Exponential decay of the solutions to nonlinear Schrödinger systems

We show that the components of finite energy solutions to general nonlinear Schrödinger systems have exponential decay at infinity. Our results apply to positive or sign-changing components, and to cooperative, competitive, or mixed-interaction systems. As an application, we use the exponential decay to derive an upper bound for the least possible energy of a solution with a prescribed number of positive and nonradial sign-changing components.


Introduction
Consider the nonlinear Schrödinger system where N ≥ 1, V i ∈ L ∞ (R N ), β ij ∈ R and 1 < p < 2 * 2 .Here 2 * is the usual critical Sobolev exponent, namely, 2 * := 2N  N −2 if N ≥ 3 and 2 * := ∞ for N = 1, 2. Systems of this type occur as models for various natural phenomena.In physics, for example, they describe the behavior of standing waves for a mixture of Bose-Einstein condensates of different hyperfine states which overlap in space [13].The coefficients β ij determine the type of interaction between the states; if β ij > 0, then there is an attractive force between u i and u j , similarly, if β ij < 0, then the force is repulsive, and if β ij = 0, then there is no direct interaction between these components.Whenever all the interaction coefficients are positive, we say that the system is cooperative.If β ii > 0 and β ij < 0 for all i = j, then the system is called competitive.And if some β ij are positive and others are negative for i = j, then we say that the system has mixed couplings.All these regimes exhibit very different qualitative behaviors and have been studied extensively in recent years, see for instance [5, 6, 8-12, 17, 19-24, 26] and the references therein.
System (1.1) has a variational structure, and therefore a natural strategy is to find weak solutions by minimizing an associated energy functional on a suitable set, under additional assumptions on the matrix (β ij ) and on the potentials V i .Using this approach, several kinds of solutions have been found in terms of their signs and their symmetries.However, there seems to be no information available about the decay of these solutions at infinity.In this paper, we show that finite energy solutions must decay exponentially at infinity, and a rate can be found in terms of the potentials V i .Our main result is the following one.
We emphasize that each component may have a different decay depending on each potential V i .The main obstacle to showing (1.2) is to handle the possibly sublinear term |u i | p−2 u i for p ∈ (1, 2) (which is always the case for N ≥ 4).To explain this point in more detail, assume that (u 1 , . . ., u ℓ ) is a solution of (1.1) and write the i-th equation of the system as Since every u j ∈ H 1 (R N ) ∩ C 0 (R N ), we know that a i and c i are bounded in R N , but |u i | p−2 → ∞ as |x| → ∞ and it is also singular at the nodal set of a sign-changing solution.As a consequence, one cannot use directly previously known results about exponential decay for scalar equations, such as those in [1,3,18].In fact, one can easily construct a one dimensional solution of a similar scalar equation that has a power-type decay.For instance, let w ∈ C 2 (R) be a positive function such that w(x) = |x| −2/3 for |x| > 1 and let , and w decays as a power at infinity.This shows that the proof of the exponential estimate in Theorem 1.1 must rely on a careful study of the system structure.In other words, although the sublinear nonlinearity |u i | p−2 u i appears in (1.1), the system is not sublinear.As a whole, it is always superlinear.
With this in mind, we adapt some of the arguments in [1,18] preserving at each step the system structure of the problem.These arguments rely basically on elliptic regularity and comparison principles.
The exponential decay of solutions is a powerful tool in their qualitative study.As an application of Theorem 1.1, we derive energy bounds of solutions having prescribed positive and nonradial signchanging components.For this, power type decay would not be enough.
We say that u is fully nontrivial if every component we look for solutions such that every component of u h is positive if h ∈ Q + and every component of u h is nonradial and changes sign if h ∈ Q − .To this end, we use variational methods in a space having suitable symmetries.As shown in [11,Section 3], to guarantee that the solutions obtained are fully nontrivial we need to assume the following two conditions: (B 2 ) For each h ∈ Q, the graph whose set of vertices is I h and whose set of edges is E h := {{i, j} : i, j ∈ I h , i = j, β ij > 0} is connected.
In [11] it is shown that, for any q, the system (1.1) has a fully nontrivial solution satisfying the sign requirements described above.Furthermore, an upper bound for its energy is exhibited, but only for systems with at most 2 blocks, i.e., for q = 1, 2.Here we use Theorem 1.1 to obtain an energy bound for any number of blocks.
Theorem 1.2.Let N = 4 or N ≥ 6, and let ), and (B 3 ).Then, there exists a fully nontrivial solution u = (u 1 , . . ., u q ) to the system (1.4) with the following properties: (c) If q ≥ 2 the following estimate holds true where and ω is the unique positive radial solution to the equation To prove Theorem 1.2, we follow the approach in [11] and impose on the variational setting some carefully constructed symmetries which admit finite orbits.This approach immediately gives energy estimates but it requires showing a quantitative compactness condition which needs precise knowledge about the asymptotic decay of the components of the system.Here is where we use Theorem 1.1.
The paper is organized as follows.Section 2 is devoted to the proof of the exponential decay stated in Theorem 1.1.The application of this result to derive energy bounds is contained in Section 3, where we also give some concrete examples.
) and (2.2), assuming without loss of generality that ρ ≥ 2 and adding over i, we get Therefore, where ⌊r⌋ denotes the floor of r.
In the rest of the paper, we write where Set f i as in (2.4).By (2.5), there is a constant where (p − 1)s + p(s − 2) > 0.Then, by [14,Theorem 9.11], there is a positive constant From the previous inequalities we derive where C 3 = C 3 (N, p) is the constant given by the Sobolev embedding , let u = (u 1 , . . ., u ℓ ) be a solution of (1.1) and let f i be as in (2.4).Then, there are constants η > 0, C 1 > 0, and C 2 > 0 such that η|x| , for all x ∈ R N and i = 1, . . ., ℓ.
Proof.For x ∈ R N with |x| ≥ 2, set r := 1 2 |x|.Then, B 1 (x) ⊂ R N B r and, by Lemma 2.1, there are positive constants K 1 = K 1 (u, σ, β, N, ρ, p) and ϑ = ϑ(σ), with ρ and σ i as in (V 2 ), such that . By Lemma 2.3 there are positive constants K 2 = K 2 (u, β, N, p, Λ, s) and Therefore, where K 4 is the positive constant given by the embedding W 2,s (B 1 2 ).Since u i is continuous, we may choose The estimate for f i follows immediately from (2.5).
The following result is a particular case of [18, Theorem 2.1].We include a simplified proof for completeness.
We are ready to prove Theorem 1.1.
u is called fully nontrivial if every component u i is different from zero.We say that u is block-wise nontrivial if at least one component in each block u h is nontrivial.Following [11], we introduce suitable symmetries to produce a change of sign in some components.Let G be a finite subgroup of the group O(N ) of linear isometries of R N and denote by Note that, if φ ≡ 1 is the trivial homomorphism and u satisfies (3.1), then u is G-invariant.On the other hand, if φ is surjective every nontrivial function satisfying (3.1) is nonradial and changes sign.Define For each h = 1, . . ., q, fix a homomorphism φ h : G → Z 2 .Take φ i := φ h for all i ∈ I h and set φ = (φ 1 , . . ., φ ℓ ).Denote by and let J φ : H φ → R be the functional given by This functional is of class C 1 and its critical points are the solutions to the system (1.4) satisfying (3.1).The block-wise nontrivial solutions belong to the Nehari set N φ := {u ∈ H φ : u h = 0 and ∂ u h J φ (u)u h = 0 for every h = 1, . . ., ℓ}.
then there exists a unique s u ∈ (0, ∞) q such that s u u ∈ N φ .Furthermore, Proof Set Q := {1, . . ., q} and fix a decomposition From now on, we consider the following symmetries.We write Define Due to the lack of compactness, c φ is not always attained; see e.g.[11, Corollary 2.8(i)].A sufficient condition for this to happen is given by the next lemma.We use the following notation.If Q ′ ⊂ Q := {1, . . ., q} we consider the subsystem of (1.4) obtained by deleting all components of u h for every h / ∈ Q ′ , and we denote by J φ Q ′ and N φ Q ′ the functional and the Nehari set associated to this subsystem.We write If Q ′ = {h} we omit the curly brackets and write, for instance, c φ h or J φ h .
To verify condition (3.4) we introduce a suitable test function.Fix m ≥ 5 and let K m be as in Definitions If h ∈ Q − we take ζ h := (1, 0, 0) and we define where ω is the positive radial solution to (1.7) and where t hR > 0 is chosen so that Lemma 3.5.If m ≥ 5, then, for each h ∈ {1, . . ., q}, there exist and apply [11, Proposition 4.1(i) and Lemma 4.4].
Proof of Theorem 1.2.Assume (B 1 ) and let φ h : G m → Z 2 be given by (3.3).For q = 1 and m ≥ 5 it is proved in [11, Corollary 4.2 and Proposition 4.5] that c φ is attained at u ∈ N φ satisfying Taking m = 5 gives statement (b).Fix m = 6.We claim that c φ is attained and that the estimate (c) holds true for every q ≥ 2. To prove this claim, we proceed by induction.Assume it is true for q − 1 with q ≥ 2.
We will show that the compactness condition (3.4) holds true.Using a change of coordinates, it suffices to argue for h = q.By induction hypothesis there exists w = (w 1 , . . ., w q−1 ) For each R > 1 let σ qR be as in (3.5) and take t q ∈ (0, ∞) ℓ−ℓ q−1 as in Lemma 3.5.Set w hR = w h for h = 1, . . ., q−1 and w qR = t q σ qR , and define w R = (w 1R , . . ., w ℓR ) := (w 1R , . . ., w qR ).Then, as w ∈ N φ Q {q} and the interaction between the components of w and σ qR tends to 0 as R → ∞, we have that w R satisfies (3.2) for large enough R and, as a consequence, there exist R 1 > 0 and (s 1R , . . ., s qR ) ∈ [1/2, 2] q such that (s 1R w 1R , . . ., s qR w qR ) ∈ N φ if R ≥ R 1 .Set u R = (u 1R , . . ., u ℓR ) := (s 1R w 1R , . . ., s qR w qR ).Using that w ∈ N φ Q {q} and t q σ qR ∈ N φ q , from the last statement in Lemma 3.1(ii) and Lemma 3.5 we derive Therefore, for every g ∈ G m , where Remark 3.8.In the proof of Theorem 1.2 we use [1, Theorem 2.3], which also characterizes the sharp decay rate for positive components by providing a bound from below.This kind of information can be useful to show uniqueness of positive solutions for some problems, see [4,Section 8.2].
To conclude, we discuss some special cases.
Assumptions (B 2 ) and (B 3 ) guarantee that u is fully nontrivial.Note that the left-hand side of the inequality in (B 3 ) depends only on the entries of the submatrices (β ij ) i,j∈I h , h = 1, . . ., q, whereas the right-hand side only depends on the other entries.So, if the former are large enough with respect to the absolute values of the latter, (B 3 ) is satisfied.For example, if we take ℓ = 2q and the matrix is

A An auxiliary result
Lemma A.1.For every r ≥ 1 there is a linear operator E r : H 1 (R N B r ) → H 1 (R N ) such that, for every u ∈ H 1 (R N B r ), (i) E r u = u a.e. in R N B r , for some positive constant C 1 depending only on N and not on r.As a consequence, given p ∈ (1, 2 * 2 ) there is a positive constant C depending only on N and p such that |u| L 2p (R N Br) ≤ C u H 1 (R N Br) for every u ∈ H 1 (R N B r ) and every r ≥ 1.
Then, E r u = E 1 u.Clearly, E r satisfies (i).Note that | u| 2 L 2 (R N B 1 ) = r −N |u| 2 L 2 (R N Br) and that Similar identities hold true when we replace R N B 1 and R N B r with R N .Therefore, which yields (ii).Furthermore, This inequality, combined with (ii), yields which gives (iii).For p ∈ (1, N N −2 ) let C 2 = C 2 (N, p) be the constant for the Sobolev embedding H 1 (R N ) ⊂ L 2p (R N ).Then, for any u ∈ H 1 (R N B r ), using statements (i) and (iii) we obtain as claimed.
. See [8, Lemma 2.2] or [11, Lemma 2.2].Lemma 3.2.If c φ is attained, then the system (1.4) has a block-wise nontrivial solution u= (u 1 , ..., u ℓ ) ∈ H φ .Furthermore, if u i is nontrivial, then u i is positive if φ i ≡ 1 and u i is nonradial and changes sign if φ i is surjective.Proof.It is shown in [8,Lemma 2.4] that any minimizer of J φ on N φ is a block-wise nontrivial solution to(1.4).If u i = 0 and φ i is surjective, then u i is nonradial and changes sign.If φ i ≡ 1 then |u i | is G-invariant and replacing u i with |u i | we obtain a solution with the required properties.
[11, by Lemmas 3.4 and 3.2, c φ is attained at a block-wise nontrivial solution u of (1.4) such that every component of u h is positive if h ∈ Q + and every component of u h is nonradial and changes sign if h ∈ Q − .Furthermore, since we are assuming (B 2 ) and (B 3 ) with C * as in (3.7) below,[11, Theorem 3.3] asserts that u is fully nontrivial.Finally, note thatp > 1 = d m because m = 6.As |G m ζ h | = 6 if h ∈ Q + and |G m ζ h | = 12 if h ∈ Q − ,the estimate in statement (c) follows by induction.Remark 3.6.If m = 5 and p > d m we arrive to a similar conclusion, where, in this case, the constant b h in statement (b) is 5 if h ∈ Q + and it is 10 if h ∈ Q − .Note, however, that numbers p satisfying d 5 = 2 sin π 5 < p < N N −2 exist only for N ≤ 13.Remark 3.7.For φ h as in (3.3), the constant C * > 0 appearing in (B 3 ) depends on N , p, q, and Q + .It is explicitly defined in[11, Equation (3.1)] as β 13 β 14