaves like 1/ ln s 2 for |s| small.
Remark 1.3.Under the assumptions (g0)-(g4), if there exists a solution (u, λ) Moreover, we obtain the relative compactness in H 1 (R N ) of minimizing sequences for inf D(ρ) J up to translations.Here and in the sequel, with a small abuse of notations, if (u n ) n is a sequence in X, we write u n ∈ X instead of (u n ) n ⊂ X.
Theorem 1.5.If (g0)-(g4) hold, then there exists ρ > 0 such that for every ρ > ρ, if 0 < ρ n → ρ and It is well known that such a relative compactness is an important step toward the orbital stability of solutions to (1.1)-(1.2).Unfortunately, we currently do not have the necessary tools to investigate the dynamics of (1.3) in this general setting, hence this issue is postponed to future work.Moreover, unlike the classical approach when J is of class C 1 , where the existence of a constrained minimizer for J and, consequently, a solution to (1.1)-(1.2) is a by-product of the aforementioned relative compactness, here we prove first the existence of a solution to (1.1)-(1.2) and then use this fact to prove the compactness result.
When g is as in (1.5), we additionally have the following non-existence result.
Theorem 1.6.Let g be given by (1.5) with α > 0, µ ∈ R, and Remark 1.7.In Theorem 1.1, we look for (ground state) solutions with sufficiently large ρ because this implies that the energy is negative, which is used, in turn, to ensure such an existence.Then, as we pointed out in Remark 1.3, a consequence is that λ > 0. On the other hand, in the context of Theorem 1.6 (i) and µ > 0, ground state solutions still exist for all ρ > 0 as in Corollary 1.9 below, but with no information about the Lagrange multiplier or the energy.
We follow the approach of [24] and consider a family of approximating problems.Note that the approximations are different from those in [15], where the nonlinearity is modified in a neighbourhood of infinity, and are more similar to those in [14], although not the same as the nonlinearity therein is of the form (1.4).
Let g − (s) := g + (s) − g(s) and G − (s) := G + (s) − G(s) ≥ 0 for s ∈ R. In view of (g1) and (g3), ), but it is not of class C 1 in general.In order to overcome this problem, for every ε ∈ (0, 1) let us take an even function dt, s ∈ R, and now observe that G ε − (s) ≤ c ε |s| 2 for every |s| ≤ 1 and some constant c ε > 0 depending only on ε > 0. Hence, for ε ∈ (0, 1), J ε is of class The idea of considering the L 2 -disc D instead of the L 2 -sphere S was introduced in [5] in a context where the nonlinear term is mass-supercritical and Sobolev-subcritical at infinity (i.e., lim |s|→∞ G(s)/|s| 2+4/N = ∞, lim |s|→∞ G(s)/|s| 2 * = 0) and mass-critical or -supercritical at the origin (i.e., lim sup s→0 G(s)/|s| 2+4/N < ∞); later on, it was exploited again in [25], in a similar context that allows systems with Sobolev-critical nonlinearities, in [28], for equations and systems of equations in the mass-critical or -subcritical setting, and in [6], where polyharmonic operators and Hardy-type potentials are considered.One of the reasons for this choice was that the embedding H 1 (R N ) ֒→ L 2 (R N ) is not compact, even if considering radial functions or with symmetries as in [20], which, in turn, causes that the limit point of a weakly convergent sequence in S need not belong to S; on the contrary, limit points of weakly convergent sequences in D stay in D, and the additional information that such a weak limit point belongs to the set one is working with proves to be useful.In this paper, this strategy is fundamental: it allows the critical point found in Theorem 2.1 below -the existence result for the perturbed problem -to be a minimizer of the corresponding energy functional over the whole set D, and this fact plays an important role in the proof of Theorem 1.1 because the "candidate minimizer" of J| D might not, initially, belong to S. Now, we turn to the problem of finding multiple solutions (in fact, infinitely many) to (1.1)-(1.2).In this case, we need to change both the assumptions about g and our approach.The idea is to build a subspace of H 1 (R N ) in the spirit of H, but with a more general setting.The issues with using the previous approach to obtain an infinite sequence of solutions are mainly two.First, even at the level of the perturbed functional, one can obtain an arbitrary large number of solutions if ρ is sufficiently large, but it is unclear whether infinitely many solutions exist; this is caused by the lack of compact embedding into L 2 (R N ), which in turn raises the need for each of the usual minimax values (cf.Theorem 4.9 below) to be negative, for which we need ρ to be large (and how large depends differently on each minimax level).Second, fixed a family (depending on ε) of critical values of the perturbed functional obtained in the way we have just briefly mentioned, and considering the corresponding family of critical points, it is not clear whether this family converges to a critical point of the unperturbed functional, unlike what happens in the Proof of Theorem 1.1, where we can exploit the information that the critical levels are minima.
We assume that the right-hand side in (1.1) is given by (1.9) together with the following assumptions, where that satisfies the ∆ 2 and ∇ 2 conditions globally and such that If N = 2, then for all q ≥ 2 and α > 4π there holds |f (s)| a(s) Observe that (f3) implies

Let us define
If N = 4 or N ≥ 6, with the purpose of finding non-radial solutions (cf.[1,23] For 2 < p < 2 * , let us denote by C N,p the best constant in the Gagliardo-Nirenberg inequality: (1.11) Our multiplicity result reads as follows.

By setting
with α > 0, and µ, p in the right range, we check that (A) and (f0)-(f3) hold and we recover the case (1.5).
Corollary 1.9.Let g be given by (1.5) with α > 0, µ ≥ 0, and 2 < p ≤ 2 + 4/N .If (1.12) holds, then there exists a solution (u, λ) In particular, we get a solution to (1.1)-(1.2) with (1.5) for any ρ > 0 provided that 2 < p < 2 + 4/N .Moreover, (f3) allows us to handle terms g that grow "too fast" to be included in the usual H 1 (R N ) setting.For example, letting with α > 0, we can allow f − (s) ≈ |s| p−2 s for |s| ≫ 1 for every p > 2, even when N ≥ 3 and p > 2 * .At the same time, we can treat nonlinearities that are "more singular" at the origin, for example, letting and taking any f that satisfies (f0)-(f3).The price to pay for this different approach is that we need (1.9) and (A) to construct a proper functional setting (i.e., an Orlicz one); this prevents us from dealing, e.g., with terms that behave like 1/ ln s 2 near the origin, which instead are allowed by the use of the approximating problems to find least-energy solutions.On the other hand, we can weaken the assumptions about f , the "remaining" part of the nonlinearity: it can be controlled by a instead of the usual conditions with an H 1 (R N )-setting (in particular, we can deal with Sobolevsupercritical nonlinearities), we do not need F > A anywhere, f + can have a mass-critical growth at infinity (provided ρ is small), and there are no restrictions on ρ if f + has a mass-subcritical growth at infinity; none of these scenarios can be handled with the previous approach.Except for [30, Theorems 1.2 and 1.3], where the nonlinear term has the very specific form (1.5), Theorem 1.8 seems to be the first multiplicity result when normalized solutions are sought in the strongly sublinear regime, and also the first time at all that more than two solutions are obtained.The paper is structured al follows: in Section 2, we obtain least-energy solutions to the perturbed problem; in Section 3, we prove Theorems 1.1, 1.5, 1.6, and Proposition 1.4; in Section 4, we obtain infinitely many solutions to (1.1)-(1.2).

Solutions to the perturbed problem
Recall from (1.8) the definition of J ε : H 1 (R N ) → R. The main purpose of this section is to prove the following result, where we denote c ε (ρ) := inf D(ρ) J ε .
Proof.Arguing as in [2, Proof of Theorem 2] and observing that There holds From the definition of c ε , which is finite due to Lemma 2.2, there exist In virtue of the translation invariance, we can assume that u 1 and u 2 have disjoint supports.Then and so The following lemma is inspired from [29, Lemma 3.2 (ii)].
Remark 2.8.The proof of Theorem 2.1 shows that every minimizing sequence for c ε (ρ) is relatively compact in H 1 (R N ) up to a translation.As a matter of fact, from

Multiple solutions
4.1.Functional setting.Let us recall from [27] that a function A : R → R is called an N-function if and only if it is nonnegative, even, convex, and satisfies It is said to satisfy the ∆ 2 condition globally if and only if there exists K > 0 such that A(2s) ≤ KA(s) for all s ∈ R.
It is said to satisfy the ∇ 2 condition globally if and only if there exist ℓ > 1 such that 2ℓA(s) ≤ A(ℓs) for all s ∈ R.
From now on, A shall denote the same function as in (A).We can define the Orlicz space associated with A as ) .If we define the norm u V := inf κ > 0 : [27,Theorem IV.I.10]).Moreover, the following holds true.

Note that
We obtain the following variant of Lions' lemma.We prove the convergence in L 2 (R N ).Take any p ∈ (2, 2 * ) and ε > 0. We find δ > 0 such that Hence, we get In view of Lemma 4.1 (ii), we conclude by letting ε → 0.
Let O be any subgroup O(N ) such that R N is compatible with O (cf. [20]), i.e., lim |y|→∞ m(y, r) = ∞ for some r > 0, where, for y ∈ R N , m(y, r) := sup n ∈ N : there exist g 1 , . . ., g n ∈ O such that B(g i y, r) ∩ B(g j y, r) = ∅ for i = j .
In view of [20], for some sequence y n ∈ R N and a constant c.Observe that in the family {B(hy n , 1)} h∈O we find an increasing number of disjoint balls provided that |y n | → ∞.Since u n is bounded in L 2 (R N ) and invariant with respect to O, by (4.3) (y n ) must be bounded.Then for sufficiently large r one obtains and we get a contradiction with the convergence of u n in L 2 loc (R N ).Therefore by Lemma 4.2 we conclude.
Finally, we have the following result.where C > 1 is the constant given in the characterization of the ∆ 2 condition -i.e., (4.1).Then, for every u, v ∈ V , we have Now we can use the same argument as in [12,Lemma 2.1] to obtain that the functionals u → R N A(u) dx and u → R N F (u) dx belong to C 1 (W ).The remaining part is obvious.4.2.Proof of Theorem 1.8.We recall the definitions F − := F + − F and f − := F ′ − .Lemma 4.5.If (A), (f0)-(f3), and (1.12) hold, then J| W ∩S is coercive and bounded below.
Proof.For every δ > 0, there exists C δ > 0 such that for every s ∈ R For every u ∈ W ∩ S there holds We conclude by Lemma 4.1 (ii) and taking δ so small that 2 O is the dual space of W O .Testing (4.4) with u n , we obtain that λ n is bounded as well, hence there exists λ ∈ R such that λ n → λ up to a subsequence, and (u, λ) is a solution to (1.1).Finally, from the Nehari identity and the fact that λ n → λ, we obtain which, together with Remark 4.6, implies that |∇u n | 2 2 → |∇u| 2 2 and R N a(u n )u n dx → R N a(u)u dx.It remains to prove that u n − u V → 0. In virtue of Lemma 4.1(iii), it suffices to prove that R N A(u n ) dx → R N A(u) dx.Additionally, since A satisfies the ∇ 2 condition globally, this occurs if the sequence a(u n )u n is bounded above by an integrable function, which holds true from (A) and [7,Example (b)].
We make use of the following abstract theorem, where G stands for the Krasnoselsky genus [32,Chapter 5].Proof.This theorem is basically (part of) [17, Theorem 2.1] (see also [26]), so we omit the proof.The only difference is that the values c k are critical regardless of their sign, which is a consequence of I| M satisfying the Palais-Smale condition at any level.
Proof of Theorem 1.8.Let us set E = W O(N ) (respectively, E = X ), H = L 2 (R N ), R = ρ 2 , M = S ∩E, and I = J| E .From Lemmas 4.5 and 4.7, I| M = J S∩E is bounded below and satisfies the Palais-Smale condition; moreover, from Lemma 4.8, π k (S k−1 ) ∈ Γ k (respectively, πk (S k−1 ) ∈ Γ k ) for every k, so the numbers c k are finite.Applying Theorem 4.9, we conclude the part about the existence of infinitely many solutions.Concerning the existence of a least-energy solution, we consider a sequence u n ∈ S ∩E such that lim n J(u n ) = inf S∩E J. From Ekeland's variational principle, we can assume that u n is a Palais-Smale sequence for J| S∩E , hence argue as above to obtain a solution (ū, λ) ∈ R×(S ∩E) to (1.1)-(1.2) such that J(ū) = min S∩E J.The fact that min S∩W O(N) J = min S∩W J follows from the properties of the Schwartz rearrangement [19,Chapter 3].

Remark 3 . 2 .
(i) The Proof of Theorem 1.1 contains the relevant result that c ε (ρ) → c(ρ) as ε → 0 + .(ii) Unlike the proof of Theorem 2.1, we cannot use the information λ > 0 to deduce u ∈ S because we do not know whether u is a critical point of J| D .Proof of Proposition 1.4.It follows from Lemma 2.3 and Remark 3.2 (i).

Proof.
The first two points follow from [27, Theorem III.IV.12 and Corollary III.IV.15] respectively, while the third one is a consequence of the previous two and [7, Theorem 2, Example (b)].

Lemma 4 . 7 .
If (A) and (f0)-(f3) hold, then J| W O ∩ S satisfies the Palais-Smale condition.Proof.Let u n ∈ W O ∩ S such that J(u n ) is bounded and J| ′ W O ∩S (u n ) → 0. From Lemma 4.5, u n is bounded in W , therefore there exists u ∈ W O such that u n ⇀ u in W O up to a subsequence.Then, from Corollary 4.3, u n → u in L p (R N ) for every p ∈ [2, 2 * ); in particular, u ∈ S. Up to a second subsequence, we can assume that u n → u a.e. in R N .Additionally, from [3, Lemma 3], there exist λ n ∈ R such that (4.4) −∆u n + λ n u n − g(u n )u n → 0 in W ′ O , where W ′
Let ρ > 0 be determined by Lemma 2.3 and consider u n and ρ n as in the statement.Then u n is bounded in view of Lemma 2.2.If lim n max y∈R N B 1 (y) |u n | 2 dx = 0, then lim n |u n | p = 0 for every 2 < p < 2 * due to Lions' lemma [21, Lemma I.1 in Part 2], hence, taking into account (g1) and (g3), 2 < p < 2 * .Proof.