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Least energy solutions to a cooperative system of Schrödinger equations with prescribed \(L^2\)-bounds: at least \(L^2\)-critical growth

Abstract

We look for least energy solutions to the cooperative systems of coupled Schrödinger equations

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u)\quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3,\\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \end{array} \right. i\in \{1,\ldots ,K\} \end{aligned}$$

with \(G\ge 0\), where \(\rho _i>0\) is prescribed and \((\lambda _i, u_i) \in {\mathbb {R}}\times H^1 ({\mathbb {R}}^N)\) is to be determined, \(i\in \{1,\dots ,K\}\). 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 \(L^2({\mathbb {R}}^N)\) of radii \(\rho _i\), 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 \(L^2\)-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 \(K=2\), \(N\in \{3,4\}\), and G satisfies further assumptions, then \(u=(u_1,u_2)\) is normalized, i.e., \(\int _{{\mathbb {R}}^N} |u_i|^2 \, dx=\rho _i^2\) for \(i\in \{1,2\}\).

Introduction

We consider the following system of autonomous nonlinear Schrödinger equations of gradient type

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_1 + \lambda _1 u_1 = \partial _1 G(u)\\ \cdots \\ -\Delta u_K + \lambda _K u_K = \partial _K G(u)\\ \end{array} \quad \hbox {in } {\mathbb {R}}^N\right. \end{aligned}$$
(1.1)

with \(u=(u_1,\dots ,u_K):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^K\), 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 \(\Phi _j(x,t)=u_j(x)e^{-\mathrm {i}\lambda _j t}\) to a system of time-dependent Schrödinger equations of the form

$$\begin{aligned} \mathrm {i}\frac{\partial \Phi _j}{\partial t} -\Delta \Phi _j = g_j(\Phi )\quad \hbox {for }j=1,\dots ,K, \end{aligned}$$
(1.2)

where, for instance, \(g_j\) are responsible for the nonlinear polarization in a photonic crystal [2, 34] and \(\lambda _j\) are the external electric potentials.

Another field of application is condensed matter physics, where (1.1) comes from the system of coupled Gross–Pitaevski equations (1.2) with nonlinearities of the form

$$\begin{aligned} g_j(\Phi )=\left( \sum _{k=1}^K\beta _{j,k}|\Phi _k|^2\right) \Phi _j\quad \hbox {for }j=1,\dots ,K. \end{aligned}$$

The following \(L^2\)-bounds for \(\Phi \) will be studied:

$$\begin{aligned} \int _{{\mathbb {R}}^N}\left| \Phi _j(t,x)\right| ^2\,dx=\rho _j^2 \quad \text { and } \quad \int _{{\mathbb {R}}^N}\left| \Phi _j(t,x)\right| ^2\,dx\le \rho _j^2. \end{aligned}$$

Problems with prescribed masses \(\rho _j^2\) (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.

Our aim is to provide a general class of nonlinearities and to find solutions to the nonlinear Schrödinger problems

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u) \quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3, \\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \end{array} \right. \text {for every }i\in \{1,\dots ,K\} \end{aligned}$$
(1.3)

and

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i + \lambda _i u_i = \partial _iG(u) \quad \mathrm {in} \ {\mathbb {R}}^N, \ N \ge 3, \\ u_i \in H^1({\mathbb {R}}^N), \\ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx = \rho _i^2 \end{array} \right. \text {for every }i\in \{1,\dots ,K\}, \end{aligned}$$
(1.4)

where \(\rho =(\rho _1,\dots ,\rho _K)\in (0,\infty )^K\) is prescribed and \((\lambda ,u) \in {\mathbb {R}}^K\times H^1({\mathbb {R}}^N)^K\) is the unknown.

Let us introduce the sets

$$\begin{aligned}\begin{aligned} {{\mathcal {D}}}&:= \left\{ u \in H^1({\mathbb {R}}^N)^K \ : \ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \le \rho _i^2 \text { for every } i\in \{1,\dots ,K\} \right\} ,\\ {{\mathcal {S}}}&:= \left\{ u \in H^1({\mathbb {R}}^N)^K \ : \ \int _{{\mathbb {R}}^N} |u_i|^2 \, dx = \rho _i^2 \text { for every } i\in \{1,\dots ,K\} \right\} \end{aligned}\end{aligned}$$

and note that \({{\mathcal {S}}}\subset \partial {{\mathcal {D}}}\).

We shall provide suitable assumptions under which the solutions to (1.3) (resp. (1.4)) are critical points of the energy functional \(J:H^1({\mathbb {R}}^N)^K \rightarrow {\mathbb {R}}\) defined as

$$\begin{aligned} J(u) := \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx - \int _{{\mathbb {R}}^N} G(u) \, dx \end{aligned}$$

restricted to the constraint \({{\mathcal {D}}}\) (resp. \({{\mathcal {S}}}\)) with Lagrange multipliers \(\lambda _i \in {\mathbb {R}}\), i.e., they are critical points of

$$\begin{aligned} H^1({\mathbb {R}}^N)^K \ni u \mapsto J(u) + \frac{1}{2} \sum _{i=1}^K \lambda _i \int _{{\mathbb {R}}^N} |u_i|^2 \, dx \in {\mathbb {R}}\end{aligned}$$

for some \(\lambda =(\lambda _1,\dots ,\lambda _K) \in {\mathbb {R}}^K\). Let us recall that, under mild assumptions on G, see [12, Theorem 2.3], every critical point of the functional above belongs to \(W^{2,q}_loc ({\mathbb {R}}^N)^K\) for all \(q < \infty \) and satisfies the Pohožaev [10, 22, 31, 33]

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx = 2^* \int _{{\mathbb {R}}^N} G(u) - \frac{1}{2} \sum _{i=1}^K \lambda _i |u_i|^2 \, dx \end{aligned}$$

and Nehari

$$\begin{aligned} J'(u)(u)+\sum _{i=1}^K\lambda _i \int _{{\mathbb {R}}^N} |u_i|^2 \, dx = 0 \end{aligned}$$

identities. By a linear combination of the two equalities above it is easily checked that every solution satisfies

$$\begin{aligned} M(u):= \int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx - \frac{N}{2} \int _{{\mathbb {R}}^N} H(u) \, dx = 0, \end{aligned}$$

where \(H(u):= \langle g(u),u\rangle -2G(u)\) (\(\langle \cdot ,\cdot \rangle \) is the scalar product in \({\mathbb {R}}^K\)) and \(g:=\nabla G\), see e.g. [22]. Hence we introduce the constraint

$$\begin{aligned} {{\mathcal {M}}}:= \left\{ u \in H^1({\mathbb {R}}^N)^K \setminus \{0\}: M(u)=0 \right\} , \end{aligned}$$

which contains all the nontrivial solutions to (1.3) or (1.4) and does not depend on \(\lambda \). Observe that every nontrivial solution to (1.3) belongs to \({{\mathcal {M}}}\cap {{\mathcal {D}}}\) and every (nontrivial) solution to (1.4) belongs to \({{\mathcal {M}}}\cap {{\mathcal {S}}}\subset {{\mathcal {M}}}\cap {{\mathcal {D}}}\). By a ground state solution to (1.3) we mean a nontrivial solution which minimizes J among all the nontrivial solutions. In particular, if \((\lambda ,u)\) solves (1.3) and \(J(u)=\inf _{{{\mathcal {M}}}\cap {{\mathcal {D}}}}J\), then \((\lambda ,u)\) is a ground state solution (cf. Theorems 1.1 and 1.2). By a ground state solution to (1.4) we mean that \((\lambda ,u)\) solves (1.4) and \(J(u)=\inf _{{{\mathcal {M}}}\cap {{\mathcal {D}}}}J\) (cf. Theorems 1.2, 1.3, and Corollary 1.4). Note that this is more than just requiring \(J(u)=\inf _{{{\mathcal {M}}}\cap {{\mathcal {S}}}}J\), which, on the other hand, appears as a more “natural” requirement.

Working with the set \({{\mathcal {D}}}\) instead of the set \({{\mathcal {S}}}\) 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 \(\lambda _i\). 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 \(\lambda _i\) 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 \(L^2({\mathbb {R}}^N)\), 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., \(\lambda _i\ge 0\); therefore it is enough to rule out the case \(\lambda _i=0\) in order to prove that \(\lambda _i>0\) for every \(i\in \{1,\dots ,K\}\); note that ruling out the case \(\lambda _i=0\) is simpler than ruling out the case \(\lambda _i\le 0\), 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 \(\rho \mapsto \inf _{{{\mathcal {M}}}\cap {{\mathcal {S}}}}J\) 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 \({{\mathcal {D}}}\) 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 \(N\in \{3,4\}\) the Aubin–Talenti instantone is not \(L^2\)-integrable, we need to truncate it by a cut-off function and then project it into \({{\mathcal {D}}}\); however, unless \(K=1\), we cannot ensure that such a projection lies on \({{\mathcal {S}}}\), hence the use of \({{\mathcal {D}}}\) is necessary for this argument. See the proof of Proposition 2.6(ii) for further details.

Recall that, when \(K=1\) and

$$\begin{aligned} G(u)=\frac{1}{p}|u|^{p}, \quad 2<p<2^*,\;p\ne 2_N:=2+\frac{4}{N}, \end{aligned}$$
(1.5)

(1.4) is equivalent to the corresponding problem with fixed \(\lambda > 0\) (and without the \(L^2\)-bound) via a scaling-type argument. This approach fails in the case of nonhomogeneous nonlinearities or when \(K\ge 2\). In the \(L^2\)-subcritical case, i.e., when \(G(u)\sim |u|^{p}\) with \(2< p < 2_N\), one can obtain the existence of a global minimizer by minimizing directly on \({{\mathcal {S}}}\), cf. [28, 39]. In the \(L^2\)-critical (\(p = 2_N\)) and the \(L^2\)-supercritical and Sobolev-subcritical (\(2_N< p < 2^*:=\frac{2N}{N-2}\)) cases this method does not work; in particular, if \(p>2_N\) in (1.5), then \(\inf _{{\mathcal {S}}}J=-\infty \). 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 \(\frac{4}{N}<a\le b<2^*-2\) such that

$$\begin{aligned} 0 < aG(u)\le H(u)\le b G(u)\hbox { for }u \in {\mathbb {R}}\setminus \{0\}. \end{aligned}$$
(1.6)

In [22] the author used a mountain pass argument, while in [7, 8] a mini-max approach in \({{\mathcal {M}}}\) based on the \(\sigma \)-homotopy stable family of compact subsets of \({{\mathcal {M}}}\) 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 \(H_rad ^1({\mathbb {R}}^N)\subset L^2({\mathbb {R}}^N)\) 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 \(\rho >0\) and decreasing in a subinterval of \((0,\infty )\), see e.g. [9, 23].

In our approach we do not work in \(H_rad ^1\), 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 \({{\mathcal {M}}}\) involving a technical topological argument based on [20], which has been recently intensively exploited by many authors e.g. in [5,6,7,8, 23,24,25, 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 \(J|_{{{\mathcal {M}}}\cap {{\mathcal {D}}}}\) 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 \({{\mathcal {M}}}\cap {{\mathcal {D}}}\) is attained. If G is even, we exploit the Schwarz rearrangement \(u^*:=(u_1^*,\dots ,u_K^*)\) of \((|u_1|,\dots ,|u_K|)\) because, if \(u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\), 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 \(u\in {{\mathcal {M}}}\cap \partial {{\mathcal {D}}}\) 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 \(u\in {{\mathcal {S}}}\), 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 \(2<p\le 2^*\), let \(C_{N,p}>0\) be the optimal constant in the Gagliardo-Nirenberg inequality

$$\begin{aligned} |u|_p \le C_{N,p} |\nabla u|_2^{\delta _p} |u|_2^{1-\delta _p}\quad \hbox {for }u\in H^1({\mathbb {R}}^N), \end{aligned}$$
(1.7)

where \(\delta _p = N \big ( \frac{1}{2} - \frac{1}{p} \big )\) and \(\delta _pp>2\) (resp. \(\delta _pp=2\), \(\delta _pp<2\)) if and only if \(p>2_N\) (resp. \(p=2_N\), \(p<2_N\)). Here and in what follows we denote by \(|u|_k\) the \(L^k\)-norm of u, \(1\le k\le \infty \).

We assume there exists \(\theta \in (0,\infty )^K\) or \(\theta =0\) such that G is of the form

$$\begin{aligned} G(u)=\widetilde{G}(u)+\frac{1}{2^*}\sum _{j=1}^K\theta _j|u_j|^{2^*} \end{aligned}$$

for some \(\widetilde{G}:{\mathbb {R}}^K\rightarrow {\mathbb {R}}^N\). We set \(\widetilde{g}=\nabla \widetilde{G}\), \(\widetilde{H}(u)=\langle \widetilde{g}(u),u\rangle -2\widetilde{G}(u)\), \(\widetilde{h}=\nabla \widetilde{H}\), \(h:=\nabla H\), and consider the following assumptions:

  1. (A0)

    \(\widetilde{g}\) and \(\widetilde{h}\) are continuous and there exists \({\tilde{c}}>0\) such that \(|\widetilde{h}(u)|\le {\tilde{c}}(|u|+|u|^{2^*-1})\).

  2. (A1)

    \(\displaystyle \eta :=\limsup _{u\rightarrow 0}\frac{\widetilde{G}(u)}{|u|^{2_N}}<\infty \).

  3. (A2)

    If \(\theta =0\), then \(\displaystyle \lim _{|u|\rightarrow \infty }\frac{\widetilde{G}(u)}{|u|^{2_N}}=\infty \); if \(\theta \in (0,\infty )^K\), then \(\displaystyle \liminf _{|u|\rightarrow \infty }\frac{\widetilde{G}(u)}{|u|^{2_N}}>0\).

  4. (A3)

    \(\displaystyle \lim _{|u|\rightarrow \infty }\frac{\widetilde{G}(u)}{|u|^{2^*}}=0\).

  5. (A4)

    \(\displaystyle 2_N \widetilde{H}(u)\le \langle \widetilde{h}(u),u\rangle \).

  6. (A5)

    \(\displaystyle \frac{4}{N} \widetilde{G}\le \widetilde{H}\le (2^*-2) \widetilde{G}\).

Of course, \(\displaystyle \lim _{|u|\rightarrow \infty }\frac{G(u)}{|u|^{2_N}}=\infty \) if (A2) holds and GH satisfy (A1) – resp. (A4), (A5) – if so do \(\widetilde{G},\widetilde{H}\). Note that (A5) implies \(\widetilde{G}, \widetilde{H}\ge 0\). Note also that J and M are of class \({{\mathcal {C}}}^1\) if (A0) and (A5) are satisfied. For every \(u\in H^1({\mathbb {R}}^N)^K\) such that \(\int _{{\mathbb {R}}^N}H(u)\,dx>0\) we define

$$\begin{aligned} R:=R_u:=\sqrt{\frac{N\int _{{\mathbb {R}}^N}H(u)\,dx}{2\int _{{\mathbb {R}}^N}|\nabla u|^2\,dx}}>0 \end{aligned}$$

and note that \(u(R\cdot )\in {{\mathcal {M}}}\).

Observe that in view of (A2) and (A5), \(G(u)\ge \widetilde{G}(u)>0\) and \(H(u)\ge \widetilde{H}(u)>0\) for \(u\ne 0\). Indeed, take any \(v\in {\mathbb {R}}^K\) such that \(|v|=1\) and note that (A5) implies that

$$\begin{aligned}\begin{aligned} \widetilde{G}(v)t^{2^*} \ge \widetilde{G}(tv)&\ge \widetilde{G}(v)t^{2_N}\quad \hbox {if }t\ge 1,\\ \widetilde{G}(v)t^{2_N} \ge \widetilde{G}(tv)&\ge \widetilde{G}(v)t^{2^*}\quad \hbox { if }0<t\le 1. \end{aligned}\end{aligned}$$

Since (A2) holds, we get \(\widetilde{G}(tv)>0\) for sufficiently large \(t>0\), hence taking into account the above inequalities we obtain that \(\widetilde{G}(tv)>0\) for all \(t>0\) and we conclude. In particular, \({{\mathcal {M}}}\ne \emptyset \). Moreover, \({{\mathcal {M}}}\) is a \({{\mathcal {C}}}^1\)-manifold, since \(M'(u)\ne 0\) for \(u\in {{\mathcal {M}}}\), cf. [33]. As a matter of fact, if \(M'(u)=0\), then u solves \(-\Delta u = \frac{N}{4} h(u)\) and satisfies the Pohožaev identity \(\int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx = 2^* \frac{N}{4} \int _{{\mathbb {R}}^N} H(u) \, dx\). If \(M(u)=0\), then we infer \(u=0\).

We introduce the following relation:

  • Let \(f_1,f_2:{\mathbb {R}}^K\rightarrow {\mathbb {R}}\). Then \(f_1\preceq f_2\) if and only if \(f_1\le f_2\) and for every \(\varepsilon >0\) there exists \(u\in {\mathbb {R}}^K\), \(|u|<\varepsilon \), such that \(f_1(u)<f_2(u)\),

and for better outcomes we need the following stronger variant of (A4):

$$\begin{aligned} (\mathrm{A}4,\preceq ) 2_N\widetilde{H}(u)\preceq \langle \widetilde{h}(u),u\rangle \,\mathrm{if }\,\theta =0. \end{aligned}$$

Notice that (A4,\(\preceq \)) implies that \(2_N H(u)\preceq \langle h(u),u\rangle \).

From now on we assume the following condition

$$\begin{aligned} 2^*C_{N,2_N}^{2_N}\eta |\rho |^{4/N}<1, \end{aligned}$$
(1.8)

and the first main result concerning (1.3) reads as follows.

Theorem 1.1

Suppose (A0)–(A5) and (1.8) hold and, if \(\theta \in (0,\infty )^K\),

$$\begin{aligned} \inf _{{\mathcal {M}}\cap {\mathcal {D}}}J< \frac{1}{N}S^{N/2}\sum _{i=1}^K\theta _i^{1-N/2}. \end{aligned}$$
(1.9)
  1. (a)

    There exists \(u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\) such that \(J(u)=\inf _{{\mathcal {M}}\cap {\mathcal {D}}}J\). In addition, u is a K-tuple of radial, nonnegative and radially nonincreasing functions of class \(\mathfrak {C}^2\) provided that G is of the form

    $$\begin{aligned} G(u)=\sum _{i=1}^KG_i(u_i)+\sum _{j=1}^L\beta _j\prod _{i=1}^K|u_i|^{r_{i,j}}, \end{aligned}$$
    (1.10)

    where \(L\ge 1\), \(G_i:{\mathbb {R}}\rightarrow [0,\infty )\) is even, \(r_{i,j}>1\) or \(r_{i,j}=0\), \(\beta _j\ge 0\), \(2_N\le \sum _{i=1}^Kr_{i,j}<2^*\), and for every j there exists \(i_1\ne i_2\) such that \(r_{i_1,j}>1\) and \(r_{i_2,j}>1\).

  2. (b)

    If, moreover, (A4,\(\preceq \)) holds, then there exists \(\lambda =(\lambda _1,\dots .\lambda _K)\in [0,\infty )^K\) such that \((\lambda ,u)\) is a ground state solution to (1.3).

As we shall see in Sect. 2, (1.9) is verified if \(N\ge 5\) or if \(N\in \{3,4\}\) 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 \(G(u)\sim |u|^{2_N}\) at 0, but (A2) excludes the same behaviour at infinity. Moreover, \(\widetilde{G}\) 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,\(\preceq \)), i.e., \(G(u)=\widetilde{G}(u)\) cannot be of the form (1.10) with \(G_i(u)=\alpha _i |u|^{2_N}\), \(\alpha _i \ge 0\), and \(\sum _{i=1}^Kr_{i,j}=2_N\) for every j.

Here and later on, when we say G is of the form (1.10), we also mean the additional conditions on \(G_i\), \(\beta _j\), 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 \(i\in \{1,\ldots ,K\}\). If, in addition, \(G_i\) satisfies (A4,\(\preceq \)) for some i, then G satisfies (A4,\(\preceq \)) as well.

More can be said if \(N\in \{3,4\}\).

Theorem 1.2

Assume that (A0)–(A3), (A4,\(\preceq \)), (A5), and (1.8) are satisfied, G is of the form (1.10), \(N\in \{3,4\}\), and (1.9) holds if \(\theta \in (0,\infty )^K\). Then there exist \(u\in {{\mathcal {M}}}\cap \partial {{\mathcal {D}}}\) of class \({{\mathcal {C}}}^2\) and \(\lambda =(\lambda _1,\dots ,\lambda _K)\in [0,\infty )^K\) such that \((\lambda ,u)\) is a ground state solution to (1.3). In addition, each \(u_i\) is radial, nonnegative, and radially nonincreasing. Moreover, for every \(i\in \{1,\dots ,K\}\) either \(u_i=0\) or \(\int _{{\mathbb {R}}^N}|u_i|^2\,dx=\rho _i^2\) and, if \(u_i\ne 0\), then \(\lambda _i>0\) and \(u_i>0\). In particular, if \(u\in {{\mathcal {S}}}\), then \(\lambda \in (0,\infty )^K\) and \((\lambda ,u)\) is a ground state solution to (1.4).

Note that the obtained ground state solution u belongs to \(\partial {{\mathcal {D}}}\), i.e., at least one of the \(L^2\)-bounds must be the equality \(\int _{{\mathbb {R}}^N} |u_i|^2 \, dx=\rho _i^2\). In particular, ground states solutions can be semitrivial.

If \(K=2\), \(L=1\), and the coefficient of the coupling term is large, then we find ground state solutions to (1.4).

Theorem 1.3

Assume that (A0)–(A3), (A4,\(\preceq \)), (A5), and (1.8) are satisfied, \(N\in \{3,4\}\), \(K=2\), \(L=1\), and (1.9) holds if \(\theta \in (0,\infty )\). If G is of the form (1.10) and \(r_{1,1}+r_{2,1}>2_N\), then for every sufficiently large \(\beta _1>0\) there exists a ground state solution \((\lambda ,u)\in (0,\infty )^2\times {{\mathcal {S}}}\) to (1.4). Moreover, each component of u is positive, radial, radially nonincreasing and of class \({{\mathcal {C}}}^2\).

Observe that, if in Theorem 1.3\(G_i(t)=\mu _i|t|^{p_i}/p_i\) for some \(\mu _i>0\) and \(p_i\in (2_N,2^*)\), \(i\in \{1,2\}\), then clearly \(\eta =0\) in (1.8) and this result was very recently obtained by Li and Zou in [24, Theorem 1.3], again, unlike this paper, by means of the involved topological argument due to Ghoussoub [20], cf. [5,6,7,8, 23, 25, 29, 35, 36]. If \(\eta >0\) or \(\theta \in (0,\infty )^K\), the result seems to be new and we obtain a ground state solution to (1.4) for sufficiently small \(|\rho |\) in the former case, see (1.8), or under rather mild additional assumptions about \(\widetilde{G}\) in the latter, see Proposition 2.6. Furthermore, to our knowledge, this is the first result about normalized solutions to a system of Schrödinger equations where the nonlinearity is rather general, in particular not (entirely) of power-type, e.g.

$$\begin{aligned} \widetilde{G}_i(u_i)=\frac{\mu _i}{p_i} |u_i|^{p_i}\ln (1+|u_i|),\,\quad p_i\in [2_N,2^*-1],\mu _i>0, i\in \{1,2\} \end{aligned}$$
(1.11)

as well as where the nonlinearity is the sum of power-type nonlinerites including the Sobolev critical terms of the form

$$\begin{aligned} G_i(u_i)= & {} \frac{\nu _i}{2_N}|u_i|^{2_N}+\frac{\mu _i}{p_i}|u_i|^{p_i}\nonumber \\&+\frac{\theta _i}{2^*}|u_i|^{2^*},\quad p_i\in (2_N,2^*),\mu _i,\nu _i\ge 0, \mu _i+\nu _i>0, i\in \{1,2\}, \end{aligned}$$
(1.12)

where \(\displaystyle \eta =\frac{\max \{\nu _1,\nu _2\}}{2_N}\ge 0\). 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 \(\mu _i+\nu _i>0\) and \(\theta _i>0\), \(i=1,2\). As for other possible examples of scalar functions \(\widetilde{G}_1,\widetilde{G}_2\) 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.

Corollary 1.4

Assume that \(K=1\), (A0)–(A3), (A4,\(\preceq \)), (A5), and (1.8) are satisfied, and (1.9) holds if \(\theta \in (0,\infty )\). Assume as well that \(H\preceq (2^*-2)G\) or that \(N\in \{3,4\}\) and G is even. Then there exist \(u\in {{\mathcal {M}}}\cap {{\mathcal {S}}}\) and \(\lambda \in (0,\infty )\) such that \((\lambda ,u)\) is a ground state solution to (1.4). If G is even, then u is radial, positive, radially decreasing, and of class \(\mathfrak {C}^2\).

Recently, Soave considered (1.12) with \(\theta _1=0\) in [35] and with \(\theta _1>0\) but \(\nu _1=0\) or \(\mu _1=0\) in [36], with, additionally, an upper bound on \(\mu _1>0\) if \(N\ge 5\). In other recent papers, Wei and Wu [43] considered (1.12) with \(\theta _1>0\), \(\nu _1=0\), and no upper bound on \(\mu _1\), while Alves, Ji, and Miyagaki [3] considered (1.12) with \(\theta _1>0\), \(\nu _1=0\), and a lower bound on \(\mu _1\). Corollary 1.4 generalizes the results from [3, 36, 43] both because no bound on \(\mu _1\) is needed (upper or lower) and because the Sobolev-subcritical term \(\widetilde{G}\) can be \(L^2\)-critical, \(L^2\)-supercritical, or even both, without the need of consisting of (sums of) power functions. Of course, Corollary 1.4 also generalizes the results from [11, 23], which do not deal with the Sobolev-critical case.

Finally, observe that conditions (A0)–(A5) and (A4,\(\preceq \)) are positively additive, i.e., if \(\widetilde{G}\) and \(\widetilde{G}'\) satisfy the conditions with \(\eta \) and \(\eta '\) in (A1) respectively and \(\alpha ,\alpha '>0\), then \(\alpha \widetilde{G}+ \alpha '\widetilde{G}'\) satisfy the corresponding conditions with \(\displaystyle \limsup _{u\rightarrow 0}\frac{\alpha \widetilde{G}+ \alpha '\widetilde{G}'}{|u|^{2_N}}\le \alpha \eta +\alpha '\eta '\).

The proof

Lemma 2.1

Let \(f_1,f_2\in {{\mathcal {C}}}({\mathbb {R}}^K)\) and assume there exists \(C>0\) such that \(|f_1(u)|+|f_2(u)|\le C(|u|^2+|u|^{2^*})\) for every \(u\in {\mathbb {R}}^K\). Then \(f_1\preceq f_2\) if and only if \(f_1\le f_2\) and

$$\begin{aligned} \int _{{\mathbb {R}}^N}f_1(u)-f_2(u)\,dx<0 \end{aligned}$$

for every \(u\in H^1({\mathbb {R}}^N)^K\setminus \{0\}\).

Proof

We argue similarly as in the case \(K=1\) provided in [11, Lemma 2.1]. \(\square \)

We will always assume that (A0) holds. Lemmas 2.22.5 are variants of the results contained in [11, 23] with some improvements and adapted to the system of equations.

Lemma 2.2

If (A1)–(A3), (A5), and (1.8) hold, then \(\inf \{|\nabla u|_2^2:u\in {\mathcal {M}}\cap {\mathcal {D}}\}>0\).

Proof

Recall that, if \(p\in [2,2^*]\), then

$$\begin{aligned} \big ||u|\big |_p=|u|_p \text { and } \big |\nabla |u|\big |_2\le |\nabla u|_2 \text { for every }u\in H^1({\mathbb {R}}^N)^K. \end{aligned}$$

For every \(\varepsilon >0\) there exists \(c_\varepsilon >0\) such that for every \(u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\)

$$\begin{aligned}\begin{aligned} |\nabla u|_2^2&=\frac{N}{2}\int _{{\mathbb {R}}^N}H(u)\,dx\le 2^*\bigl (c_\varepsilon |u|_{2^*}^{2^*}+(\varepsilon +\eta )|u|_{2_N}^{2_N}\bigr )=2^*\bigl (c_\varepsilon \big ||u|\big |_{2^*}^{2^*}+(\varepsilon +\eta )\big ||u|\big |_{2_N}^{2_N}\bigr )\\&\le 2^*\Bigl (c_\varepsilon C_{N,2^*}^{2^*}\big |\nabla |u|\big |_2^{2^*}+(\varepsilon +\eta )C_{N,2_N}^{2_N}|\rho |^{4/N}\big |\nabla |u|\big |_2^2\Bigr )\\&\le 2^*\bigl (c_\varepsilon C_{N,2^*}^{2^*}|\nabla u|_2^{2^*}+(\varepsilon +\eta )C_{N,2_N}^{2_N}|\rho |^{4/N}|\nabla u|_2^2\bigr ) \end{aligned}\end{aligned}$$

i.e.,

$$\begin{aligned} 0\le 2^*c_\varepsilon C_{N,2^*}^{2^*}|\nabla u|_2^{2^*}+\bigl (2^*(\varepsilon +\eta )C_{N,2_N}^{2_N}|\rho |^{4/N}-1\bigr )|\nabla u|_2^2 \end{aligned}$$
(2.1)

Taking \(\varepsilon \) sufficiently small so that

$$\begin{aligned} 2^*(\varepsilon +\eta )C_{N,2_N}^{2_N}|\rho |^{4/N}<1 \end{aligned}$$

we conclude. \(\square \)

For \(u\in H^1({\mathbb {R}}^N)^K\setminus \{0\}\) and \(s>0\) define \(s\star u(x):=s^{N/2}u(sx)\) and \(\varphi (s):=J(s\star u)\).

Lemma 2.3

Assume that (A1)–(A5) hold and let \(u\in H^1({\mathbb {R}}^N)^K\setminus \{0\}\) such that

$$\begin{aligned} \eta <\frac{|\nabla u|_2^2}{2|u|_{2_N}^{2_N}}. \end{aligned}$$
(2.2)

Then there exist \(a=a(u)>0\) and \(b=b(u)\ge a\) such that each \(s\in [a,b]\) is a global maximizer for \(\varphi \) and \(\varphi \) is increasing on (0, a) and decreasing on \((b,\infty )\). Moreover, \(s\star u\in {{\mathcal {M}}}\) if and only if \(s\in [a,b]\), \(M(s\star u)>0\) if and only if \(s\in (0,a)\), and \(M(s\star u)<0\) if and only if \(s>b\). If (A4,\(\preceq \)) holds, then \(a=b\).

Note that (1.8) implies (2.2) provided that \(u\in {{\mathcal {D}}}\). Indeed, from (1.7)

$$\begin{aligned} 2\eta |u|_{2_N}^{2_N}\le 2\eta C_{N,2_N}^{2_N} |\nabla u|_2^{2} |u|_2^{4/N}\le 2\eta C_{N,2_N}^{2_N} |\nabla u|_2^{2} |\rho |^{4/N}<|\nabla u|_2^{2}. \end{aligned}$$

Proof

Notice that from (A1)

$$\begin{aligned} \varphi (s)=\int _{{\mathbb {R}}^N}\frac{s^2}{2}|\nabla u|^2-\frac{G(s^{N/2}u)}{s^N}\,dx\rightarrow 0 \end{aligned}$$

as \(s\rightarrow 0^+\) and from (A2) \(\lim _{s\rightarrow \infty }\varphi (s)=-\infty \). From (A1) and (A3) for every \(\varepsilon >0\) there exists \(c_\varepsilon >0\) such that

$$\begin{aligned} G(u)\le (\varepsilon +\eta )|u|^{2_N}+c_\varepsilon |u|^{2^*}, \end{aligned}$$

therefore,

$$\begin{aligned} \varphi (s)\ge s^2\Big (\int _{{\mathbb {R}}^N}\frac{1}{2}|\nabla u|^2-(\eta +\varepsilon )|u|^{2_N}\,dx\Big )-c_\varepsilon s^{2^*}\int _{{\mathbb {R}}^N}|u|^{2^*}\,dx>0 \end{aligned}$$

for sufficiently small \(\varepsilon \) and s. It follows that there exists an interval \([a,b]\subset (0,\infty )\) such that \(\varphi |_{[a,b]}=\max \varphi \). Moreover

$$\begin{aligned} \varphi '(s)=s\int _{{\mathbb {R}}^N}|\nabla u|^2-\frac{N}{2}\frac{H(s^{N/2}u)}{s^{N+2}}\,dx \end{aligned}$$

and the function

$$\begin{aligned} s\in (0,\infty )\mapsto \int _{{\mathbb {R}}^N}\frac{H(s^{N/2}u)}{s^{N+2}}\,dx \end{aligned}$$

is nondecreasing (resp. increasing) due to (A4) (resp. (A4,\(\preceq \)) and Lemma 2.1) and tends to \(\infty \) as \(s\rightarrow \infty \) due to (A2) and (A5). There follows that \(\varphi '(s)>0\) if \(s\in (0,a)\) and \(\varphi '(s)<0\) if \(s>b\) and that \(a=b\) if (A4,\(\preceq \)) holds. Finally, observe that

$$\begin{aligned} s\varphi '(s)=\int _{{\mathbb {R}}^N}s^2|\nabla u|^2-\frac{N}{2}\frac{H(s^{N/2}u)}{s^N}\,dx=M(s\star u). \end{aligned}$$

\(\square \square \)

Lemma 2.4

If (A1)–(A5) and (1.8) are verified, then J is coercive on \({{\mathcal {M}}}\cap {{\mathcal {D}}}\).

Proof

First of all note that, if \(u\in {{\mathcal {M}}}\), then due to (A5)

$$\begin{aligned} J(u)=J(u)-\frac{1}{2}M(u)=\int _{{\mathbb {R}}^N}\frac{N}{4}H(u)-G(u)\,dx\ge 0 \end{aligned}$$

and so, a fortiori, J is nonnegative on \({{\mathcal {M}}}\cap {{\mathcal {D}}}\). Let \((u^{(n)})\subset {{\mathcal {M}}}\cap {{\mathcal {D}}}\) such that \(\Vert u^{(n)}\Vert \rightarrow \infty \), i.e., \(\lim _n|\nabla u^{(n)}|_2=\infty \), and define

$$\begin{aligned} s_n:=|\nabla u^{(n)}|_2^{-1}>0\quad \text {and}\quad w^{(n)}:=s_n\star u^{(n)}. \end{aligned}$$

Note that \(s_n\rightarrow 0\), \(|w^{(n)}_i|_2=|u^{(n)}_i|_2\le \rho _i\) for \(i\in \{1,\dots ,K\}\), and \(|\nabla w^{(n)}|_2^2=1\), in particular \((w^{(n)})\) is bounded in \(H^1({\mathbb {R}}^N)^K\). Suppose by contradiction that

$$\begin{aligned} \limsup _n\max _{y\in {\mathbb {R}}^N}\int _{B(y,1)}|w^{(n)}|^2\,dx>0. \end{aligned}$$

Then there exist \((y^{(n)})\subset {\mathbb {R}}^N\) and \(w\in H^1({\mathbb {R}}^N)^K\) such that, up to a subsequence, \(w^{(n)}(\cdot +y^{(n)})\rightharpoonup w\ne 0\) in \(H^1({\mathbb {R}}^N)^K\) and \(w^{(n)}(\cdot +y^{(n)})\rightarrow w\) a.e. in \({\mathbb {R}}^N\). Thus, owing to (A2),

$$\begin{aligned}\begin{aligned} 0&\le \frac{J(u^{(n)})}{|\nabla u^{(n)}|_2^2}\le \frac{1}{2}-\int _{{\mathbb {R}}^N}\frac{G(u^{(n)})}{|\nabla u^{(n)}|_2^2}\,dx=\frac{1}{2}-s_n^{N+2}\int _{{\mathbb {R}}^N}G\bigl (u^{(n)}(s_nx)\bigr )\,dx\\&=\frac{1}{2}-s_n^{N+2}\int _{{\mathbb {R}}^N}G(s_n^{-N/2}w^{(n)})=\frac{1}{2}-\int _{{\mathbb {R}}^N}\frac{G(s_n^{-N/2}w^{(n)})}{|s_n^{-N/2}w^{(n)}|^{2_N}}|w^{(n)}|^{2_N}\,dx\\&=\frac{1}{2}-\int _{{\mathbb {R}}^N}\frac{G\bigl (s_n^{-N/2}w^{(n)}(x+y^{(n)})\bigr )}{|s_n^{-N/2}w^{(n)}(x+y^{(n)})|^{2_N}}|w^{(n)}(x+y^{(n)})|^{2_N}\,dx\rightarrow -\infty . \end{aligned}\end{aligned}$$

It follows that

$$\begin{aligned} \lim _n\max _{y\in {\mathbb {R}}^N}\int _{B(y,1)}|w^{(n)}|^2\,dx=0 \end{aligned}$$

and so, from Lions’ Lemma [28], \(w^{(n)}\rightarrow 0\) in \(L^{2_N}({\mathbb {R}}^N)^K\). Since

$$\begin{aligned} s_n^{-1}\star w^{(n)}=u^{(n)}\in {{\mathcal {M}}}, \end{aligned}$$

Lemma 2.3 yields

$$\begin{aligned} J(u^{(n)})=J(s_n^{-1}\star w^{(n)})\ge J(s\star w^{(n)})=\frac{s^2}{2}-s^N\int _{{\mathbb {R}}^N}G\bigl (s^{N/2}w^{(n)}(s\cdot )\bigr )\,dx \end{aligned}$$

for every \(s>0\). Taking into account that

$$\begin{aligned} \lim _n\int _{{\mathbb {R}}^N}G\bigl (s^{N/2}w^{(n)}(s\cdot )\bigr )\,dx=0, \end{aligned}$$

we have that \(\liminf _nJ(u^{(n)})\ge s^2/2\) for every \(s>0\), i.e., \(\lim _nJ(u^{(n)})=\infty \). \(\square \)

Lemma 2.5

If (A1)–(A5) and (1.8) are verified, then \(c:=\inf _{{{\mathcal {M}}}\cap {{\mathcal {D}}}}J>0\).

Proof

We prove that there exists \(\alpha >0\) such that

$$\begin{aligned} |\nabla u|_2\le \alpha \Rightarrow J(u)\ge \frac{|\nabla u|_2^2}{2N}. \end{aligned}$$
(2.3)

From (1.7) and (1.8), for every \(\varepsilon >0\) there exists \(c_\varepsilon >0\) such that

$$\begin{aligned}\begin{aligned} \int _{{\mathbb {R}}^N}G(u)\,dx&\le c_\varepsilon C_{N,2^*}^{2^*}|\nabla u|_2^{2^*}+(\varepsilon +\eta )C_{N,2_N}^{2_N}|\rho |^{4/N}|\nabla u|_2^2\\&\le \left( c_\varepsilon C_{N,2^*}^{2^*}|\nabla u|_2^{2^*-2}+\varepsilon C_{N,2_N}^{2_N}|\rho |^{4/N}+\frac{1}{2}-\frac{1}{N}\right) |\nabla u|_2^2. \end{aligned}\end{aligned}$$

Choosing

$$\begin{aligned} \varepsilon =\frac{1}{4NC_{N,2_N}^{2_N}|\rho |^{4/N}} \quad \text {and} \quad \alpha =\frac{1}{(4Nc_\varepsilon C_{N,2^*}^{2^*})^\frac{1}{2^*-2}} \end{aligned}$$

we obtain, provided \(|\nabla u|_2\le \alpha \),

$$\begin{aligned} \int _{{\mathbb {R}}^N}G(u)\,dx\le \left( \frac{1}{2}-\frac{1}{2N}\right) |\nabla u|_2^2 \end{aligned}$$

and so \(\displaystyle J(u)\ge \frac{|\nabla u|_2^2}{2N}\). Now take \(u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\) and \(\alpha >0\) such that (2.3) holds and define

$$\begin{aligned} s:=\frac{\alpha }{|\nabla u|_2}\quad \text {and}\quad w:=s\star u. \end{aligned}$$

Clearly \(|w_i|_2=|u_i|_2\le \rho _i\) for \(i\in \{1,\dots ,K\}\) and \(|\nabla w|_2=\alpha \), whence in view of Lemma 2.3

$$\begin{aligned} J(u)\ge J(w)\ge \frac{|\nabla w|_2^2}{2N}=\frac{\alpha ^2}{2N}>0. \end{aligned}$$

\(\square \square \)

From now on, \(c>0\) will stand for the infimum of J over \({{\mathcal {M}}}\cap {{\mathcal {D}}}\).

Proposition 2.6

Assume that \(\theta \in (0,\infty )^K\) and that (A1)–(A5), (1.8), and (at least) one of the following conditions hold:

  1. (i)

    \(N\ge 5\);

  2. (ii)

    there exist \(2_N\le p\le 2^*\) and \(2_N\le q<2^*\) such that

    $$\begin{aligned} \liminf _{|u|\rightarrow 0}\frac{\widetilde{G}(u)}{|u|^p}>0 \quad \text {and} \quad \liminf _{|u|\rightarrow \infty }\frac{\widetilde{G}(u)}{|u|^q}>0 \end{aligned}$$
    (2.4)

    and \(\max \{p,q\}/2-\min \{p,q\}<-1\) if \(N=3\).

Then (1.9) holds.

Recall that, from (A2), the second condition in (2.4) always holds with \(q=2_N\). Notice that the restriction on the relation between pq is always satisfied if \(p=q\).

Proof

Define \(u_0^1\) as the Aubin–Talenti instanton [4, 40]

$$\begin{aligned} u_0^1(x):=\left( \frac{\sqrt{N(N-2)}}{1+|x|^2}\right) ^\frac{N-2}{2} \end{aligned}$$

and, for \(\varepsilon >0\),

$$\begin{aligned} u_0^\varepsilon (x):=\varepsilon ^{1-N/2}u_0^1(x/\varepsilon )=\left( \frac{\varepsilon \sqrt{N(N-2)}}{\varepsilon ^2+|x|^2}\right) ^\frac{N-2}{2}. \end{aligned}$$

Recall that, for every \(\varepsilon >0\), \(|\nabla u_0^\varepsilon |_2=|\nabla u_0^1|_2\), \(|u_0^\varepsilon |_{2^*}=|u_0^1|_{2^*}\), and \(u_0^\varepsilon \) is a minimizer for

$$\begin{aligned} S:=\inf \left\{ \int _{{\mathbb {R}}^N}|\nabla v|^2\,dx:v\in {{\mathcal {D}}}^{1,2}({\mathbb {R}}^N),\int _{{\mathbb {R}}^N}|v|^{2^*}\,dx=1\right\} . \end{aligned}$$

(i) For every \(\varepsilon >0\) and \(j\in \{1,\dots ,K\}\) define \({\bar{u}}_j^\varepsilon :=\theta _j^{(2-N)/4}u_0^\varepsilon \). Since \(u_0^\varepsilon \in L^2({\mathbb {R}}^N)\) for every \(\varepsilon >0\) and \(|u_0^\varepsilon |_2\rightarrow 0\) as \(\varepsilon \rightarrow 0^+\), we have \({\bar{u}}^\varepsilon :=({\bar{u}}^\varepsilon _1,\ldots ,{\bar{u}}^\varepsilon _K)\in {{\mathcal {D}}}\) for sufficiently small \(\varepsilon \). Moreover, in view of Lemma B.1, \({\bar{u}}^\varepsilon \) is such that

$$\begin{aligned} \frac{|\nabla {\bar{u}}^\varepsilon |_2^2}{\left( \sum _{j=1}^K\theta _j|{\bar{u}}_j^\varepsilon |_{2^*}^{2^*}\right) ^{2/2^*}}=\inf _{u\in {{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\setminus \{0\}}\frac{|\nabla u|_2^2}{\left( \sum _{j=1}^K\theta _j|u_j|_{2^*}^{2^*}\right) ^{2/2^*}}=\left( \sum _{j=1}^K\theta _j^{1-N/2}\right) ^{2/N}S. \end{aligned}$$

Recall that \(\widetilde{G}(u)>0\) for \(u\ne 0\) and then, taking \(\varepsilon \) sufficiently small,

$$\begin{aligned}\begin{aligned} c\le J(s_\varepsilon \star {\bar{u}}^\varepsilon )&\le -\int _{{\mathbb {R}}^N} \widetilde{G}(s_\varepsilon \star {\bar{u}}^\varepsilon ) \,dx+ \max _{s>0}\frac{s^2}{2}\int _{{\mathbb {R}}^N}|\nabla {\bar{u}}^\varepsilon |^2\,dx-\frac{s^{2^*}}{2^*} \sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N}|{\bar{u}}_j^\varepsilon |^{2^*}\,dx\\&<\max _{s>0}\frac{s^2}{2}\int _{{\mathbb {R}}^N}|\nabla {\bar{u}}^\varepsilon |^2\,dx-\frac{s^{2^*}}{2^*} \sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N}|{\bar{u}}_j^\varepsilon |^{2^*}\,dx\\&=\frac{1}{N}\frac{|\nabla {\bar{u}}^\varepsilon |_2^N}{\left( \sum _{j=1}^K\theta _j|{\bar{u}}_j^\varepsilon |_{2^*}^{2^*}\right) ^{N/2-1}}=\sum _{j=1}^K\theta _j^{1-N/2}\frac{S^{N/2}}{N}. \end{aligned}\end{aligned}$$

(ii) If \(N\ge 5\), then the statement follows form (i), therefore we can assume \(N\in \{3,4\}\). Since \(u_0^1\not \in L^2({\mathbb {R}}^N)\), let \(0\le \phi \in {{\mathcal {C}}}_0^\infty ({\mathbb {R}}^N)\) radial such that \(\phi \equiv 1\) in \(B_1\) and \(\phi \equiv 0\) in \({\mathbb {R}}^N\setminus B_2\), where \(B_r\) stands for the closed ball centred at 0 of radius r. For every \(\varepsilon >0\) define

$$\begin{aligned} u_j^\varepsilon :=\theta _j^\frac{2-N}{4}\phi u_0^\varepsilon \quad \text {and} \quad v^\varepsilon :=\frac{{\bar{\rho }}}{|u^\varepsilon |_2}(u_1^\varepsilon ,\dots ,u_K^\varepsilon )\in {{\mathcal {D}}}, \end{aligned}$$

where \(\bar{\rho }:=\min _{j\in \{1,\dots ,K\}}\rho _j\), and recall (cf., e.g., [38, p. 179], [36, Lemma A.1]) that

$$\begin{aligned}\begin{aligned} |\nabla (\phi u_0^\varepsilon )|_2^2&=S^{N/2}+O(\varepsilon ^{N-2})\\ |\phi u_0^\varepsilon |_{2^*}^2&= {\left\{ \begin{array}{ll} S+O(\varepsilon ^4) \quad \text {if } N=4\\ S^{1/2}+O(\varepsilon ^2) \quad \text {if } N=3 \end{array}\right. }\\ |\phi u_0^\varepsilon |_2^2&= {\left\{ \begin{array}{ll} C_4\varepsilon ^2|\ln \varepsilon |+O(\varepsilon ^2) \quad \text {if } N=4\\ C_3\varepsilon +O(\varepsilon ^2) \quad \text {if } N=3, \end{array}\right. } \end{aligned}\end{aligned}$$

where \(C_N>0\) depends only on N and \(\phi \). Note that

$$\begin{aligned} \int _{{\mathbb {R}}^N}|\phi u_0^\varepsilon |^r\chi _{\{\phi u_0^\varepsilon \ge 1\}}\,dx\ge C\varepsilon ^{N-(N/2-1)r} \end{aligned}$$

for some constant \(C>0\) and sufficiently small \(\varepsilon >0\), where \(r\in \{p,q\}\) and \(\chi _A\) stands for the characteristic function of A. Indeed, let \(|x|^2\le \varepsilon \sqrt{N(N-2)}-\varepsilon ^2\). If \(\varepsilon \) is sufficiently small, then \(x\in B_1\) and, consequently, \(\phi (x) u_0^\varepsilon (x)=u_0^\varepsilon (x)\ge 1\), whence

$$\begin{aligned}\begin{aligned} \int _{{\mathbb {R}}^N}|\phi u_0^\varepsilon |^r\chi _{\{\phi u_0^\varepsilon \ge 1\}}\,dx&\ge \int _{\left\{ |x|\le \left( \varepsilon \sqrt{N(N-2)}-\varepsilon ^2\right) ^{1/2}\right\} }|u^\varepsilon _0|^r\,dx\\&=\varepsilon ^{N-(N/2-1)r}\int _{\left\{ |y|\le \left( \sqrt{N(N-2)}/\varepsilon -1\right) ^{1/2}\right\} }|u^1_0|^r\,dy \end{aligned}\end{aligned}$$

and we conclude, since \(u^1_0\in L^r({\mathbb {R}}^N)\). Define \(s_\varepsilon >0\) such that \(s_\varepsilon *v^\varepsilon \in {{\mathcal {M}}}\). In a similar way to the proof of Lemma 2.2, for every \(\delta >0\) there exists \(C_\delta >0\) not depending on \(\varepsilon \) such that

$$\begin{aligned} \frac{1}{2^*}|\nabla v^\varepsilon |_2^2\le & {} (\eta +\delta )|v^\varepsilon |_{2_N}^{2_N}+C_\delta s_\varepsilon ^{2^*-2}\sum _{j=1}^K\theta _j|v_j^\varepsilon |_{2^*}^{2^*}\le (\eta +\delta )C_{N,2_N}^{2_N}|\rho |^{4/N}|\nabla v^\varepsilon |_2^2\\&+\,\,C_\delta s_\varepsilon ^{2^*-2}\sum _{j=1}^K\theta _j|v_j^\varepsilon |_{2^*}^{2^*} \end{aligned}$$

(note that \(u\mapsto \left( \sum _{j=1}^K\theta _j|u_j|_{2^*}^{2^*}\right) ^{1/2^*}\) is an equivalent norm in \(L^{2^*}({\mathbb {R}}^N)^K\)), i.e., taking \(\delta \) sufficiently small and denoting \(m:=\bigl (1/2^*-(\eta +\delta )C_{N,2_N}^{2_N}|\rho |^{4/N}\bigr )/C_\delta >0\),

$$\begin{aligned} s_\varepsilon ^{2^*-2}\ge \frac{m|\nabla v^\varepsilon |_2^2}{\sum _{j=1}^K\theta _j|v_j^\varepsilon |_{2^*}^{2^*}}=m\bar{\rho }^{2-2^*}\frac{|\nabla (\phi u_0^\varepsilon )|_2^2|u^\varepsilon |_2^{2^*-2}}{|\phi u_0^\varepsilon |_{2^*}^{2^*}}. \end{aligned}$$

In a similar way to point (i),

$$\begin{aligned} c\le -\int _{{\mathbb {R}}^N}\widetilde{G}(s_\varepsilon \star v^\varepsilon )\,dx+\frac{1}{N}\frac{|\nabla v^\varepsilon |_2^N}{\left( \sum _{j=1}^K\theta _j|v_j^\varepsilon |_{2^*}^{2^*}\right) ^{N/2-1}}. \end{aligned}$$

There holds

$$\begin{aligned}\begin{aligned} |\nabla v^\varepsilon |_2^2 = \frac{\bar{\rho }^2|\nabla (\phi u_0^\varepsilon )|_2^2\sum _{j=1}^K\theta _j^{1-N/2}}{|u^\varepsilon |_2^2} \quad \text {and} \quad \sum _{j=1}^K\theta _j|v_j^\varepsilon |_{2^*}^{2^*}= \frac{\bar{\rho }^{2^*}|\phi u_0^\varepsilon |_{2^*}^{2^*}\sum _{j=1}^K\theta _j^{1-N/2}}{|u^\varepsilon |_2^{2^*}}, \end{aligned}\end{aligned}$$

thus, denoting \(k=2\) (resp. \(k=4\)) if \(N=3\) (resp. \(N=4\)),

$$\begin{aligned} \begin{aligned} \frac{|\nabla v^\varepsilon |_2^N}{\left( \sum _{j=1}^K\theta _j|v_j^\varepsilon |_{2^*}^{2^*}\right) ^{N/2-1}}&=\sum _{j=1}^K\theta _j^{1-N/2}\left( \frac{|\nabla (\phi u_0^\varepsilon )|_2}{|\phi u_0^\varepsilon |_{2^*}}\right) ^N\\&=\sum _{j=1}^K\theta _j^{1-N/2}\left( \frac{S^{N/2}+O(\varepsilon ^{N-2})}{S^{(N-2)/2}+O(\varepsilon ^k)}\right) ^{N/2}\\&=\sum _{j=1}^K\theta _j^{1-N/2}\bigl (S+O(\varepsilon ^{N-2})\bigr )^{N/2}\\&=\sum _{j=1}^K\theta _j^{1-N/2}S^{N/2}+O(\varepsilon ^{N-2}). \end{aligned}\end{aligned}$$

Now we estimate \(\int _{{\mathbb {R}}^N}\widetilde{G}(s_\varepsilon \star v^\varepsilon )\,dx\) as \(\varepsilon \rightarrow 0^+\). From (2.4) and the fact, due to (A2) and (A5), that \(\widetilde{G}(u)>0\) if \(u\ne 0\), we deduce there exists \(C>0\) such that \(\widetilde{G}(u)\ge C|u|^p\) if \(|u|\le 1\) and \(\widetilde{G}(u)\ge C|u|^q\) if \(|u|>1\).

$$\begin{aligned}\begin{aligned} \int _{{\mathbb {R}}^N}\widetilde{G}(s_\varepsilon \star v^\varepsilon )\,dx&\ge C s_\varepsilon ^{N(p/2-1)}\int _{{\mathbb {R}}^N}|v^\varepsilon |^p\chi _{\{|s_\varepsilon ^{N/2} v^\varepsilon |\le 1\}}\,dx\\&\quad +C s_\varepsilon ^{N(q/2-1)}\int _{{\mathbb {R}}^N}|v^\varepsilon |^q\chi _{\{|s_\varepsilon ^{N/2} v^\varepsilon |> 1\}}\,dx\\&\ge C' |\phi u_0^\varepsilon |_2^{N(p/2-1)-p}\int _{{\mathbb {R}}^N}|\phi u_0^\varepsilon |^p\chi _{\{|s_\varepsilon ^{N/2} v^\varepsilon |\le 1\}}\,dx\\&\quad +C' |\phi u_0^\varepsilon |_2^{N(q/2-1)-q}\int _{{\mathbb {R}}^N}|\phi u_0^\varepsilon |^q\chi _{\{|s_\varepsilon ^{N/2} v^\varepsilon |> 1\}}\,dx\\&\ge C' |\phi u_0^\varepsilon |_2^{(N/2-1)p-N}\int _{{\mathbb {R}}^N}|\phi u_0^\varepsilon |^p\chi _{\{|s_\varepsilon ^{N/2}v^\varepsilon |\le 1\}}\chi _{\{\phi u_0^\varepsilon \ge 1\}}\,dx\\&\quad +C' |\phi u_0^\varepsilon |_2^{(N/2-1)q-N}\int _{{\mathbb {R}}^N}|\phi u_0^\varepsilon |^q\chi _{\{|s_\varepsilon ^{N/2} v^\varepsilon |> 1\}}\chi _{\{\phi u_0^\varepsilon \ge 1\}}\,dx\\&\ge C' \min \big \{|\phi u_0^\varepsilon |_2^{(N/2-1)p-N},|\phi u_0^\varepsilon |_2^{(N/2-1)q-N}\big \}\\&\qquad \int _{{\mathbb {R}}^N}|\phi u_0^\varepsilon |^{\min \{p,q\}}\chi _{\{\phi u_0^\varepsilon \ge 1\}}\,dx\\&\ge C'' |\phi u_0^\varepsilon |_2^{(N/2-1)\max \{p,q\}-N}\varepsilon ^{N-(N/2-1)\min \{p,q\}} \end{aligned}\end{aligned}$$

as \(\varepsilon \rightarrow 0^+\) because \((N/2-1)r-N<0\), \(r\in \{p,q\}\), where \(C',C''>0\) are constants. There follows that

$$\begin{aligned} c\le \sum _{j=1}^K\theta _j^{1-N/2}\frac{S^{N/2}}{N}+O(\varepsilon ^{N-2})-C''|\phi u_0^\varepsilon |_2^{(N/2-1)\max \{p,q\}-N}\varepsilon ^{N-(N/2-1)\min \{p,q\}}. \end{aligned}$$

If \(N=3\), then

$$\begin{aligned}\begin{aligned} |\phi u_0^\varepsilon |_2^{(N/2-1)\max \{p,q\}-N}\varepsilon ^{N-(N/2-1)\min \{p,q\}}&=\frac{\varepsilon ^{3-\min \{p,q\}/2}}{(C_3\varepsilon )^{3/2-\max \{p,q\}/4}+O(\varepsilon ^{3-\max \{p,q\}/2})}\\&\ge C\varepsilon ^{(3+\max \{p,q\}/2-\min \{p,q\})/2} \end{aligned}\end{aligned}$$

and \(0<(3+\max \{p,q\}/2-\min \{p,q\})/2<1=N-2\). If \(N=4\), then

$$\begin{aligned}\begin{aligned} |\phi u_0^\varepsilon |_2^{(N/2-1)\max \{p,q\}-N}\varepsilon ^{N-(N/2-1)\min \{p,q\}}&=\frac{\varepsilon ^{4-\min \{p,q\}}}{(\sqrt{C_4|\ln \varepsilon |}\,\varepsilon )^{4-\max \{p,q\}}+O(\varepsilon ^{4-\max \{p,q\}})}\\&\ge C\varepsilon ^{|p-q|}|\ln \varepsilon |^{\max \{p,q\}/2-2} \end{aligned}\end{aligned}$$

and \(|p-q|<2=N-2\), \(\max \{p,q\}-4\le 0\), and \(|p-q|>0\) or \(\max \{p,q\}-4<0\). Either way, \(O(\varepsilon ^{N-2})-C''|\phi u_0^\varepsilon |_2^{(N/2-1)\max \{p,q\}-N}\varepsilon ^{N-(N/2-1)\min \{p,q\}}<0\) for sufficiently small \(\varepsilon \) and

$$\begin{aligned} c<\sum _{j=1}^K\theta _j^{1-N/2}\frac{S^{N/2}}{N}. \end{aligned}$$

Since there exist nonlinearities that do not satisfy the assumptions of Proposition 2.6(ii), we provide other sufficient conditions for (1.9) to hold.

Lemma 2.7

Assume that (A1)–(A5) are satisfied and \(\theta \in (0,\infty )^K\).

(a) If \(K=1\), \(\eta =0\), and \(\displaystyle \lim _{u\rightarrow 0}\widetilde{G}(u)/|u|^{2^*}=\infty \), then there exists \(\rho _0>0\) such that (1.9) is satisfied provided that \(\rho >\rho _0\).

(b) If (1.8) holds and \(\displaystyle \lim _{|u|\rightarrow \infty }\widetilde{G}(u)/|u|^{2_N}=\infty \), then there exists \(\theta _0>0\) such that (1.9) is satisfied provided that \(\theta _i<\theta _0\) for some \(i\in \{1,\ldots ,K\}\).

Proof

(a) We prove that \(c\rightarrow 0\) as \(\rho \rightarrow \infty \) (note that (1.8) is satisfied for every \(\rho >0\) because \(\eta =0\)). Let \(\rho _n \rightarrow \infty \) and take \(u \in L^\infty ({\mathbb {R}}^N)\) such that \(|u|_2=1\). Without loss of generality we may assume that \(\rho _n > 1\) and define \(u_n := \rho _n u\) so that \(|u_n|_2=\rho _n\). From Lemma 2.3 there exists \(s_n >0\) such that \(v_n :=s_n^{N/2} u_n (s_n \cdot ) \in {{\mathcal {M}}}\). Moreover, \(|v_n|_2=|u_n|_2\), hence

$$\begin{aligned} 0 < \inf \{J(v) : v\in {{\mathcal {M}}}, \, |v|_2\le \rho _n\} \le J(v_n) \le \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla v_n|^2 \, dx = \frac{1}{2} (s_n\rho _n)^2 \int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx, \end{aligned}$$

so it is enough to show that \(s_n \rho _n \rightarrow 0\). Note that

$$\begin{aligned} \left( s_n\rho _n\right) ^2 \int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx = \int _{{\mathbb {R}}^N} |\nabla v_n|^2 \, dx = \frac{N}{2} \int _{{\mathbb {R}}^N} H(v_n) \, dx = \frac{N}{2} s_n^{-N} \int _{{\mathbb {R}}^N} H(s_n^{N/2} \rho _n u) \, dx \end{aligned}$$

and

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx = \frac{N}{2} s_n^{-N-2} \rho _n^{-2} \int _{{\mathbb {R}}^N} H(s_n^{N/2} \rho _n u) \, dx = \frac{N}{2} \rho _n^{4/N} \int _{{\mathbb {R}}^N} \frac{H(s_n^{N/2} \rho _n u)}{ \left| s_n^{N/2} \rho _n u \right| ^{2_N} } |u|^{2_N} \, dx. \end{aligned}$$

There follows that

$$\begin{aligned} \lim _n\int _{{\mathbb {R}}^N} \frac{H(s_n^{N/2} \rho _n u)}{ \left| s_n^{N/2} \rho _n u \right| ^{2_N} } |u|^{2_N} \, dx = 0, \end{aligned}$$

whence \(s_n^{N/2} \rho _n \rightarrow 0\). Fix \(\varepsilon > 0\). From (A5) and the fact that \(\lim _{t\rightarrow 0}G(t)/|t|^{2^*}=\infty \), there follows that

$$\begin{aligned} H(s) \ge \frac{4}{N} G(s) \ge \varepsilon ^{-1} |s|^{2^*} \end{aligned}$$

for sufficiently small |s|. Then, taking into account that \(u \in L^\infty ({\mathbb {R}}^N)\), for sufficiently large n

$$\begin{aligned}\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx&= \frac{N}{2} s_n^{-N-2} \frac{1}{\rho _n^2} \int _{{\mathbb {R}}^N} H(s_n^{N/2} \rho _n^2 u) \, dx \ge \varepsilon ^{-1} \frac{N}{2} s_n^{-N-2} \frac{1}{\rho _n^2} \left| s_n^{N/2} \rho _n \right| ^{2^*} |u|_{2^*}^{2^*}\\&= \varepsilon ^{-1} \frac{N}{2} (s_n \rho _n)^{\frac{4}{N-2}}|u|_{2^*}^{2^*} \end{aligned}\end{aligned}$$

and \(s_n \rho _n \rightarrow 0\) as \(n \rightarrow \infty \), which completes the proof.

(b) Take any \(u_0\in {{\mathcal {D}}}\setminus \{0\}\) and note that (2.2) holds. In view of Lemma 2.3 there exists \(s_0>0\) such that \(s_0\star u_0\in {{\mathcal {M}}}\) and

$$\begin{aligned} c\le J(s_0\star u_0)\le \max _{s>0}J(s\star u_0)\le \max _{s>0}\frac{s^2}{2}\int _{{\mathbb {R}}^N}|\nabla u_0|^2\,dx-\int _{{\mathbb {R}}^N}\frac{\widetilde{G}(s^{N/2}u_0)}{s^N}\,dx. \end{aligned}$$

Observe that the latter expression is finite due to Lemma 2.3 with \(\theta =0\). Hence we can take \(\theta _0>0\) so small that, if \(\theta _i<\theta _0\), then \(\sum _{j=1}^K\theta _j^{1-N/2}S^{N/2}/N\ge \theta _i^{1-N/2}S^{N/2}/N\) is greater than the right-hand side of the formula above. \(\square \)

We give explicit examples of nonlinearities that do not satisfy the assumptions of Proposition 2.6. Let \(N=3\) and \(\varepsilon >0\) be sufficiently small. If \(\widetilde{g}(u)=\widetilde{g}_1(u)=\min \{|u|^{4-\varepsilon },|u|^{4/3}\}u\) and if \(\theta =\theta _1\) is not sufficiently small, then we can use Lemma 2.7 (a) provided that \(\rho =\rho _1\) is sufficiently large, but not part (b). If G is of the form (1.10) and

$$\begin{aligned} \widetilde{g}_i(u)=\min \{|u|^{4},|u|^{4/3+\varepsilon }\}u \end{aligned}$$
(2.5)

and if \(K=2\) or \(\rho \) is not sufficiently large, then we can use Lemma 2.7 (b) provided that \(\theta _i\) is sufficiently small for some \(i\in \{1\dots ,K\}\), but not part (a).

In view of Lemma 2.4, any minimizing sequence \((u^{(n)})\subset {{\mathcal {M}}}\cap {{\mathcal {D}}}\) such that \(J(u^{(n)})\rightarrow c>0\) is bounded. By the standard concentration-compactness argument [28], \(u^{(n)}\rightharpoonup {{\tilde{u}}}\) for some \({{\tilde{u}}}\ne 0\) up to a subsequence and up to translations. It is not clear, however, if \(J({{\tilde{u}}})=c\) or \({{\tilde{u}}}\in {{\mathcal {M}}}\cap {\mathcal {D}}\). Note that we can find \(R>0\) such that \( {{\tilde{u}}}(R\cdot )\in {{\mathcal {M}}}\) and in order to ensure that \(J({{\tilde{u}}})=c\) and \({{\tilde{u}}}\in {{\mathcal {D}}}\) we need to know that \(R\ge 1\). The latter crucial condition requires the profile decomposition analysis of \((u^{(n)})\) provided by the following lemma.

Lemma 2.8

Let \((u^{(n)})\subset H^1({\mathbb {R}}^N)^K\) be bounded. Then there exist sequences \(({\tilde{u}}^{(i)})_{i=0}^\infty \subset H^1({\mathbb {R}}^N)^K\) and \((y^{(i,n)})_{i=0}^\infty \subset {\mathbb {R}}^N\) such that \(y^{(0,n)}=0\), \(\lim _n|y^{(i,n)}-y^{(j,n)}|=0\) if \(i\ne j\), and for every \(i\ge 0\) and every \(F:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) of class \({{\mathcal {C}}}^1\) such that

$$\begin{aligned} \lim _{u\rightarrow 0}\frac{F(u)}{|u|^2}=\lim _{|u|\rightarrow \infty }\frac{F(u)}{|u|^{2^*}}=0 \end{aligned}$$

there holds (up to a subsequence)

$$\begin{aligned} u^{(n)}(\cdot +y^{(i,n)})&\rightharpoonup&{\tilde{u}}^{(i)}\text { as }n\rightarrow \infty \end{aligned}$$
(2.6)
$$\begin{aligned} \lim _n\int _{{\mathbb {R}}^N}|\nabla u^{(n)}|^2\,dx= & {} \sum _{j=0}^i\int _{{\mathbb {R}}^N}|\nabla {\tilde{u}}^{(j)}|^2\,dx+\lim _n\int _{{\mathbb {R}}^N}|\nabla v^{(i,n)}|^2\,dx \end{aligned}$$
(2.7)
$$\begin{aligned} \limsup _n\int _{{\mathbb {R}}^N}F(u^{(n)})\,dx= & {} \sum _{i=0}^\infty \int _{{\mathbb {R}}^N}F({\tilde{u}}^{(i)})\,dx, \end{aligned}$$
(2.8)

where \(v^{(i,n)}(x):=u^{(n)}(x)-\sum _{j=0}^{i}{\tilde{u}}^{(j)}(x-y^{(j,n)})\).

Proof

We argue similarly as in the case \(K=1\) provided in [31, Theorem 1.4]. \(\square \)

Lemma 2.9

If (A1)–(A5) and (1.8) hold and either \(\theta =0\) or \(\theta \in (0,\infty )^K\) and (1.9) is satisfied, then c is attained.

Proof

Let \((u^{(n)})\subset {{\mathcal {M}}}\cap {{\mathcal {D}}}\) such that \(\lim _nJ(u^{(n)})=c\). Then \((u^{(n)})\) is bounded due to Lemma 2.4 and, in view of Lemma 2.8, we find \(({\tilde{u}}^{(i)})_{i=0}^\infty \subset H^1({\mathbb {R}}^N)^K\) and \((y_n^{(i,n)})_{i=0}^\infty \subset {\mathbb {R}}^N\) such that (2.6)–(2.8) hold. Let \(I:=\{i\ge 0:{\tilde{u}}^{(i)}\ne 0\}\).

Suppose that \(\theta \in (0,\infty )^K\) and (1.9) is satisfied.

Claim 1. \(I\ne \emptyset \). By contradiction suppose that \({\tilde{u}}^{(i)}=0\) for every \(i\ge 0\). Then

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla u^{(n)}|^2 \, dx=\frac{N}{2} \int _{{\mathbb {R}}^N} H(u^{(n)}) \, dx =\frac{N}{2} \int _{{\mathbb {R}}^N} \widetilde{H}(u^{(n)}) \, dx+\sum _{j=1}^K\theta _j \int _{{\mathbb {R}}^N} |u^{(n)}_j|^{2^*} \, dx. \end{aligned}$$

Observe that (A1), (A3), and (A5) imply that

$$\begin{aligned} \lim _{u\rightarrow 0}\frac{\widetilde{H}(u)}{|u|^2}=\lim _{|u|\rightarrow \infty }\frac{\widetilde{H}(u)}{|u|^{2^*}}=0 \end{aligned}$$

and

$$\begin{aligned} o(1)+\int _{{\mathbb {R}}^N} |\nabla u^{(n)}|^2 \, dx=\sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N} |u^{(n)}_j|^{2^*} \, dx. \end{aligned}$$
(2.9)

For the sake of simplicity, let us denote \({\bar{S}}:=\left( \sum _{j=1}^K\theta _j^{1-N/2}\right) ^{2/N}S\), cf. “Appendix B”. Then

$$\begin{aligned} o(1)+\int _{{\mathbb {R}}^N} |\nabla u^{(n)}|^2 \, dx\le {\bar{S}}^{-2^*/2}\left( \int _{{\mathbb {R}}^N} |\nabla u^{(n)}|^{2} \, dx\right) ^{2^*/2}. \end{aligned}$$

Passing to a subsequence we set \(\nu :=\lim _n\int _{{\mathbb {R}}^N} |\nabla u^{(n)}|^2 \, dx>0\) from Lemma 2.2 and we get \(\nu ^{2/(N-2)}\ge {\bar{S}}^{N/(N-2)}\). Then

$$\begin{aligned} c=\lim _nJ(u^{(n)})=\lim _nJ(u^{(n)})-\frac{1}{2^*}M(u^{(n)})=\frac{1}{N}\nu \ge \frac{1}{N} {\bar{S}}^{N/2}, \end{aligned}$$
(2.10)

so we obtain a contradiction and \(I\ne \emptyset \).

Claim 2. For every \(i\in I\) there holds \(u^{(n)}(\cdot +y^{(i,n)})\rightarrow {\tilde{u}}^{(i)}\) in \({{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\) or \(\int _{{\mathbb {R}}^N} |\nabla {\tilde{u}}^{(i)}|^2 \, dx<\frac{N}{2}\int _{{\mathbb {R}}^N}H({\tilde{u}}^{(i)})\, dx\). Suppose that there exists \(i\in I\) such that \(\nu :=\lim _n\int _{{\mathbb {R}}^N} |\nabla v^{(n)}|^{2} \, dx>0\) (passing to a subsequence) and the reverse inequality holds, where \(v^{(n)}:=u^{(n)}(\cdot +y^{(i,n)})-{\tilde{u}}^{(i)}\). By Vitali’s convergence theorem

$$\begin{aligned} \int _{{\mathbb {R}}^N}\big (H(u^{(n)})-H(v^{(n)})\big )\, dx= & {} \int _{{\mathbb {R}}^N}\int _0^1 -\frac{d}{ds}H(u^{(n)}-s{\tilde{u}}^{(i)})\, ds\,dx\\= & {} \int _{{\mathbb {R}}^N}\int _0^1 h(u^{(n)}-s{\tilde{u}}^{(i)}){\tilde{u}}^{(i)}\,ds\, dx\\\rightarrow & {} \int _0^1 \int _{{\mathbb {R}}^N} h({\tilde{u}}^{(i)}-s{\tilde{u}}^{(i)}){\tilde{u}}^{(i)}\,dx\,ds\\= & {} \int _{{\mathbb {R}}^N}\int _0^1 -\frac{d}{ds}H({\tilde{u}}^{(i)}-s{\tilde{u}}^{(i)})\, ds\, dx\\= & {} \int _{{\mathbb {R}}^N}H({\tilde{u}}^{(i)})\, dx \end{aligned}$$

as \(n\rightarrow \infty \). Again, passing to a subsequence,

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla v^{(n)}|^{2} \, dx+\int _{{\mathbb {R}}^N} |\nabla {\tilde{u}}^{(i)}|^2 \, dx=\frac{N}{2}\Big (\int _{{\mathbb {R}}^N}H(v^{(n)})\, dx+\int _{{\mathbb {R}}^N}H({\tilde{u}}^{(i)})\, dx\Big )+o(1) \end{aligned}$$

and, since \(\int _{{\mathbb {R}}^N} |\nabla {\tilde{u}}^{(i)}|^2 \, dx\ge \frac{N}{2}\int _{{\mathbb {R}}^N}H({\tilde{u}}^{(i)})\, dx\), we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla v^{(n)}|^{2} \, dx\le \frac{N}{2}\int _{{\mathbb {R}}^N}H(v^{(n)})\, dx+o(1) \end{aligned}$$
(2.11)

and define \(R_n>0\) such that \(v^{(n)}(R_n\cdot )\in {{\mathcal {M}}}\). We want to prove that \(R_n\rightarrow 1\). If

$$\begin{aligned} \frac{N}{2}\int _{{\mathbb {R}}^N}H(v^{(n)})\,dx<\int _{{\mathbb {R}}^N}|\nabla v^{(n)}|^2\,dx \end{aligned}$$

holds for a.e. n, then from (2.11) and the fact that \(\nu >0\) we get \(R_n\rightarrow 1\). If, passing to a subsequence,

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla v^{(n)}|^{2} \, dx\le \frac{N}{2}\int _{{\mathbb {R}}^N}H(v^{(n)})\, dx \end{aligned}$$

holds, then we infer \(R_n\ge 1\). Note that \(\lim _n | u^{(n)}|_2^2-| v^{(n)}|_2^2=| {\tilde{u}}^{(i)}|_2^2>0\), hence \(v^{(n)}\in {{\mathcal {D}}}\) and \(v^{(n)}(R_n\cdot )\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\) for a.e. n. Hence the Brezis–Lieb Lemma yields

$$\begin{aligned} \begin{aligned} c&\le J\bigl (v^{(n)}(R_n\cdot )\bigr )=J\bigl (v^{(n)}(R_n\cdot )\bigr )-\frac{1}{2}M\bigl (v^{(n)}(R_n\cdot )\bigr )\,dx\\&=\frac{1}{R^N_n}\int _{{\mathbb {R}}^N}\frac{N}{4}H(v^{(n)})-G(v^{(n)})\,dx\\&\le \int _{{\mathbb {R}}^N}\frac{N}{4}H(v^{(n)})-G(v^{(n)})\,dx\le \int _{{\mathbb {R}}^N}\frac{N}{4}H(u^{(n)})-G(u^{(n)})\,dx+o(1)\\&=J(u^{(n)})-\frac{1}{2}M(u^{(n)})+o(1) =J(u^{(n)})+o(1)=c+o(1), \end{aligned}\end{aligned}$$
(2.12)

which implies that \(R_n\rightarrow 1\) as claimed. Therefore we have that

$$\begin{aligned} \int _{{\mathbb {R}}^N} |\nabla v^{(n)}|^{2} \, dx=o(1)+\frac{N}{2}\int _{{\mathbb {R}}^N}H(v^{(n)})\, dx=o(1)+\sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N} |v_j^{(n)}|^{2^*} \, dx\nonumber \\ \end{aligned}$$
(2.13)

and as in Claim 1 we get \(\nu ^{2/(N-2)}\ge {\bar{S}}^{N/(N-2)}\). Since \(J(u^{(n)})-J(v^{(n)})=J({\tilde{u}}^{(i)})+o(1)\) and \(J({\tilde{u}}^{(i)})\ge \int _{{\mathbb {R}}^N}\frac{N}{4}H({\tilde{u}}^{(i)})-G({\tilde{u}}^{(i)})\,dx\ge 0\), we have

$$\begin{aligned} c= & {} \lim _nJ({\tilde{u}}^{(i)})+J(v^{(n)})=J({\tilde{u}}^{(i)})+\frac{1}{2}\nu -\frac{1}{2^*}\lim _n\sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N}|v_j^{(n)}|^{2^*}\,dx\nonumber \\\ge & {} \frac{1}{N}\nu \ge \frac{1}{N}{\bar{S}}^{N/2}, \end{aligned}$$
(2.14)

a contradiction.

Conclusion. Let \(i\in I\) and, for simplicity, let us denote \({\tilde{u}}^{(i)}=:{\tilde{u}}\). If \(\int _{{\mathbb {R}}^N} |\nabla {\tilde{u}}|^2 \, dx<\frac{N}{2}\int _{{\mathbb {R}}^N}H({\tilde{u}})\, dx\), then there exists \(R>1\) such that \({{\tilde{u}}}(R\cdot )\in {{\mathcal {M}}}\), whence \({\tilde{u}}(R\cdot )\in {{\mathcal {D}}}\). Hence Fatou’s Lemma yields

$$\begin{aligned} \begin{aligned} c&\le J\bigl ({{\tilde{u}}}(R\cdot )\bigr )=J\bigl ({{\tilde{u}}}(R\cdot )\bigr )-\frac{1}{2}M\bigl ({{\tilde{u}}}(R\cdot )\bigr )\,dx=\frac{1}{R^N}\int _{{\mathbb {R}}^N}\frac{N}{4}H({{\tilde{u}}})-G({\tilde{u}})\,dx\\&<\liminf _n\int _{{\mathbb {R}}^N}\frac{N}{4}H(u^{(n)})-G(u^{(n)})\,dx=\liminf _nJ(u^{(n)})-\frac{1}{2}M(u^{(n)})\\&=\liminf _nJ(u^{(n)})=c, \end{aligned} \end{aligned}$$
(2.15)

which is a contradiction. Therefore \(u^{(n)}(\cdot +y^{(i,n)})\rightarrow {\tilde{u}}\) in \({{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\) (which, together with (2.7), implies that I is a singleton) and, consequently, in \(L^{2^*}({\mathbb {R}}^N)^K\). Moreover, in virtue of the Brezis–Lieb lemma, \(\int _{{\mathbb {R}}^N}H(u^{(n)})\,dx\rightarrow \int _{{\mathbb {R}}^N}H({\tilde{u}})\,dx\) because, from the interpolation inequality,

$$\begin{aligned}\begin{aligned} \int _{{\mathbb {R}}^N}H(u^{(n)}-{\tilde{u}})\,dx&\le C(|u^{(n)}-{\tilde{u}}|_{2_N}^{2_N}+|u^{(n)}-{\tilde{u}}|_{2^*}^{2^*})\\&\le C(|u^{(n)}-{\tilde{u}}|_2^{2t}|u^{(n)}-{\tilde{u}}|_{2^*}^{2^*(1-t)}+|u^{(n)}-{\tilde{u}}|_{2^*}^{2^*})\rightarrow 0 \end{aligned}\end{aligned}$$

for some \(C>0\) and \(t=\frac{2^*-2_N}{2^*-2}\). Hence \({\tilde{u}}\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\), and, arguing as before but with \(R=1\), \(J({\tilde{u}})=c\).

Now we consider the case \(\theta =0\) 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 \(\theta =0\) as well. \(\square \)

For \(f:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) measurable we denote by \(f^*\) the Schwarz rearrangement of |f|. Likewise, if \(A\subset {\mathbb {R}}^N\) is measurable, we denote by \(A^*\) the Schwarz rearrangement of A [10, 26].

Lemma 2.10

Assume that (A1)–(A5) and (1.8) are verified, G is of the form (1.10), and either \(\theta =0\) or \(\theta \in (0,\infty )^K\) and (1.9) holds. Then c is attained by a K-tuple of radial, nonnegative and radially nonincreasing functions.

Proof

Let \({\tilde{u}}\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\) such that \(J({\tilde{u}})=c\) be given by Lemma 2.9. For every \(j\in \{1,\dots ,K\}\) let \(u_j\) be the Schwarz rearrangement of \(|{\tilde{u}}_j|\) and denote \(u:=(u_1,\dots ,u_K)\). Let \(a=a(u)\) be determined by Lemma 2.3. In view of the properties of the Schwarz rearrangement [10, 26], we obtain

$$\begin{aligned} M(1\star u)=M(u)\le M({\tilde{u}})=0, \end{aligned}$$

therefore in view of Lemma 2.3 we have that \(a\le 1\) and, consequently, \(M(a\star {\tilde{u}})\ge 0\). Let

$$\begin{aligned} d:=\frac{N}{2}\max _{j=1,\dots , L}\Big (\sum _{i=1}^Kr_{i,j}-2\Big )\ge 2. \end{aligned}$$

Then

$$\begin{aligned}\begin{aligned} c&\le \, J(a\star u)=J(a\star u)-\frac{1}{d}M(a\star u)\\&= \, \int _{{\mathbb {R}}^N}\sum _{i=1}^Ka^2\biggl (\frac{1}{2}-\frac{1}{d}\biggr )|\nabla u_i|^2+\frac{1}{a^N}\biggl (\frac{N}{2d}H_i(a^{N/2}u_i)-G_i(a^{N/2}u_i)\biggr )\,dx\\&\quad -\frac{1}{a^N} \sum _{j=1}^L\beta _j\bigg (1-\frac{1}{d}\Big (\sum _{i=1}^Kr_{i,j}-2\Big )\bigg )\prod _{i=1}^K|a^{N/2}u_i|^{r_{i,j}}\\&\le \, \int _{{\mathbb {R}}^N}\sum _{i=1}^Ka^2\biggl (\frac{1}{2}-\frac{1}{d}\biggr )|\nabla {\tilde{u}}_i|^2+\frac{1}{a^N}\biggl (\frac{N}{2d}H_i(a^{N/2}|{\tilde{u}}_i|)-G_i(a^{N/2}|{\tilde{u}}_i|)\biggr )\,dx\\&\quad -\frac{1}{a^N} \sum _{j=1}^L\beta _j\bigg (1-\frac{1}{d}\Big (\sum _{i=1}^Kr_{i,j}-2\Big )\bigg )\prod _{i=1}^K|a^{N/2}{\tilde{u}}_i|^{r_{i,j}}\\&= \, J(a\star {\tilde{u}})-\frac{1}{d}M(a\star {\tilde{u}})\le J(a\star {\tilde{u}})\le J({\tilde{u}})=c, \end{aligned}\end{aligned}$$

i.e., \(J(a\star u)=c\). \(\square \)

Lemma 2.11

  1. (a)

    Assume that (A1)–(A3), (A4,\(\preceq \)), (A5), and (1.8) hold and let \(u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\) such that \(J(u)=c\) and \(u_i\) is radial for every \(i\in \{1,\dots ,K\}\). Then u is of class \({{\mathcal {C}}}^2\).

  2. (b)

    If, in addition, \(N\in \{3,4\}\), G is of the form (1.10), and \(u_i\) is nonnegative for every \(i\in \{1,\dots ,K\}\), then \(u\in \partial {{\mathcal {D}}}\). Moreover, for every \(i\in \{1,\dots ,K\}\), either \(|u_i|_2=\rho _i\) or \(u_i=0\).

Proof

(a) In Proposition A.1 we set \(f=J\), \(\phi _i(v)=|v_i|_2^2-\rho _i^2\), \(1\le i\le m=K\), \(\psi _1(v)=M(v)\), \(n=1\), \(v\in {{\mathcal {H}}}=H^1({\mathbb {R}}^N)^K\). Then there exist \((\lambda _1,\dots ,\lambda _K)\in [0,\infty )^K\) and \(\sigma \in {\mathbb {R}}\) such that

$$\begin{aligned} -(1-2\sigma )\Delta u_i+\lambda _iu_i=\partial _iG(u)-\sigma \frac{N}{2}\partial _iH(u) \end{aligned}$$
(2.16)

for every \(i\in \{1,\dots ,K\}\) and u satisfies the Nehari identity

$$\begin{aligned} (1\!-\!2\sigma )\!\int _{{\mathbb {R}}^N}|\nabla u|^2\,dx\!+\!\sum _{i=1}^K\int _{{\mathbb {R}}^N}\lambda _i|u_i|^2\,dx\!+\!\int _{{\mathbb {R}}^N}\sigma \frac{N}{2}\langle h(u),u\rangle \!-\!\langle g(u),u\rangle \,dx\!=\!0.\qquad \end{aligned}$$
(2.17)

If \(\sigma =\frac{1}{2}\), then (A4,\(\preceq \)), (A5), and (2.17) yield

$$\begin{aligned}\begin{aligned} 0&\ge \int _{{\mathbb {R}}^N}\frac{N}{4}\langle h(u),u\rangle -\langle g(u),u\rangle \,dx=\int _{{\mathbb {R}}^N}\frac{N}{4}\langle h(u),u\rangle -H(u)-2G(u)\,dx\\&>\int _{{\mathbb {R}}^N}\frac{N}{2}H(u)-2G(u)\,dx\ge 0, \end{aligned}\end{aligned}$$

a contradiction. Hence \(\sigma \ne \frac{1}{2}\) and u satisfies also the Pohožaev identity

$$\begin{aligned} (1-2\sigma )\int _{{\mathbb {R}}^N}|\nabla u|^2\,dx+\frac{2^*}{2}\sum _{i=1}^K\int _{{\mathbb {R}}^N}\lambda _i|u_i|^2\,dx+2^*\int _{{\mathbb {R}}^N}\sigma \frac{N}{2}H(u)-G(u)\,dx=0.\nonumber \\ \end{aligned}$$
(2.18)

Combining (2.17) and (2.18) we obtain

$$\begin{aligned} (1-2\sigma )\int _{{\mathbb {R}}^N}|\nabla u|^2\,dx+\frac{N}{2}\int _{{\mathbb {R}}^N}\sigma N\Bigl (\frac{1}{2}\langle h(u),u\rangle -H(u)\Bigr )-H(u)\,dx=0 \end{aligned}$$

and, using the fact that \(u\in {{\mathcal {M}}}\),

$$\begin{aligned} (1-2\sigma )\int _{{\mathbb {R}}^N}H(u)\,dx+\int _{{\mathbb {R}}^N}\sigma N\Bigl (\frac{1}{2}\langle h(u),u\rangle -H(u)\Bigr )-H(u)\,dx=0, \end{aligned}$$

that is

$$\begin{aligned} \sigma \int _{{\mathbb {R}}^N}\langle h(u),u\rangle -2_NH(u)\,dx=0, \end{aligned}$$

which together with (A4,\(\preceq \)) yields \(\sigma =0\). In view of [12, Theorem 2.3], \(u\in W^{2,q}_loc ({\mathbb {R}}^N)^K\) for all \(q < \infty \), hence \(u\in {{\mathcal {C}}}^{1,\alpha }_loc ({\mathbb {R}}^N)^K\) for all \(\alpha <1\). Then, arguing as in the proof of [10, Lemma 1], we have that u is of class \({{\mathcal {C}}}^2\).

(b) First we show that \(u\in \partial {{\mathcal {D}}}\). Suppose by contradiction that \(|u_i|_2<\rho _i\) for every i. Then \(\lambda _1=\dots =\lambda _K=0\) and from (2.17) and (2.18) (with \(\sigma =0\) as in proof of (a)) there follows

$$\begin{aligned} \int _{{\mathbb {R}}^N}\langle g(u),u\rangle -2^*G(u)\,dx=0. \end{aligned}$$
(2.19)

In view of (A5)

$$\begin{aligned} 2^* G\bigl (u(x)\bigr )=\langle g\bigl (u(x)\bigr ),u(x)\rangle \end{aligned}$$
(2.20)

for every \(x\in {\mathbb {R}}^N\). Since \(G_i\) satisfies (A5), we get \(2^* G_i(u_i(x))\ge g_i(u_i(x))u_i(x)\) for all \(i\in \{1,\dots ,K\}\) and note that

$$\begin{aligned} 2^*\sum _{j=1}^L\beta _j\prod _{i=1}^K|u_i(x)|^{r_{i,j}}\ge \sum _{j=1}^L\beta _j\sum _{k=1}^K r_{k,j}\prod _{i=1}^K|u_i(x)|^{r_{i,j}}, \end{aligned}$$

since \(\sum _{k=1}^K r_{k,j}<2^*\). Hence, from (2.20), the inequalities above are actually equalities. On the other hand, for every \(j\in \{1,\dots ,L\}\), \(\sum _{i=1}^Kr_{i,j}<2^*\), which yields \(\beta _j=0\) or \(\prod _{i=1}^K|u_i(x)|^{r_{i,j}}=0\) for every \(x\in {\mathbb {R}}^N\), so that the coupling term is zero and thus

$$\begin{aligned} 2^* G_i(u_i(x))= g_i(u_i(x))u_i(x) \end{aligned}$$

for every \(i\in \{1,\dots ,K\}\) and every \(x\in {\mathbb {R}}^N\).

Now fix \(i\in \{1,\dots ,K\}\) such that \(u_i\ne 0\). Since \(u_i\in H^1({\mathbb {R}}^N)\cap {{\mathcal {C}}}^2\), there exists an open interval \(I \subset {\mathbb {R}}\) such that \(0 \in {\overline{I}}\) and \(2^* G_i(s)=g_i(s)s\) for \(s\in {\overline{I}}\). Then \(G_i(s)=\theta _i|s|^{2^*}/2^*\) for \(s\in {\overline{I}}\) and \(u_i\) solves \(-\Delta u_i = \theta _i |u_i|^{2^* - 2 } u_i\). Hence, since \(u_i\ge 0\), \(u_i\) is an Aubin–Talenti instanton, up to scaling and translations, which is not \(L^2\)-integrable because \(N \in \{3,4\}\). Therefore \(u\in \partial {{\mathcal {D}}}\).

Now we prove the second part and suppose that there exists \(\nu \in \{1,\dots ,K-1\}\) such that, up to changing the order, \(|u_i|_2<\rho _i\) for every \(i\in \{1,\dots ,\nu \}\) and \(|u_i|_2=\rho _i\) for every \(i\in \{\nu +1,\dots ,K\}\). From Proposition A.1 there exist \(0=\lambda _1=\dots =\lambda _\nu \le \lambda _{\nu +1},\dots ,\lambda _K\) and \(\sigma \in {\mathbb {R}}\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} -(1-2\sigma )\Delta u_i=\partial _iG(u)-\sigma \frac{N}{2}\partial _iH(u) \quad \text {for every }i\in \{1,\dots ,\nu \}\\ -(1-2\sigma )\Delta u_i+\lambda _iu_i=\partial _iG(u)-\sigma \frac{N}{2}\partial _iH(u) \quad \text {for every }i\in \{\nu +1,\dots ,K\} \end{array}\right. } \end{aligned}$$
(2.21)

and as before we obtain \(\sigma =0\). Since \(G_i\) satisfies the scalar variant of (A5), \((0,\infty )\ni s\mapsto G_i(s)/s^{2_N}\in {\mathbb {R}}\) is nondecreasing, hence \(G_i\) is nondecreasing as well for all i. Then, the first \(\nu \) equations in (2.21) with \(\sigma =0\) yield \(-\Delta u_i\ge 0\) for \(i\in \{1,\dots ,\nu \}\). Since \(u\in L^{\frac{N}{N-2}}({\mathbb {R}}^N)^K\) as \(N\in \{3,4\}\), u is of class \({{\mathcal {C}}}^2\), and \(u_i\ge 0\), [21, Lemma A.2] implies \(u_i=0\) for every \(i\in \{1,\dots ,\nu \}\). Notice that we have proved that \(\lambda _i=0\) implies that \(u_i=0\). \(\square \)

Remark 2.12

We point out that in addition to the assumptions of Lemma 2.11, i.e., (A1)–(A3), (A4,\(\preceq \)), (A5), and (1.8) hold, \(u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\), and \(J(u)=c\), we can show that \(u\in \partial {{\mathcal {D}}}\) for any dimension \(N\ge 3\) and without the assumption that G is of the form (1.10) provided that \(H\preceq (2^*-2)G\) holds. Indeed, observe that (2.19) contradicts \(H\preceq (2^*-2)G\) and Lemma 2.1.

Proof of Theorem 1.1

Statement (a) follows from Lemmas 2.9 and 2.10. From Lemma 2.11 (a), u is of class \({{\mathcal {C}}}^2\), while from Proposition A.1 there exist \((\lambda _1,\dots ,\lambda _K)\in [0,\infty )^K\) and \(\sigma \in {\mathbb {R}}\) such that (2.16) holds and \(\sigma =0\) as in the proof of Lemma 2.11 (a). \(\square \)

Proof of Theorem 1.2

It follows from Lemma 2.11 (b), Theorem 1.1 (b), and the maximum principle [18, Lemma IX.V.1] (the implication \(u_i\ne 0\Rightarrow \lambda _i>0\) is proved as in the proof of Lemma 2.11 (b)). \(\square \)

Proof of Corollary 1.4

From Theorem 1.1, there exists \(u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\cap {{\mathcal {C}}}^2({\mathbb {R}}^N)\) and \(\lambda \ge 0\) such that \(J(u)=c\) and \((\lambda ,u)\) is a solution to (1.3). Observe that, from Lemma 2.10, we can assume that u is radial, nonnegative (in fact, positive owing to the maximum principle and because G is nondecreasing on \((0,\infty )\)), and radially nonincreasing provided that G is even. Next, since \(N\in \{3,4\}\) and G is even or \(H\preceq (2^*-2)G\), arguing as in the proof of Lemma 2.11 (b) – see also Remark 2.12 – we obtain that \(u\in \partial {{\mathcal {D}}}={{\mathcal {S}}}\) and \((\lambda ,u)\) is a solution to (1.4). Since u satisfies the Nehari and the Pohožaev inequalities, we get

$$\begin{aligned} \lambda \frac{2}{N-2}\int _{{\mathbb {R}}^N}|u|^2\,dx=\int _{{\mathbb {R}}^N}2^*G(u)-g(u)u\,dx \end{aligned}$$

and, again arguing as in the proof of Lemma 2.11 (b) or Remark 2.12, we obtain \(\int _{{\mathbb {R}}^N}2^*G(u)-g(u)u\,dx>0\), whence \(\lambda >0\). Finally, suppose that G is even, so u is (in particular) positive and radially nonincreasing. Note that \(u(x)\rightarrow 0\) as \(|x|\rightarrow \infty \) and that there exists \(t_0>0\) such that \(g(t)\le \lambda t\) for every \(t\in [0,t_0]\) and \(g(t)>\lambda t\) for every \(t>t_0\). If u is constant in the annulus \(A:=\{\tau _1<|x|<\tau _2\}\) for some \(\tau _2>\tau _1>0\), then \(0=-\Delta u=g(u)-\lambda u\) in A, thus \(-\Delta u\le 0\) in \(\Omega :=\{|x|>\tau _1\}\) because u is radially nonincreasing and \(u(x)\le t_0\) if \(x\in \Omega \). At the same time, u attains the maximum over \({\overline{\Omega }}\) at every point of A, which is impossible because \(u|_\Omega \) is not constant. This proves that u is radially decreasing. \(\square \)

Lemma 2.13

Suppose that \(K=2\), \(L=1\), and the assumptions in Lemma 2.11 (b) hold. If \(r_{1,1}+r_{2,1}>2_N\) and \(\beta _1\) is sufficiently large, then \(u\in {{\mathcal {S}}}\).

Proof

Since \(L=1\), we denote \(\beta _1\), \(r_{1,1}\), \(r_{2,1}\) by \(\beta \), \(r_{1}\), \(r_{2}\) respectively. Suppose by contradiction that \(u_1=0\) or \(u_2=0\), say \(u_1=0\), which implies that \(|u_2|_2=\rho _2\). We want to find a suitable \(w\in {{\mathcal {S}}}\) such that

$$\begin{aligned} J(a\star w)<c=J(0,u_2), \end{aligned}$$
(2.22)

where \(a=a(w)\) is defined in Lemma 2.3 (note that \(a(w)=b(w)\) because (A4,\(\preceq \)) holds), which is impossible. First we show that c does not depend on \(\beta \). Consider the functional

$$\begin{aligned} J_*:v\in H^1({\mathbb {R}}^N)\mapsto \int _{{\mathbb {R}}^N}\frac{1}{2}|\nabla v|^2-G_2(v)\,dx\in {\mathbb {R}}\end{aligned}$$

and the sets

$$\begin{aligned}\begin{aligned} {{\mathcal {D}}}_*&:= \left\{ v \in H^1({\mathbb {R}}^N) \ : \ \int _{{\mathbb {R}}^N} |v|^2 \, dx \le \rho _2^2 \right\} ,\\ {{\mathcal {M}}}_*&:= \left\{ v \in H^1({\mathbb {R}}^N) \setminus \{0\} \ : \ \int _{{\mathbb {R}}^N} |v|^2 \, dx = \frac{N}{2} \int _{{\mathbb {R}}^N}H_2(v)\,dx \right\} . \end{aligned}\end{aligned}$$

Observe that \(J(0,v)=J_*(v)\) for \(v\in H^1({\mathbb {R}}^N)\). Moreover \((0,v)\in {{\mathcal {D}}}\) if and only if \(v\in {{\mathcal {D}}}_*\), and \((0,v)\in {{\mathcal {M}}}\) if and only if \(v\in {{\mathcal {M}}}_*\). In particular,

$$\begin{aligned} c=J(0,u_2)=J_*(u_2)\ge \inf _{{{\mathcal {M}}}_*\cap {{\mathcal {D}}}_*}J_*=\inf \{J(0,v):(0,v)\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\}\ge c, \end{aligned}$$

i.e., \(c=\inf _{{{\mathcal {M}}}_*\cap {{\mathcal {D}}}_*}J_*\), and the claim follows because \(J_*\), \({{\mathcal {D}}}_*\), and \({{\mathcal {M}}}_*\) do not depend on \(\beta \).

In view of Corollary 1.4, there exists \({\bar{v}}\in {{\mathcal {M}}}_*\cap \partial {{\mathcal {D}}}_*\) such that

$$\begin{aligned} J_*({\bar{v}})=\inf _{{{\mathcal {M}}}_*\cap {{\mathcal {D}}}_*}J_*=c=\inf _{{{\mathcal {M}}}_*\cap \partial {{\mathcal {D}}}_*}J_*. \end{aligned}$$

Note that \({\bar{v}}\) does not depend on \(\beta \). Define \(w=(w_1,w_2):=\bigl (\frac{\rho _1}{\rho _2}{\bar{v}},{\bar{v}}\bigr )\). From Lemma 2.3, \(a=a_\beta \) is implicitly defined by

$$\begin{aligned}\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla w|^2\,dx&=\,\frac{N}{2}\int _{{\mathbb {R}}^N}\frac{G_1'(a_\beta ^{N/2}w_1)a_\beta ^{N/2}w_1-2G_1(a_\beta ^{N/2}w_1)}{a_\beta ^{N+2}}\\&\quad +\frac{G_2'(a_\beta ^{N/2}w_2)a_\beta ^{N/2}w_2-2G_2(a_\beta ^{N/2}w_2)}{a_\beta ^{N+2}}\\&\quad +\beta (r_1+r_2-2)a_\beta ^{N(r_1+r_2-2)/2-2}w_1^{r_1}w_2^{r_2}\,dx\\&\ge \,\beta (r_1+r_2-2)a_\beta ^{N(r_1+r_2-2)/2-2}\frac{N}{2}\int _{{\mathbb {R}}^N}w_1^{r_1}w_2^{r_2}\,dx, \end{aligned}\end{aligned}$$

hence there exist \(C>0\) not depending on \(\beta \) such that

$$\begin{aligned} 0<\beta a_\beta ^{N(r_1+r_2-2)/2-2}\le C, \end{aligned}$$
(2.23)

whence

$$\begin{aligned} \lim _{\beta \rightarrow \infty }a_\beta =0. \end{aligned}$$
(2.24)

Since \(a_\beta \star w\in {{\mathcal {M}}}\), we have from (A5)

$$\begin{aligned}\begin{aligned} J(a_\beta \star w)&=\int _{{\mathbb {R}}^N}\frac{N}{4}H(a_\beta \star w)-G(a_\beta \star w)\,dx\le \frac{2}{N-2}\int _{{\mathbb {R}}^N}G(a_\beta \star w)\,dx\\&=\frac{2}{N-2}\int _{{\mathbb {R}}^N}\frac{G_1(a_\beta ^{N/2}w_1)+G_2(a_\beta ^{N/2}w_2)}{a_\beta ^N}\,dx\\&\quad +\frac{2\beta a_\beta ^{N(r_1+r_2-2)/2}}{N-2}\int _{{\mathbb {R}}^N}w_1^{r_1}w_2^{r_2}\,dx, \end{aligned}\end{aligned}$$

therefore (2.22) holds true for sufficiently large \(\beta \) owing to (A1), (2.23), and (2.24). \(\square \)

Proof of Theorem 1.3

It follows from Lemma 2.13 and Theorem 1.2. \(\square \)

Now we investigate the behaviour of the ground state energy with respect to \(\rho \). For \(\rho =(\rho _1,\dots ,\rho _K)\in (0,\infty )^K\) we denote

$$\begin{aligned} {{\mathcal {D}}}(\rho ):= & {} \biggl \{u\in H^1({\mathbb {R}}^N)^K:\int _{{\mathbb {R}}^N}|u_i|^2\,dx\le \rho _i^2 \text { for every } i\in \{1,\dots ,K\}\biggr \}\\ {{\mathcal {S}}}(\rho ):= & {} \biggl \{u\in H^1({\mathbb {R}}^N)^K:\int _{{\mathbb {R}}^N}|u_i|^2\,dx=\rho _i^2 \text { for every } i\in \{1,\dots ,K\}\biggr \}\\ c(\rho ):= & {} \inf \{J(u):u\in {{\mathcal {M}}}\cap {{\mathcal {D}}}(\rho )\}. \end{aligned}$$

Proposition 2.14

Assume that (A0)–(A5) and (1.8) are satisfied.

  1. (i)

    If \(\theta =0\), then c is continuous and \(\lim _{\rho \rightarrow 0^+}c(\rho )=\infty \), where \(\rho \rightarrow 0^+\) means \(\rho _i\rightarrow 0^+\) for every \(i\in \{1,\dots ,K\}\).

  2. (ii)

    Let \(\theta \in (0,\infty )^K\) and \(\rho \in (0,\infty )^K\) . If (1.9) holds for every \(\rho '\in \prod _{j=1}^K(\rho _j-\varepsilon ,\rho _j)\) and some \(\varepsilon >0\), then c is continuous at \(\rho \). If (1.9) holds for every \(\rho '\in (0,\varepsilon )^K\) and some \(\varepsilon >0\), then \(\displaystyle \lim _{\rho '\rightarrow 0^+}c(\rho ')= \frac{1}{N}S^{N/2}\sum _{i=1}^K\theta _i^{1-N/2}\)

  3. (iii)

    If every ground state solution to (1.3) belongs to \({{\mathcal {S}}}(\rho )\) (e.g. if the assumptions of Theorem 1.3 are satisfied), then c is decreasing in the following sense: if \(\rho ,\rho '\in (0,\infty )^K\) are such that \(\rho _i\ge \rho _i'\) for every \(i\in \{1,\dots ,K\}\) and \(\rho _j>\rho _j'\) for some \(j\in \{1,\dots ,K\}\), then \(c(\rho )<c(\rho ')\).

Proof

Fix \(\rho \in (0,\infty )^K\) and let \(\rho ^{(n)}\rightarrow \rho \). We begin by proving the upper semicontinuity of c at \(\rho \). Let \(w\in {{\mathcal {M}}}\cap {{\mathcal {D}}}(\rho )\) such that \(J(w)=c(\rho )\), denote \(w_i^{(n)}:=\rho _i^{(n)}w_i/\rho _i\), and consider \(w^{(n)}=(w_1^{(n)},\dots ,w_K^{(n)})\in {{\mathcal {D}}}(\rho ^{(n)})\). Due to Lemma 2.3, for every n there exists \(s_n>0\) such that \(s_n\star w^{(n)}\in {{\mathcal {M}}}\). Note that

$$\begin{aligned} \frac{N}{2}\int _{{\mathbb {R}}^N}\frac{H\bigl (s_n^{N/2}(\rho _1^{(n)}w_1/\rho _1,\dots ,\rho _K^{(n)}w_K/\rho _K)\bigr )}{s_n^{N+2}}\,dx=\int _{{\mathbb {R}}^N}|\nabla w^{(n)}|^2\,dx\rightarrow \int _{{\mathbb {R}}^N}|\nabla w|^2\,dx.\nonumber \\ \end{aligned}$$
(2.25)

If \(\limsup _ns_n=\infty \), then from (A2) and (A5) the left-hand side of (2.25) tends to \(\infty \) up to a subsequence, which is a contradiction. If \(\liminf _ns_n=0\), then from (A1), (A3), (A5) and (1.8) and arguing as in Lemma 2.2 we obtain that the limit superior of the left-hand side of (2.25) is less than \(|\nabla w|_2^2\), which is again a contradiction. There follows that, up to a subsequence, \(s_n\rightarrow s\) for some \(s>0\) and \(s\star w\in {{\mathcal {M}}}\). In view of Lemma 2.3,

$$\begin{aligned} \limsup _nc(\rho ^{(n)})\le \lim _nJ(s_n\star w_n)=J(s\star w)=J(w)=c(\rho ). \end{aligned}$$

Now we prove the lower semicontinuity of c at \(\rho \). Let \(\rho ^{(n)}\rightarrow \rho \) and \(u^{(n)}\in {{\mathcal {M}}}\cap {{\mathcal {D}}}(\rho ^{(n)})\subset {{\mathcal {M}}}\cap {{\mathcal {D}}}(2\rho )\) such that \(J(u^{(n)})=c(\rho ^{(n)})\le c(\rho /2)\). In view of Lemma 2.4, \((u^{(n)})\) is bounded, hence we can consider the sequences \(({\tilde{u}}^{(i)})\) and \((y^{(i,n)})\) given by Lemma 2.8; note that \({\tilde{u}}^{(i)}\in {{\mathcal {D}}}\). We consider the case \(\theta \in (0,\infty )^K\) because the other one (i.e., \(\theta =0\)) is similar and simpler.

Claim: There exists \(i\ge 0\) such that \(\lim _nu^{(n)}(\cdot +y^{(i,n)})\rightarrow {\tilde{u}}^{(i)}\ne 0\) in \({{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\). The proof is similar to that of Lemma 2.9, thus we focus only on the differences. If \({\tilde{u}}^{(i)}=0\) for every \(i\ge 0\), then as in (2.10) we obtain the contradiction

$$\begin{aligned} \frac{{\bar{S}}^{N/2}}{N}>c(\rho _1-\varepsilon ,\dots ,\rho _K-\varepsilon )\ge \limsup _nc(\rho ^{(n)})=\limsup _nJ(u^{(n)})\ge \frac{{\bar{S}}^{N/2}}{N}. \end{aligned}$$
(2.26)

Let \(i\ge 0\) such that \({\tilde{u}}^{(i)}\ne 0\) and define \(v^{(n)}:=u^{(n)}(\cdot +y^{(i,n)})-{\tilde{u}}^{(i)}\). If \(\liminf _n|\nabla v^{(n)}|_2>0\) and \(|\nabla {\tilde{u}}^{(i)}|_2^2\ge \frac{N}{2}\int _{{\mathbb {R}}^N}H({\tilde{u}}^{(n)})\,dx\), then we prove that \(R_n\rightarrow 1\), where \(R_n>0\) is such that \(v^{(n)}(R_n\cdot )\in {{\mathcal {M}}}\). In particular, if up to a subsequence \(R_n\ge 1\), then as in (2.12) we get

$$\begin{aligned}\begin{aligned} 0<c(2\rho )\le c(\rho ^{(n)})&\le \frac{1}{R_n}\int _{{\mathbb {R}}^N}\frac{N}{4}H(v^{(n)})-g(v^{(n)})\,dx\le \int _{{\mathbb {R}}^N}\frac{N}{4}H(v^{(n)})-G(v^{(n)})\,dx\\&\le c(\rho ^{(n)})+o(1). \end{aligned}\end{aligned}$$

Next, as in (2.14) we obtain again the contradiction (2.26), which proves that \(v^{(n)}\rightarrow 0\) in \({{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\) (up to a subsequence) or \(|\nabla {\tilde{u}}^{(i)}|_2^2<\frac{N}{2}\int _{{\mathbb {R}}^N}H({\tilde{u}}^{(n)})\,dx\). In the latter case, we define \(R>1\) such that \({\tilde{u}}^{(i)}(R\cdot )\in {{\mathcal {M}}}\) as in (2.15) we get the contradiction

$$\begin{aligned} c(\rho )\le J\bigl ({\tilde{u}}^{(i)}(R\cdot )\bigr )<\limsup _nc(\rho ^{(n)})\le c(\rho ), \end{aligned}$$

where the last inequality is due to the upper semicontinuity. This proves the Claim, which yields, together with the interpolation inequality, that \({\tilde{u}}^{(i)}\in {{\mathcal {M}}}\cap {{\mathcal {D}}}\) and so

$$\begin{aligned} c(\rho )\le J({\tilde{u}}^{(i)})=\lim _nJ(u^{(n)})=\lim _nc(\rho ^{(n)}). \end{aligned}$$

Now we prove the behaviour of \(c(\rho ')\) as \(\rho '\rightarrow 0\). Let \(\rho ^{(n)}\rightarrow 0^+\) and \(u^{(n)}\in {{\mathcal {M}}}\cap {{\mathcal {D}}}(\rho ^{(n)})\) such that \(J(u^{(n)})=c(\rho ^{(n)})\). Denote \(s_n:=|\nabla u^{(n)}|_2^{-1}\) and \(w^{(n)}:=s_n\star u^{(n)}\) and note that \(s_n^{-1}\star w^{(n)}=u^{(n)}\in {{\mathcal {M}}}\), \(|\nabla w^{(n)}|_2=1\) and

$$\begin{aligned} |w^{(n)}|_2^2=|u^{(n)}|_2^2=|\rho ^{(n)}|^2\rightarrow 0 \end{aligned}$$

as \(n\rightarrow \infty \). In particular \(\bigl (w^{(n)}\bigr )\) is bounded in \(L^{2^*}({\mathbb {R}}^N)^K\) and so

$$\begin{aligned} |w^{(n)}|_{2_N}\le |w^{(n)}|_2^\frac{2}{N+2}|w^{(n)}|_{2^*}^\frac{N}{N+2}\rightarrow 0 \end{aligned}$$

as \(n\rightarrow \infty \). Suppose that \(\theta =0\). Then, in view of (A1) and (A3), for every \(s>0\)

$$\begin{aligned} \lim _n\int _{{\mathbb {R}}^N}\frac{G(s^{N/2}w^{(n)})}{s^{N}}\,dx=0 \end{aligned}$$

and, consequently,

$$\begin{aligned} J(u^{(n)})=J(s_n^{-1}\star w^{(n)})\ge J(s\star w^{(n)})=\frac{s^2}{2}-\int _{{\mathbb {R}}^N}\frac{G(s^{N/2}w^{(n)})}{s^N}\,dx=\frac{s^2}{2}+o(1), \end{aligned}$$

whence \(\lim _nJ(u^{(n)})=\infty \).

Now suppose that \(\theta \in (0,\infty )^K\). Since \(|u^{(n)}|_2^2=|\rho ^{(n)}|^2\rightarrow 0\), we get \(u^{(n)}\rightarrow 0\) in \(L^q({\mathbb {R}}^N)^K\) for \(2\le q<2^*\). Arguing as above, for every \(s>0\)

$$\begin{aligned} \lim _n\int _{{\mathbb {R}}^N}\frac{\widetilde{G}(s^{N/2}u^{(n)})}{s^{N}}\,dx=0, \end{aligned}$$

hence

$$\begin{aligned} \lim _n\int _{{\mathbb {R}}^N}\frac{G(s^{N/2}u^{(n)})}{s^{-N}}\,dx=\lim _n\int _{{\mathbb {R}}^N}\frac{\widetilde{G}(s^{N/2}u^{(n)})}{s^{-N}}\,dx+\frac{s^{2^*}}{2^*}\sum _{j=1}^K\theta _j \lim _n\int _{{\mathbb {R}}^N}|u^{(n)}_j|^{2^*}\, dx. \end{aligned}$$

Consequently,

$$\begin{aligned} J(u^{(n)})\ge & {} J(s\star u^{(n)})=\frac{s^2}{2}\int _{{\mathbb {R}}^N}|\nabla u^{(n)}|^2\,dx-\int _{{\mathbb {R}}^N}\frac{G(s^{N/2}u^{(n)})}{s^N}\,dx\\= & {} \frac{s^2}{2}\lim _n\int _{{\mathbb {R}}^N}|\nabla u^{(n)}|^2\,dx-\frac{s^{2^*}}{2^*}\sum _{j=1}^K\theta _j \lim _n\int _{{\mathbb {R}}^N}|u^{(n)}_j|^{2^*}\, dx+o(1) \end{aligned}$$

for any \(s>0\). Then, in view of Lemma B.1

$$\begin{aligned} \lim _nJ(u^{(n)})\ge & {} \max _{s>0}\; \frac{s^2}{2}\lim _n\int _{{\mathbb {R}}^N}|\nabla u^{(n)}|^2\,dx-\frac{s^{2^*}}{2^*}\sum _{j=1}^K\theta _j \lim _n\int _{{\mathbb {R}}^N}|u^{(n)}_j|^{2^*}\, dx\\= & {} \frac{1}{N}\frac{\lim _n|\nabla u^{(n)}|_2^N}{\left( \sum _{j=1}^K\theta _j\lim _n|u_j^{(n)}|_{2^*}^{2^*}\right) ^{N/2-1}}\\\ge & {} \frac{1}{N}{\bar{S}}^{\frac{N}{2}}=\frac{1}{N}S^{N/2}\sum _{i=1}^K\theta _i^{1-N/2} \end{aligned}$$

and taking into account (1.9) we obtain

$$\begin{aligned} \lim _nJ(u^{(n)})=\frac{1}{N}S^{N/2}\sum _{i=1}^K\theta _i^{1-N/2}. \end{aligned}$$

Now assume that every ground state solution to (1.3) belongs to \({{\mathcal {S}}}(\rho )\) and let \(\rho ,\rho '\) as in the statement. Let \(u\in {{\mathcal {M}}}\cap {{\mathcal {S}}}(\rho )\) and \(u'\in {{\mathcal {M}}}\cap {{\mathcal {S}}}(\rho ')\subset {{\mathcal {M}}}\cap {{\mathcal {D}}}(\rho )\setminus {{\mathcal {S}}}(\rho )\) such that \(J(u)=c(\rho )\) and \(J(u')=c(\rho ')\). Clearly \(c(\rho )\le c(\rho ')\). If \(c(\rho )=c(\rho ')\), then \(c(\rho )=J(u')\), with \(u'\in {{\mathcal {M}}}\cap {{\mathcal {D}}}(\rho )\setminus {{\mathcal {S}}}(\rho )\), which is a contradiction. \(\square \)

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Acknowledgements

The authors are grateful to the reviewer for the fruitful remarks concerning the first version of this paper. The authors were partially supported by the National Science Centre, Poland (Grant No. 2017/26/E/ST1/00817). J. Mederski was also partially supported by the Alexander von Humboldt Foundation (Germany) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 258734477—SFB 1173 during the stay at Karlsruhe Institute of Technology.

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Appendices

Appendix A. Sign of Lagrange multipliers

The following result concerns the sign of a Lagrange multiplier when the corresponding constraint is given by an inequality and the critical point of the restricted functional is a minimizer. The result is related with Clarke’s [15, Theorem 1], however it is not clear whether we can apply it directly in our situation.

Proposition A.1

Let \({{\mathcal {H}}}\) be a real Hilbert space and \(f,\phi _i,\psi _j\in {{\mathcal {C}}}^1({{\mathcal {H}}})\), \(i\in \{1,\dots ,m\}\), \(j\in \{1,\dots ,n\}\). Suppose that for every

$$\begin{aligned} x\in \bigcap _{i=1}^m\phi _i^{-1}(0)\cap \bigcap _{j=1}^n\psi _j^{-1}(0) \end{aligned}$$

the differential

$$\begin{aligned} \bigl (\phi _i'(x),\psi _j'(x)\bigr )_{1\le i\le m,1\le j\le n}:{{\mathcal {H}}}\rightarrow {\mathbb {R}}^{m+n} \end{aligned}$$

is surjective. If \({\bar{x}}\in {{\mathcal {H}}}\) minimizes f over

$$\begin{aligned} \{x\in {{\mathcal {H}}}:\phi _i(x)\le 0\text { for every }i=1,\dots ,m\text { and }\psi _j(x)=0\text { for every }j=1,\dots ,n\}, \end{aligned}$$

then there exist \((\lambda _i)_{i=1}^m\in [0,\infty )^m\) and \((\sigma _j)_{j=1}^n\in {\mathbb {R}}^n\) such that

$$\begin{aligned} f'({\bar{x}})+\sum _{i=1}^m \lambda _i\phi _i'({\bar{x}})+\sum _{j=1}^n \sigma _i\psi _j'({\bar{x}})=0. \end{aligned}$$

Proof

Fix \(\varepsilon >0\) and define the functional \(F:{{\mathcal {H}}}\rightarrow [0,\infty )\) as

$$\begin{aligned} F(x):=\max _{1\le i\le m,1\le j\le n}\{f(x)-f({\bar{x}})+\varepsilon ,\phi _i(x),|\psi _j(x)|\}. \end{aligned}$$

and observe that F is locally Lipschitz and bounded from below by 0. Since \(F({\bar{x}})=\varepsilon \), in view of the Ekeland variational principle [16, Theorem 1.1] there exists \(z=z_\varepsilon \in {{\mathcal {H}}}\) such that

$$\begin{aligned} \Vert {\bar{x}}-z\Vert\le & {} \sqrt{\varepsilon },\\ F(x)+\sqrt{\varepsilon }\,\Vert x-z\Vert\ge & {} F(z) \quad \forall x\in {{\mathcal {H}}}. \end{aligned}$$

From [15, Propositions 6, 8] there follows that \(0\in \partial F(z)+\sqrt{\varepsilon }\,\partial \Vert \cdot -z\Vert (z)\), where \(\partial \) stands for the generalized gradient [15, Definition 1]. Hence, there exists \(\xi =\xi _\varepsilon \in \partial F(z)\) such that \(-\xi \in \sqrt{\varepsilon }\,\partial \Vert \cdot -z\Vert (z)\). In view of [15, Propositions 1, 9], \(\Vert \xi \Vert \le \sqrt{\varepsilon }\) and \(\xi \) lies in the convex hull of \(f(z)-f({\bar{x}})+\varepsilon \), \(\phi _i(z)\), and \(|\psi _j(z)\)|, i.e., there exists \(\tau ,\lambda _1,\dots ,\lambda _m,{\hat{\sigma }}_1,\dots ,{\hat{\sigma }}_n\ge 0\) depending on \(\varepsilon \), such that \(\tau +\lambda _1+\dots +\lambda _m+{\hat{\sigma }}_1+\dots +{\hat{\sigma }}_n=1\),

$$\begin{aligned} \xi \in \biggl (\tau f'(z)+\sum _{i=1}^m\lambda _i\phi _i'(z)+\sum _{j=1}^n{\hat{\sigma }}_j\partial |\psi _j|(z)\biggr ), \end{aligned}$$

and \(\lambda _i=0\) (resp. \({\hat{\sigma }}_j=0\)) if \(\phi _i(z)\le 0\) (resp. \(\psi _j(z)=0\)).

For every \(j\in \{1,\dots ,n\}\) such that \(\psi _j(z)\ne 0\) we have

$$\begin{aligned} \partial |\psi _j|(z)=\{sign \bigl (\psi _j(z)\bigr )\psi _j'(z)\}. \end{aligned}$$

If \(j\in \{1,\dots ,n\}\) is as before, we define \(\sigma _j:=sign \bigl (\psi _j(z)\bigr ){\hat{\sigma }}_j\), otherwise we define \(\sigma _j:=0\). In particular, we have

$$\begin{aligned} \sum _{j=1}^n{\hat{\sigma }}_j\partial |\psi _j|(z)=\Biggl \{\sum _{j=1}^n\sigma _j\psi _j'(z)\Biggr \}. \end{aligned}$$

Summing up, we obtain the following: for every \(\varepsilon >0\) there exist \(\tau \ge 0\), \((\lambda _i)_{i=1}^m\in [0,\infty )^m\), \((\sigma _j)_{j=1}^n\in {\mathbb {R}}^n\) and \(z\in B({\bar{x}},\sqrt{\varepsilon })\) such that

$$\begin{aligned}&\xi :=\tau f'(z)+\sum _{i=1}^m\lambda _i\phi _i'(z)+\sum _{j=1}^n\sigma _j\psi _j'(z)\in B(0,\sqrt{\varepsilon }),\\&\tau +\sum _{i=1}^m\lambda _i+\sum _{j=1}^n|\sigma _j|=1. \end{aligned}$$

Letting \(\varepsilon \rightarrow 0^+\) we get

$$\begin{aligned} \tau f'({\bar{x}})+\sum _{i=1}^m\lambda _i\phi _i'({\bar{x}})+\sum _{j=1}^n\sigma _j\psi _j'({\bar{x}})=0 \end{aligned}$$
(A.1)

for some \(\tau \ge 0\), \((\lambda _i)_{i=1}^m\in [0,\infty )^m\), \((\sigma _j)_{j=1}^n\in {\mathbb {R}}^n\) such that

$$\begin{aligned} \tau +\sum _{i=1}^m\lambda _i+\sum _{j=1}^n|\sigma _j|=1. \end{aligned}$$

Suppose by contradiction that \(\tau =0\), whence

$$\begin{aligned} \sum _{i=1}^m\lambda _i\phi _i'({\bar{x}})+\sum _{j=1}^n\sigma _j\psi _j'({\bar{x}})=0. \end{aligned}$$
(A.2)

If \(\phi _i({\bar{x}})<0\) for some \(i\in \{1,\dots ,m\}\), then of course \(\lambda _i=0\), hence, up to considering a (possibly empty) subset of \(\{1,\dots ,m\}\) in (A.2), we can assume that \(\phi _1({\bar{x}})=\dots =\phi _{m_0}({\bar{x}})=0\) and \(\lambda _{m_0+1}=\ldots =\lambda _m=0\) for some \(0\le m_0\le m\), where \(m_0=0\) denotes that \(\lambda _i=0\) for all \(i\in \{1,\dots ,m\}\), whereas \(m_0=m\) denotes \(\phi _1({\bar{x}})=\dots =\phi _{m}({\bar{x}})=0\). Then the differential

$$\begin{aligned} \bigl (\phi _1'({\bar{x}}),\dots ,\phi _{m_0}'({\bar{x}}),\psi _1'({\bar{x}}),\dots ,\psi _n'({\bar{x}})\bigr ):{{\mathcal {H}}}\rightarrow {\mathbb {R}}^{m_0+n} \end{aligned}$$

is surjective and so, for every \(i\in \{1,\dots ,m_0\}\) (resp. \(j\in \{1,\dots ,n\}\)), we can choose \(y\in {{\mathcal {H}}}\) such that \(\phi _i'({\bar{x}})(y)\ne 0\), \(\phi _k'({\bar{x}})(y)=0\) for every \(k\in \{1,\dots ,m_0\}\setminus \{i\}\) and \(\psi _j'({\bar{x}})(y)=0\) for every \(j\in \{1,\dots ,n\}\) (resp. \(\psi _j'({\bar{x}})(y)\ne 0\), \(\psi _k'({\bar{x}})(y)=0\) for every \(k\in \{1,\dots ,n\}\setminus \{j\}\) and \(\phi _i'({\bar{x}})(y)=0\) for every \(i\in \{1,\dots ,m_0\}\)). This and (A.2) imply \(\lambda _i=0\) for every \(i\in \{1,\dots ,m_0\}\) and \(\sigma _j=0\) for every \(j\in \{1,\dots ,n\}\), a contradiction. We can thus divide both sides of (A.1) by \(\tau \) and, up to relabelling \(\lambda _i\) and \(\sigma _j\) (\(i\in \{1,\dots ,m_0\}\), \(j\in \{1,\dots ,n\}\)), conclude the proof. \(\square \)

Appendix B. A Sobolev-type constant

Let \(\theta =(\theta _1,\dots ,\theta _K)\in (0,\infty )^K\),

$$\begin{aligned} {{\bar{S}}}:= \inf _{u\in {{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\setminus \{0\}} \frac{\int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx}{\left( \sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N} |u_j|^{2^*} \, dx\right) ^{2/2^*}}, \end{aligned}$$

and, clearly, in view of the Sobolev embeddings, \({{\bar{S}}}>0\).

Lemma B.1

\({{\bar{S}}}\) is attained by \((\theta _1^{-\frac{N-2}{4}} u_1,\ldots ,\theta _K^{-\frac{N-2}{4}} u_K)\), where \(u_j\) are Aubin–Talenti instantons. Moreover

$$\begin{aligned} {{\bar{S}}}:=\left( \sum _{j=1}^K\theta _j^{-\frac{N-2}{2}}\right) ^{2/N}S. \end{aligned}$$

Proof

We prove that \({\bar{S}}\) is attained. Let \(I:{{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\rightarrow {\mathbb {R}}\) be defined as

$$\begin{aligned} I(u)=\int _{{\mathbb {R}}^N}\frac{1}{2}|\nabla u|^2-\frac{1}{2^*}\sum _{j=1}^K\theta _j|u_j|^{2^*}\,dx. \end{aligned}$$

If \(u=(u_1,\ldots ,u_K)\in {{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\), then

$$\begin{aligned} I'(u)=0 \Leftrightarrow -\Delta u_j=\theta _j|u_j|^{2^*-2}u_j \text { for every } j\in \{1,\dots ,K\}. \end{aligned}$$

Define the Nehari manifold for I as

$$\begin{aligned} {{\mathcal {N}}}:=\left\{ u\in {{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\setminus \{0\} \, : \, |\nabla u|_2^2=\sum _{j=1}^K\theta _j|u_j|_{2^*}^{2^*}\right\} \end{aligned}$$

ad note that, if \(u\in {{\mathcal {N}}}\), then

$$\begin{aligned}\begin{aligned} I(u)&=\frac{1}{N}|\nabla u|_2^2=\frac{1}{N}\sum _{j=1}^K\theta _j|u_j|_{2^*}^{2^*}\\ \frac{\int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx}{\left( \sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N} |u_j|^{2^*} \, dx\right) ^{2/2^*}}&=|\nabla u|_2^{2(2^*-2)/2^*}=\left( \sum _{j=1}^K\theta _j|u_j|_{2^*}^{2^*}\right) ^{(2^*-2)/2^*}, \end{aligned}\end{aligned}$$

hence \(\frac{\int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx}{\left( \sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N} |u_j|^{2^*} \, dx\right) ^{2/2^*}}=A\) if and only if \(I(u)=\frac{1}{N} A^{2^*/(2^*-2)}\). Moreover, if \(u\in {{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\setminus \{0\}\), then \(tu\in {{\mathcal {N}}}\) for some \(t>0\) and the fraction in the definition of \({\bar{S}}\) does not depend on the rescaling \(t\mapsto tu\), therefore

$$\begin{aligned} {{\bar{S}}}= \inf _{u\in {{\mathcal {N}}}} \frac{\int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx}{\left( \sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N} |u_j|^{2^*} \, dx\right) ^{2/2^*}} \end{aligned}$$

and \(\inf _{{\mathcal {N}}}I=\frac{1}{N}{\bar{S}}^{2/(2^*-2)^*}>0\). Let \(u^{(n)}\in {{\mathcal {N}}}\) be such that \(I(u^{(n)})\rightarrow \inf _{{\mathcal {N}}}I\). Up to replacing \(u^{(n)}=(u_1^{(n)},\ldots ,u_K^{(n)})\) with \((|u_1^{(n)}|,\ldots ,|u_K^{(n)}|)\), we can also assume that \(u_j^{(n)}\ge 0\) for every nj. In virtue of Ekeland’s variational principle [44], we can assume \(I'(u^{(n)})\rightarrow 0\). Since \(\inf _{{\mathcal {N}}}I>0\), \(u^{(n)}\not \rightarrow 0\) in \(L^{2^*}({\mathbb {R}}^N)^K\), thus, in view of Solimini’s theorem [37, Theorem 1], see also [42, Lemma 5.3]), there exist \((s_n)\subset (0,\infty )\), \((y_n)\subset {\mathbb {R}}^N\) and \(u\in {{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\setminus \{0\}\), such that \(s_n^{1/2}u^{(n)}(s_n\cdot +y_n))\rightharpoonup u\) in \({{\mathcal {D}}}^{1,2}({\mathbb {R}}^N)^K\) and \(s_n^{1/2}u^{(n)}(s_n\cdot +y_n))\rightarrow u\) a.e. in \({\mathbb {R}}^N\) up to a subsequence. In particular, \(I'(u)=0\) and so \(u\in {{\mathcal {N}}}\). Observe that each component of u is of the form \(u_j=\theta _j^{-\frac{N-2}{4}} u_j^0\), where \(u_j^0\) is an Aubin–Talenti instanton. Therefore

$$\begin{aligned} {{\bar{S}}}= \frac{\int _{{\mathbb {R}}^N} |\nabla u|^2 \, dx}{\left( \sum _{j=1}^K\theta _j\int _{{\mathbb {R}}^N} |u_j|^{2^*} \, dx\right) ^{2/2^*}} =\left( \sum _{j=1}^K\theta _j^{-\frac{N-2}{2}}\right) ^{2/N}S. \end{aligned}$$

\(\square \)

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Mederski, J., Schino, J. Least energy solutions to a cooperative system of Schrödinger equations with prescribed \(L^2\)-bounds: at least \(L^2\)-critical growth. Calc. Var. 61, 10 (2022). https://doi.org/10.1007/s00526-021-02116-0

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  • DOI: https://doi.org/10.1007/s00526-021-02116-0

Mathematics Subject Classification

  • Primary 35Q40
  • 35Q60
  • Secondary 35J20
  • 78A25