Least energy solutions to a cooperative system of Schrödinger equations with prescribed L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-bounds: at least L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-critical growth

We look for least energy solutions to the cooperative systems of coupled Schrödinger equations -Δui+λiui=∂iG(u)inRN,N≥3,ui∈H1(RN),∫RN|ui|2dx≤ρi2i∈{1,…,K}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u)\quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3,\\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \end{array} \right. i\in \{1,\ldots ,K\} \end{aligned}$$\end{document}with G≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\ge 0$$\end{document}, where ρi>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _i>0$$\end{document} is prescribed and (λi,ui)∈R×H1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\lambda _i, u_i) \in {\mathbb {R}}\times H^1 ({\mathbb {R}}^N)$$\end{document} is to be determined, i∈{1,⋯,K}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{1,\dots ,K\}$$\end{document}. Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in L2(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {R}}^N)$$\end{document} of radii ρi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _i$$\end{document}, which allows to provide general growth assumptions about G and to know in advance the sign of the corresponding Lagrange multipliers. We assume that G has at least L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-critical growth at 0 and admits Sobolev critical growth. The more assumptions we make about G, N, and K, the more can be said about the minimizers of the corresponding energy functional. In particular, if K=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=2$$\end{document}, N∈{3,4}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \{3,4\}$$\end{document}, and G satisfies further assumptions, then u=(u1,u2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u=(u_1,u_2)$$\end{document} is normalized, i.e., ∫RN|ui|2dx=ρi2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{{\mathbb {R}}^N} |u_i|^2 \, dx=\rho _i^2$$\end{document} for i∈{1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{1,2\}$$\end{document}.


Introduction
We consider the following system of autonomous nonlinear Schrödinger equations of gradient type (1.1) with u = (u 1 , . . ., u K ) : R N → R K , which arises in different areas of mathematical physics.In particular, the system (1.1) describes the propagation of solitons, which are special nontrivial solitary wave solutions Φ j (x, t) = u j (x)e −iλ j t to a system of time-dependent Schrödinger equations of the form (1.2) i ∂Φ j ∂t − ∆Φ j = g j (Φ) for j = 1, . . ., K, where, for instance, g j are responsible for the nonlinear polarization in a photonic crystal [2,34] and λ j are the external electric potentials.
The following L 2 -bounds for Φ will be studied: Problems with prescribed masses ρ 2 j (the former constraint) appear in nonlinear optics, where the mass represents the power supply, and in the theory of Bose-Einstein condensates, where it represents the total number of atoms (see [1,17,19,27,30,32,41]). Prescribing the masses make sense also because they are conserved quantities in the corresponding evolution equation (1.2) together with the energy (see the functional J below), cf.[13,14].As for the latter constraint, we propose it as a model for some experimental situations, e.g. when the power supply provided can oscillate without exceeding a given value.
Recall that a general class of autonomous systems of Schrödinger equations was studied by Brezis and Lieb in [12] and using a constrained minimization method they showed the existence of a least energy solution, i.e., a nontrivial solution with the minimal energy.Their method using rescaling arguments does not apply with the L 2 -bounds.
We shall provide suitable assumptions under which the solutions to (1.3) (resp.(1.4)) are critical points of the energy functional J : H 1 (R N ) K → R defined as restricted to the constraint D (resp.S) with Lagrange multipliers λ i ∈ R, i.e., they are critical points of for some λ = (λ 1 , . . ., λ K ) ∈ R K .Let us recall that, under mild assumptions on G, see [12,Theorem 2.3], every critical point of the functional above belongs to W 2,q loc (R N ) K for all q < ∞ and satisfies the Pohožaev [10,22,31,33] and Nehari identities.By a linear combination of the two equalities above it is easily checked that every solution satisfies where H(u) := g(u), u − 2G(u) ( •, • is the scalar product in R K ) and g := ∇G, see e.g.[22].Hence we introduce the constraint which contains all the nontrivial solutions to (1.3) or (1.4) and does not depend on λ.Observe that every nontrivial solution to (1.3) belongs to M∩D and every (nontrivial) solution to (1.4) belongs to M ∩ S ⊂ M ∩ D. By a ground state solution to (1.3) we mean a nontrivial solution which minimizes J among all the nontrivial solutions.In particular, if (λ, u) solves (1.3) and J(u) = inf M∩D J, then (λ, u) is a ground state solution (cf.Theorems 1.1 and 1.2).By a ground state solution to (1.4) we mean that (λ, u) solves (1.4) and J(u) = inf M∩D J (cf.Theorems 1.2, 1.3, and Corollary 1.4).Note that this is more than just requiring J(u) = inf M∩S J, which, on the other hand, appears as a more "natural" requirement.
Working with the set D instead of the set S for a system of Schrödinger equations seems to be new and has, among others, a specific advantage related to the sign of the Lagrange multipliers λ i .We begin by showing why this issue is important.First of all, from a physical point of view there are situations, e.g.concerning the eigenvalues of equations describing the behaviour of ideal gases, where the chemical potentials λ i have to be positive, see e.g.[27,32].In addition, from a mathematical point of view the (strict) positivity of such Lagrange multipliers often plays an important role in the strong convergence of minimizing sequences in L 2 (R N ), see e.g.[6, Lemma 3.9]; finally, the nonnegativity is used in some of the proofs below, e.g. the one of Lemma 2.11 (a).The aforementioned advantage is as follows: in [15], Clarke proved that, in a minimization problem, Lagrange multipliers related to a constraint given by inequalities have a sign, i.e., λ i ≥ 0; therefore it is enough to rule out the case λ i = 0 in order to prove that λ i > 0 for every i ∈ {1, . . ., K}; note that ruling out the case λ i = 0 is simpler than ruling out the case λ i ≤ 0, cf. the proof of Lemma 2.11 (b).The nonnegativity/positivity of the Lagrange multipliers of (1.4) has often been obtained by means of involved tools (or at the very minimum in a not-so-straightforward way), such as stronger variants of Palais-Smale sequences in the spirit of [22] as in [6,Lemma 3.6, proof of Theorem 1.1] or preliminary properties of the ground state energy map ρ → inf M∩S J as in [24, Lemma 2.1, proof of Lemma 4.5].Our argument, based on [15], is simple, does not seem to be exploited in the theory of normalized solutions, and is demonstrated in Proposition A.1 in an abstract way for future applications, e.g. for different operators in the normalized solutions setting like the fractional Laplacian [25,29].
A second, but not less important, advantage of considering the set D concerns the property that the ground state energy in the Sobolev-critical case is below the ground state energy of the limiting problem, cf.(1.9).More precisely, since in dimension N ∈ {3, 4} the Aubin-Talenti instantone is not L 2 -integrable, we need to truncate it by a cut-off function and then project it into D; however, unless K = 1, we cannot ensure that such a projection lies on S, hence the use of D is necessary for this argument.See the proof of Proposition 2.6 (ii) for further details.
Recall that, when K = 1 and ) is equivalent to the corresponding problem with fixed λ > 0 (and without the L 2 -bound) via a scaling-type argument.This approach fails in the case of nonhomogeneous nonlinearities or when K ≥ 2. In the L 2 -subcritical case, i.e., when G(u) ∼ |u| p with 2 < p < 2 N , one can obtain the existence of a global minimizer by minimizing directly on S, cf.[28,39].In the L 2critical (p = 2 N ) and the L 2 -supercritical and Sobolev-subcritical (2 N < p < 2 * := 2N N −2 ) cases this method does not work; in particular, if p > 2 N in (1.5), then inf S J = −∞.The purpose of this work is to find general growth conditions on G in the spirit of Berestycki, Lions [10] and Brezis, Lieb [12] as well as involving the Sobolev critical terms, and to provide a direct approach to obtain ground state solutions to (1.3), (1.4), and similar elliptic problems.The problem (1.4) for one equation was studied by Jeanjean [22] and by Bartsch and Soave [7,8] with a general nonlinear term satisfying the following condition of Ambrosetti-Rabinowitz type: there exist 4 In [22] the author used a mountain pass argument, while in [7,8] a mini-max approach in M based on the σ-homotopy stable family of compact subsets of M and the Ghoussoub minimax principle [20] were adopted.The same topological principle has been recently applied to the system (1.4) with particular power-like nonlinearities, e.g. in [5][6][7][8], and by Jeanjean and Lu [23] for K = 1 and a general nonlinearity without (1.6), but with L 2 -supercritical growth.We stress that the lack of compactness of the embedding H 1 rad (R N ) ⊂ L 2 (R N ) causes troubles in the analysis of L 2 -supercritical problems and makes the argument quite involved, see e.g.[7,8,22].A possible strategy to recover the compactness of Palais-Smale sequences, at least when K = 1, is to show that the ground state energy map is nonincreasing with respect to ρ > 0 and decreasing in a subinterval of (0, ∞), see e.g.[9,23].
In our approach we do not work in H 1 rad , with Palais-Smale sequences, or with (1.6), nor the monotonicity of the ground state energy map is required, so that we avoid the mini-max approach in M involving a technical topological argument based on [20], which has been recently intensively exploited by many authors e.g. in [5-8, 23-25, 29, 35, 36].
In particular, we work with a weaker version of (1.6), see the condition (A5) below, and we admit L 2 -critical growth at 0. We make use of a minimizing sequence of J| M∩D and we are able to consider a wide class of nonlinearities G.In the first part of this work, we adapt the techniques of [11] to the system (1.3) and the Sobolev-critical case, which ensure that the minimum of J on M ∩ D is attained.If G is even, we exploit the Schwarz rearrangement u * := (u * 1 , . . ., u * K ) of (|u 1 |, . . ., |u K |) because, if u ∈ M ∩ D, then u * can be projected onto the same set without increasing the energy.Next, we point out that dealing with systems (1.3) and (1.4) one has to involve more tools in order to find a ground state u ∈ M ∩ ∂D and some additional restrictions imposed on G, N, or K will be required.In particular, if we want to ensure that the Lagrange multipliers are positive and u ∈ S, we use the elliptic regularity results contained in [10,12], the Liouville type result [21], and Proposition A.1.Finally, a multi-dimensional version of the strict monotonicity of the ground state energy map is simply obtained in Proposition 2.14 as a consequence of our approach.
For 2 < p ≤ 2 * , let C N,p > 0 be the optimal constant in the Gagliardo-Nirenberg inequality where δ p = N We assume there exists θ ∈ (0, ∞) K or θ = 0 such that G is of the form Observe that in view of (A2) and (A5), G(u) ≥ G(u) > 0 and H(u) ≥ H(u) > 0 for u = 0. Indeed, take any v ∈ R K such that |v| = 1 and note that (A5) implies that Since (A2) holds, we get G(tv) > 0 for sufficiently large t > 0, hence taking into account the above inequalities we obtain that G(tv) > 0 for all t > 0 and we conclude.In particular, M = ∅.Moreover, M is a C 1 -manifold, since M ′ (u) = 0 for u ∈ M, cf.[33].As a matter of fact, if M ′ (u) = 0, then u solves −∆u = N 4 h(u) and satisfies the Pohožaev identity We introduce the following relation: and for better outcomes we need the following stronger variant of (A4): h(u), u .From now on we assume the following condition and the first main result concerning (1.3) reads as follows.
(a) There exists u ∈ M ∩ D such that J(u) = inf M∩D J.In addition, u is a K-tuple of radial, nonnegative and radially nonincreasing functions provided that G is of the form , and for every j there exists As we shall see in Section 2, (1.9) is verified if N ≥ 5 or if N ∈ {3, 4} and an additional mild condition holds, see Proposition 2.6 (see also Lemma 2.7).We point out that part (b) holds regardless of whether G is of the form (1.10) or not.If this is the case, then u has the additional properties as in part (a).
Notice that (A1) allows G to have L 2 -critical growth G(u) ∼ |u| 2 N at 0, but (A2) excludes the same behaviour at infinity.Moreover, G consists of the Sobolev-subcritical part in view of (A3).Finally, the pure L 2 -critical case for |u| small is ruled out by (A4, ), i.e., G(u) = G(u) cannot be of the form (1.10) with G i (u) = α i |u| 2 N , α i ≥ 0, and K i=1 r i,j = 2 N for every j.
Here and later on, when we say G is of the form (1.10), we also mean the additional conditions on G i , β j , and r i,j listed in Theorem 1.1 (a).Observe that G of the form (1.10) satisfies (A4) if and only if G i satisfies the scalar variant of (A4) for all i ∈ {1, . . ., K}. If, in addition, G i satisfies (A4, ) for some i, then G satisfies (A4, ) as well.
If K = 2, L = 1, and the coefficient of the coupling term is large, then we find ground state solutions to (1.4).
as well as where the nonlinearity is the sum of power-type nonlinerites including the Sobolev critical terms of the form In view of Proposition 2.6 (ii), taking p = 2 N or p = 2 * we easily check that (1.11) and (1.12) satisfy (1.9) and we obtain a ground state solution to (1.4) for any µ i + ν i > 0 and θ i > 0, i = 1, 2. As for other possible examples of scalar functions G 1 , G 2 we refer to (E1)-(E4) in [11].See also example (2.5).
Moreover, if K = 1 and L = 0 (i.e., there is no coupling term), then we find ground state solutions to the scalar problem (1.4) taking into account a general nonlinearity involving at least L 2 -critical and at most Sobolev-critical growth.
Proof.First of all note that, if u ∈ M, then due to (A5) and define Then there exist (y It follows that and so, from Lions' Lemma [28], for every s > 0. Taking into account that we have that lim inf n J(u (n) ) ≥ s 2 /2 for every s > 0, i.e., lim n J(u (n) ) = ∞.Proof.We prove that there exists α > 0 such that From (1.7) and (1.8), for every ε > 0 there exists c ε > 0 such that Choosing and so J(u) ≥ |∇u| 2 2

2N
. Now take u ∈ M ∩ D and α > 0 such that (2.3) holds and define and w := s ⋆ u.
From now on, c > 0 will stand for the infimum of J over M ∩ D.
, where B r stands for the closed ball centred at 0 of radius r.For every ε > 0 define where ρ := min j∈{1,...,K} ρ j , and recall (cf., e.g., [38, p. 179], [36, Lemma A.1]) that where C N > 0 depends only on N and φ.Note that for some constant C > 0 and sufficiently small ε > 0, where r ∈ {p, q} and χ A stands for the characteristic function of A. Indeed, let and we conclude, since u 1 0 ∈ L r (R N ).Define s ε > 0 such that s ε * v ε ∈ M. In a similar way to the proof of Lemma 2.2, for every δ > 0 there exists C δ > 0 not depending on ε such that is an equivalent norm in L 2 * (R N ) K , i.e., taking δ sufficiently small and denoting m .

Proof. (a)
We prove that c → 0 as ρ → ∞ (note that (1.8) is satisfied for every ρ > 0 because η = 0).Let ρ n → ∞ and take u ∈ L ∞ (R N ) such that |u| 2 = 1.Without loss of generality we may assume that ρ n > 1 and define u n := ρ n u so that |u n | 2 = ρ n .From Lemma 2.3 there exists s n > 0 such that v n := s so it is enough to show that s n ρ n → 0. Note that There follows that From (A5) and the fact that lim t→0 G(t)/|t| 2 * = ∞, there follows that for sufficiently small |s|.Then, taking into account that u ∈ L ∞ (R N ), for sufficiently large n and s n ρ n → 0 as n → ∞, which completes the proof.
(b) Take any u 0 ∈ D \ {0} and note that (2.2) holds.In view of Lemma 2.3 there exists Observe that the latter expression is finite due to Lemma 2.3 with θ = 0. Hence we can take θ 0 > 0 so small that, if θ i < θ 0 , then K j=1 θ S N/2 /N is greater than the right-hand side of the formula above.
We give explicit examples of nonlinearities that do not satisfy the assumptions of Proposition 2.6.Let N = 3 and ε > 0 be sufficiently small.If g(u) = g 1 (u) = min{|u| 4−ε , |u| 4/3 }u and if θ = θ 1 is not sufficiently small, then we can use Lemma 2.7 (a) provided that ρ = ρ 1 is sufficiently large, but not part (b).If G is of the form (1.10) and (2.5) g i (u) = min{|u| 4 , |u| 4/3+ε }u and if K = 2 or ρ is not sufficiently large, then we can use Lemma 2.7 (b) provided that θ i is sufficiently small for some i ∈ {1 . . ., K}, but not part (a).In view of Lemma 2.4, any minimizing sequence (u (n) ) ⊂ M∩D such that J(u (n) ) → c > 0 is bounded.By the standard concentration-compactness argument [28], u (n) ⇀ ũ for some ũ = 0 up to a subsequence and up to translations.It is not clear, however, if J(ũ) = c or ũ ∈ M ∩ D. Note that we can find R > 0 such that ũ(R•) ∈ M and in order to ensure that J(ũ) = c and ũ ∈ D we need to know that R ≥ 1.The latter crucial condition requires the profile decomposition analysis of (u (n) ) provided by the following lemma.
Passing to a subsequence we set ν so we obtain a contradiction and I = ∅.Claim 2. For every i ∈ I there holds (passing to a subsequence) and the reverse inequality holds, where as n → ∞.Again, passing to a subsequence, and, since holds for a.e.n, then from (2.11) and the fact that ν > 0 we get R n → 1. If, passing to a subsequence, which implies that R n → 1 as claimed.Therefore we have that (2.13) which is a contradiction.Therefore u (n) (• + y (i,n) ) → ũ in D 1,2 (R N ) K (which, together with (2.7), implies that I is a singleton) and, consequently, in for some C > 0 and t = 2 * −2 N 2 * −2 .Hence ũ ∈ M ∩ D, and, arguing as before but with R = 1, J(ũ) = c.Now we consider the case θ = 0 and in a similar way we prove Claim 1 and Claim 2 by getting a contradiction in (2.9) and (2.13).Finally note that arguments of Conclusion apply in the case θ = 0 as well.
For f : R N → R measurable we denote by f * the Schwarz rearrangement of |f |.Likewise, if A ⊂ R N is measurable, we denote by A * the Schwarz rearrangement of A [10,26].

.
Then there exist a = a(u) > 0 and b = b(u) ≥ a such that each s ∈ [a, b] is a global maximizer for ϕ and ϕ is increasing on (0, a) and decreasing on (b, ∞).Moreover, s ⋆ u ∈ M if and only if s ∈ [a, b], M(s ⋆ u) > 0 if and only if s ∈ (0, a), and M(s ⋆ u) < 0 is and only if s > b.If (A4, ) holds, then a = b.

1 / 2 n−
u (n) (s n •+y n )) ⇀ u in D 1,2 (R N ) K and s 1/2 n u (n) (s n •+y n )) → u a.e. in R N up to a subsequence.In particular, I ′ (u) = 0 and so u ∈ N .Observe that each component of u is of the formu j = θ u 0 j is an Aubin-Talenti instanton.Therefore S = R N |∇u| 2 dx K j=1 θ j R N |u j | 2 * dx 1 2 − 1 p and δ p p > 2 (resp.δ p p = 2, δ p p < 2) if and only if p > 2 N (resp.p = 2 N , p < 2 N ).Here and in what follows we denote by |u| k