Ground States for a Nonlinear Schrödinger System with Sublinear Coupling Terms

1 Introduction

In this paper we consider the system of d equations

with λi,μi>0, and bi⁢j=bj⁢i>0, which appears in several physical contexts, namely in nonlinear optics (see for instance [1] and the references therein). We also assume that

Observe that the system is variational, more precisely of gradient type, since solutions can be obtained as critical points of the C1-functional

I d : E := ( H 1 ⁢ ( ℝ n ) ) d → ℝ

defined by

I d ⁢ ( 𝐮 ) = I d ⁢ ( u 1 , … , u d ) := 1 2 ⁢ ∑ i = 1 d ∥ u i ∥ λ i 2 - 1 2 ⁢ q ⁢ ∑ i = 1 d μ i ⁢ | u i | 2 ⁢ q 2 ⁢ q - 1 q ⁢ ∑ i , j = 1 , i < j d b i ⁢ j ⁢ | u i ⁢ u j | q q ,

where |⋅|q denotes the standard Lq⁢(ℝn) norm, while

∥ u i ∥ λ i 2 := ∫ ℝ n ( | ∇ ⁡ u i | 2 + λ i ⁢ u i 2 ) ⁢ 𝑑 x , i = 1 , … , d .

We will focus on the existence of ground state solutions of (1.1), that is, solutions of the system that achieve the ground state level

A very interesting question is whether, when c is achieved, the ground state 𝐮 is nontrivial, meaning that all its components ui are nonzero.

This problem has attracted a lot of attention in the last decade, specially in the particular case of d=2 equations:

{ - Δ ⁢ u 1 + λ 1 ⁢ u 1 = μ 1 ⁢ | u 1 | 2 ⁢ q - 2 ⁢ u 1 + b ⁢ | u 2 | q ⁢ | u 1 | q - 2 ⁢ u 1 , - Δ ⁢ u 2 + λ 2 ⁢ u 2 = μ 2 ⁢ | u 2 | 2 ⁢ q - 2 ⁢ u 2 + b ⁢ | u 1 | q ⁢ | u 2 | q - 2 ⁢ u 2 .

For μ1=μ2=1, Maia, Montefusco and Pellacci proved in [18] that c is always achieved, while there exists a positive ground state (i.e., u1,u2>0 in ℝn) if b≥Λ, for a certain Λ>0 depending on λ2/λ1. The same type of result was proved by Ambrosetti and Colorado [2, 3] for q=2, n=2,3, and by de Figueiredo and Lopes [11] for n=1. On the other hand, for q≥2, there are regions where all nonnegative solutions must have a null component, as it was observed for instance by Bartsch and Wang [4], Sirakov [21] and Chen and Zou [7].

The optimal bounds for the existence of nontrivial ground states were found by Mandel [19] for every q as in (1.2). More precisely, in [19, Theorem 1] it is shown that there exists b¯:=b⁢(λ2/λ1,q,n) such that for b<b¯ all ground states have a trivial component (we will call them semitrivial ground states), while for b>b¯ all ground states are nontrivial. For μ1=μ2=1 and λ2/λ1=ω2≥1, the threshold is given by the expression (see [19, (5)])

b ¯ = inf ⁡ { c ^ 0 - 2 ⁢ q ⁢ ( ∥ u ∥ 1 2 + ∥ v ∥ w 2 2 ) q - | u | 2 ⁢ q 2 ⁢ q - | v | 2 ⁢ q 2 ⁢ q 2 ⁢ | u ⁢ v | q q : u , v ∈ H 1 ⁢ ( ℝ n ) } ,

where c^0:=∥u0∥1⁢|u0|2⁢q-1 and u0 is the unique positive radially decreasing solution of -Δ⁢u0+u0=|u0|2⁢q-2⁢u0 in ℝn (for the uniqueness result, see [13]). It is also shown that b¯=0 if 1<q<2 (see [19, Lemmas 1 (i) and 2 (i)]).

Our aim is to generalize this last result for an arbitrary number of equations. In order to state our results, let us first introduce some notations.

We will study the minimization problem

inf ⁡ { I d ⁢ ( u ) : u ∈ 𝒩 d } ,

where the so-called Nehari manifold 𝒩d is defined by

𝒩 d := { 𝐮 ∈ E : 𝐮 ≠ 0 , ∇ ⁡ I d ⁢ ( 𝐮 ) ⊥ 𝐮 } ,

that is, 𝐮∈𝒩d if and only if 𝐮≠0 and

τ d ⁢ ( 𝐮 ) := 〈 ∇ ⁡ I d ⁢ ( 𝐮 ) , 𝐮 〉 L 2 = ∑ i = 1 d ∥ u i ∥ λ i 2 - ( ∑ i = 1 d μ i ⁢ | u i | 2 ⁢ q 2 ⁢ q + 2 ⁢ ∑ j < i b i ⁢ j ⁢ | u i ⁢ u j | q q ) .

Under condition (1.2) it is classical to check that 𝒩d is a manifold, and that minimizers on the Nehari manifold are ground state solutions.

When dealing with system (1.1) it is often necessary to treat the case n=1 separately due to the lack of compactness of the injection Hr1⁢(ℝ)↪Lq⁢(ℝ), q>2, where Hr1⁢(ℝ) denotes the space of the radially symmetric functions of H1⁢(ℝ). This lack of compactness is, in a sense, a consequence of the inequality

for u∈Hr1⁢(ℝn). Indeed, (1.4) gives no decay in the case n=1. However, if u is also radially decreasing, it is easy to establish that

| u ⁢ ( x ) | ≤ C ⁢ | x | - n 2 ⁢ ∥ u ∥ L 2 ⁢ ( ℝ n ) ,

which provides decay in all space dimensions, hence the compactness follows by applying the classical Strauss’ compactness lemma [24]. Hence, putting

H r ⁢ d 1 ⁢ ( ℝ n ) = { u ∈ H r 1 ⁢ ( ℝ n ) : u ⁢ is radially decreasing } ,

we get the compactness of the injection Hr⁢d1⁢(ℝn)↪Lq⁢(ℝn) for alln≥1 (see the appendix of [5] for more details), a fact that does not seem very well known. We will use this result to present a unified approach for the problem of the energy minimization of (1.1), valid in all space dimensions. In fact, by putting Er⁢d=(Hr⁢d1⁢(ℝn))d, the cone of symmetric radially decreasing nonnegative functions of E, we will prove the following result (see also [4, 17, 18]):

Concerning the existence of ground states with nontrivial components, we will show our main result:

We recall that such theorem was shown by Mandel for systems with d=2 equations, as a consequence of a more general result, namely the characterization of the optimal threshold b¯ defined before. In general, extending results from 2 to 3 or more equations is not straightforward, as systems with d≥3 equations often present a more complex structure with respect to its d=2 counterpart (see for instance the recent results in [10]). However, by simplifying Mandel’s approach, we will be able to prove Theorem 1.2, arguing by induction in the number of equations. Roughly, assuming that a subsystem of (1.1) with d-1 equations has a certain ground state solution with nontrivial components, we will construct an element (U1,…,Ud)∈𝒩d with lower energy Id.

For d≥3 equations, the first results concerning the properties of ground states seem to be the papers by Sirakov [21] (check Theorem 4 (iv) therein) and Liu and Wang [17]. In the latter, a nontrivial ground state is proved to exist in the case n=2,3, q=2, λ1=⋯=λd, μ1=⋯=μd and for bi⁢j=b sufficiently large (see also [6] for d=3, or [22, Theorem 1.6 and Remark 3]).

Recently, in [10], the authors joint with S. Correia presented, for q=2, optimal qualitative conditions under which the ground states are nontrivial or, conversely, semitrivial. Theorem 1.2 states that, for 1<q<2, for all values of the parameters, the ground states are nontrivial. This corresponds to an important difference with respect to the case q≥2, where there are values of the parameters for which all ground states are semitrivial. In such a situation, it becomes an interesting question to study if there exist least energy nontrivial solutions of (1.1), that is, solutions minimizing the energy among the set of all nontrivial solutions. This has been done for d=2 equations in [2, 7, 14, 21], and for d≥2 in [20, 22, 23], among others. For some recent results in this directions concerning a Schrödinger-KdV system, see also [8, 9]. For multiplicity results for (1.1), we refer to [16].

2 Proof of Proposition 1.1

We begin by observing that, for (u1,…,ud)∈E with (u1,…,ud)≠(0,…,0) and τd⁢(u1,…,ud)≤0, there exists t∈]0,1] such that (t⁢u1,…,t⁢ud)∈𝒩d. Indeed, if τd⁢(u1,…,ud)=0, we choose t=1. If τd⁢(u1,…,ud)<0 we simply notice that

τ ⁢ ( t ⁢ u 1 , … , t ⁢ u d ) = t 2 ⁢ ( ∑ i = 1 d ∥ u i ∥ λ i 2 - t 2 ⁢ q - 2 ⁢ ( ∑ i = 1 d | u i | 2 ⁢ q 2 ⁢ q + 2 ⁢ ∑ i < j b i ⁢ j ⁢ | u i ⁢ u j | q q ) ) := t 2 ⁢ T 𝐮 ⁢ ( t ) ,

with T𝐮⁢(0)>0 and T𝐮⁢(1)<0.

Also, we notice that if (u1,…,ud)∈𝒩d, then

We now take a minimizing sequence (u1,k,…,ud,k)∈𝒩d for the problem

inf ⁡ { I d ⁢ ( u 1 , … , u d ) : ( u 1 , … , u d ) ∈ 𝒩 d } .

From (2.1), it is clear that this infimum is nonnegative, hence (u1,k,…,ud,k) is a bounded sequence in E.

We put ui,k* the decreasing radial rearrangements of |ui,k|, i=1,…,d. It is well known that this rearrangement preserves the Lp norm (1≤p≤+∞). Furthermore, the Pólya–Szegö inequality

| ∇ ⁡ f * | 2 ≤ | ∇ ⁡ | f | | 2

in addition with the inequality |∇⁡|f||2≤|∇⁡f|2 (see [15]) shows that

∑ i = 1 d ∥ u i , k * ∥ λ i 2 ≤ ∑ i = 1 d ∥ u i ∥ λ i 2 .

On the other hand, the Hardy–Littlewood inequality

∫ | f ⁢ g | ≤ ∫ f * ⁢ g *

combined with the monotonicity of the map λ↦λq (see for instance [12] for details) yields ∥f⁢g∥q≤∥f*⁢g*∥q and, finally (as bi⁢j>0),

τ d ⁢ ( u 1 * , … , u d * ) ≤ τ d ⁢ ( u 1 , … , u n ) = 0 .

Next, let tk∈]0,1] such that (tk⁢u1,k*,…,tk⁢ud,k*)∈𝒩d. We obtain

I ⁢ ( t k ⁢ u 1 , k * , … , t k ⁢ u d , k * ) = t k 2 ⁢ ( 1 2 - 1 2 ⁢ q ) ⁢ ∑ i = 1 d ∥ u i , k * ∥ λ i 2 ≤ ( 1 2 - 1 2 ⁢ q ) ⁢ ∑ i = 1 d ∥ u i , k ∥ λ i 2 = I ⁢ ( u 1 , k , … , u d , k ) .

This way, we obtain a minimizing sequence (tk⁢u1,k*,…,tk⁢ud,k*) in Er⁢d, denoted again, in what follows, by (u1,k,…,ud,k). Since this sequence is bounded in Er⁢d, up to a subsequence, ui,k⇀ui* in H1⁢(ℝn) weak. Also, since the injection Er⁢d→L2⁢q⁢(ℝn) is compact, up to a subsequence, ui,k→ui* in L2⁢q⁢(ℝn) strong, for all n≥1. Moreover, (u1*,…,ud*)≠(0,…,0), since (from the definition of 𝒩d and by Sobolev and Cauchy–Schwarz inequalities)

∑ i = 1 d | u i , k | 2 ⁢ q 2 ≤ C 1 ⁢ ∑ i = 1 d ∥ u i , k ∥ λ i 2 ≤ C 2 ⁢ ∑ i = 1 d | u i , k | 2 ⁢ q 2 ⁢ q .

Thus there exists δ>0, independent from k, such that ∑i=1d|ui,k|2⁢q≥δ, and by the strong convergence also ∑i=1d|ui*|2⁢q≥δ>0.

Since

τ ⁢ ( u 1 * , … , u d * ) ≤ lim inf ⁡ τ ⁢ ( u 1 , k , … , u d , k ) = 0 ,

once again we can take t∈]0,1] such that (t⁢u1*,…,t⁢ud*)∈𝒩d. Then,

This implies that (t⁢u1*,…,t⁢ud*) is a minimizer. In particular, all inequalities above are in fact equalities, thus t=1, (u1*,…,ud*)∈𝒩d and ui,k→ui* in H1⁢(ℝn) strong.

It is then clear that (u1*,…,ud*) is a ground state solution, which concludes the proof of Proposition 1.1. Observe also that ui*≥0, and by the strong maximum principle either ui>0 or ui≡0.

3 Ground States with Nontrivial Components. Proof of Theorem 1.2

As stated in the introduction, the general result will be obtained by induction on the number of equations d. We begin by considering the case d=2. In this case, the result stated in Theorem 1.2 was recently obtained by Mandel in [19] in the case μ1=μ2=1. Here, we will cover this case by a different (and more direct) method, considering also arbitrary μ1,μ2>0. Furthermore, as stated previously, our method will also extend easily to more general systems of d≥3 equations.

Denote by ci the energy level of the (unique) positive ground state ui of

- Δ ⁢ u + λ i ⁢ u = μ i ⁢ | u | 2 ⁢ q - 2 ⁢ u .

Without loss of generality, we may assume that c1≤c2. Hence, in order to prove our result, it is sufficient to exhibit (U1,U2)∈𝒩2, with U1,U2≠0, such that I2⁢(U1,U2)<I2⁢(u1,0)=c1.

For a fixed w∈H1⁢(ℝn)∖{0} and for θ>0 that will be chosen later, we begin by computing t>0 such that (t⁢u1,t⁢θ⁢w)∈𝒩2, that is,

τ 2 ⁢ ( t ⁢ u 1 , t ⁢ θ ⁢ w ) = t 2 ⁢ ∥ u 1 ∥ λ 1 2 + t 2 ⁢ θ 2 ⁢ ∥ w ∥ λ 2 2 - t 2 ⁢ q ⁢ ( μ 1 ⁢ | u 1 | 2 ⁢ q 2 ⁢ q + μ 2 ⁢ θ 2 ⁢ q ⁢ | w | 2 ⁢ q 2 ⁢ q + 2 ⁢ b 12 ⁢ θ q ⁢ | u 1 ⁢ w | q q ) = 0 ,

from where we obtain that

t 2 ⁢ q - 2 = ∥ u 1 ∥ λ 1 2 + θ 2 ⁢ ∥ w ∥ λ 2 2 μ 1 ⁢ | u 1 | 2 ⁢ q 2 ⁢ q + μ 2 ⁢ θ 2 ⁢ q ⁢ | w | 2 ⁢ q 2 ⁢ q + 2 ⁢ b 12 ⁢ θ q ⁢ | u 1 ⁢ w | q q .

Since u1∈𝒩1, we have ∥u1∥λ12=μ1⁢|u1|2⁢q2⁢q, and we obtain

where

C 1 = ∥ w ∥ λ 2 2 ∥ u 1 ∥ λ 1 2 , C 2 = | w | 2 ⁢ q 2 ⁢ q ∥ u 1 ∥ λ 1 2 , C 3 = | u 1 ⁢ w | q q ∥ u 1 ∥ λ 1 2 .

Since (t⁢u1,t⁢θ⁢w)∈𝒩2, we have

I ⁢ ( t ⁢ u , t ⁢ θ ⁢ w ) = ( 1 2 - 1 2 ⁢ q ) ⁢ ( ∥ t ⁢ u 1 ∥ λ 1 2 + θ 2 ⁢ ∥ t ⁢ w ∥ λ 2 2 ) = t 2 ⁢ ( 1 2 - 1 2 ⁢ q ) ⁢ ( 1 + C 1 ⁢ θ 2 ) ⁢ ∥ u 1 ∥ λ 1 2 ,

and condition I2⁢(t⁢u1,t⁢θ⁢w)<I2⁢(u1,0) is equivalent to

t 2 ⁢ ( 1 + θ 2 ⁢ C 1 ) < 1 ,

that is, in view of (3.1),

( 1 + θ 2 ⁢ C 1 1 + μ 2 ⁢ θ 2 ⁢ q ⁢ C 2 + 2 ⁢ b 12 ⁢ θ q ⁢ C 3 ) 1 q - 1 ⁢ ( 1 + C 1 ⁢ θ 2 ) < 1

and

( 1 + θ 2 ⁢ C 1 ) q 1 + μ 2 ⁢ θ 2 ⁢ q ⁢ C 2 + 2 ⁢ b 12 ⁢ θ q ⁢ C 3 < 1 .

Thus, we obtain

( 1 + θ 2 ⁢ C 1 ) q - 1 - μ 2 ⁢ θ 2 ⁢ q ⁢ C 2 θ q < 2 ⁢ b 12 ⁢ C 3 .

By noticing that, for 1<q<2,

lim θ → 0 + ⁡ ( 1 + θ 2 ⁢ C 1 ) q - 1 θ q = 0 ,

we conclude that this condition holds for small θ, which concludes the proof for d=2.

We now consider system (1.1) with d>2 equations. Given I⊊{1,2,…,d} denote by cI the ground state level of the system

- Δ ⁢ u i + λ i ⁢ u i = μ i ⁢ | u i | 2 ⁢ q - 2 ⁢ u i + ∑ j ∈ I , j ≠ i b i ⁢ j ⁢ | u j | q ⁢ | u i | q - 2 ⁢ u i , i ∈ I .

Let us now assume, by induction hypothesis, that there exists a ground state level cI with #⁢I=d-1 and cI<cJ for all J with #⁢J<d-1. Without loss of generality, we assume that

c := c { 1 , … , d - 1 } = min ⁡ { c I : # ⁢ I = d - 1 } ,

where c is achieved by the nontrivial ground state (u1,…,ud-1)∈𝒩d-1, solution of

- Δ ⁢ u i + λ i ⁢ u i = μ i ⁢ | u i | 2 ⁢ q - 2 ⁢ u i + ∑ j = 1 , j ≠ i d - 1 b i ⁢ j ⁢ | u j | q ⁢ | u i | q - 2 ⁢ u i , i = 1 , … , d - 1 .

Noticing that Id⁢(u1,…,ud-1,0)=Id-1⁢(u1,…,ud-1), we will prove our assertion by exhibiting (U1,…,Ud)∈𝒩d, Ui≠0, such that Id⁢(U1,…,Ud)<Id⁢(u1,…,ud-1,0)=c, which guarantees that the energy level of (U1,…,Ud) is inferior to the energy level of any solution of (1.1) with trivial components.

In this regard, for fixed w∈H1⁢(ℝn), w≠0, and θ>0, we choose t>0 such that

( t ⁢ u 1 , … , t ⁢ u d - 1 , t ⁢ θ ⁢ w ) ∈ 𝒩 d .

This condition is equivalent to τd⁢(t⁢u1,…,t⁢ud-1,t⁢θ⁢w)=0, that is,

t 2 ⁢ ( ∑ i = 1 d - 1 ∥ u i ∥ λ i 2 + θ 2 ⁢ ∥ w ∥ λ d 2 ) = t 2 ⁢ q ⁢ ( ∑ i = 1 d - 1 μ i ⁢ | u i | 2 ⁢ q 2 ⁢ q + μ d ⁢ θ 2 ⁢ q ⁢ | w | 2 ⁢ q 2 ⁢ q + 2 ⁢ ∑ i , j = 1 , j < i d - 1 b i ⁢ j ⁢ | u i ⁢ u j | q q + 2 ⁢ ∑ i = 1 d - 1 b i ⁢ d ⁢ θ q ⁢ | u i ⁢ w | q q ) .

Since (u1,…,ud-1)∈𝒩d-1, we have

∑ i = 1 d - 1 ∥ u i ∥ λ i 2 = ∑ i = 1 d - 1 μ i ⁢ | u i | 2 ⁢ q 2 ⁢ q + 2 ⁢ ∑ i , j = 1 , j < i d - 1 b i ⁢ j ⁢ | u i ⁢ u j | q q ,

which yields

where we have put

C 1 = ∥ w ∥ λ d 2 ∑ i = 1 d - 1 ∥ u i ∥ λ i 2 , C 2 = | w | 2 ⁢ q 2 ⁢ q ∑ i = 1 d - 1 ∥ u i ∥ λ i 2 , D i = | u i ⁢ w | q q ∑ i = 1 d - 1 ∥ u i ∥ λ i 2 .

Now, observe that, since (t⁢u1,…,t⁢ud-1,t⁢θ⁢w)∈𝒩d, we have

I d ⁢ ( t ⁢ u 1 , … , t ⁢ u n - 1 , t ⁢ θ ⁢ w ) = ( 1 2 - 1 2 ⁢ q ) ⁢ ( ∑ i = 1 d - 1 ∥ t ⁢ u i ∥ λ i 2 + θ 2 ⁢ ∥ t ⁢ w ∥ λ d 2 ) = t 2 ⁢ ( 1 2 - 1 2 ⁢ q ) ⁢ ( 1 + C 1 ⁢ θ 2 ) ⁢ ∑ i = 1 d - 1 ∥ u i ∥ λ i 2 .

Since

I d ⁢ ( u 1 , … , u d - 1 , 0 ) = I d - 1 ⁢ ( u 1 , … , u d - 1 ) = ( 1 2 - 1 2 ⁢ q ) ⁢ ∑ i = 1 d - 1 ∥ u i ∥ λ i 2 ,

we obtain that the condition Id⁢(U1,…,Ud)<c is equivalent to t2⁢(1+C1⁢θ2)<1, and, in view of (3.2), to

( 1 + θ 2 ⁢ C 1 ) q 1 + μ d ⁢ θ 2 ⁢ q ⁢ C 2 + 2 ⁢ ∑ i = 1 d - 1 b i ⁢ d ⁢ θ q ⁢ D i < 1

or

( 1 + θ 2 ⁢ C 1 ) q - 1 - μ d ⁢ θ 2 ⁢ q ⁢ C 2 θ q < 2 ⁢ ∑ i = 1 d - 1 b i ⁢ d ⁢ D i .

which holds for θ small enough, and the proof is complete.