A note on coupled nonlinear Schr\"odinger systems under the effect of general nonlinearities

We prove the existence of non-trivial solutions to a system of coupled, nonlinear, Schroedinger equations with general nonlinearity.


Introduction
In the last years, nonlinear Schrödinger systems have been widely investigated by several authors. These systems are models for different physical phenomena: the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics, and Bose-Einstein condensates. Roughly speaking, two ore more semilinear Schrödinger equations like (1) − ∆u + au = u 3 in R 3 are coupled together. Equation (1) describes the propagation of pulse in a nonlinear optical fiber, and the existence of a unique (up to translation) least energy solution has been proved. It turns out that this ground state solution is radially symmetric with respect to some point, positive and exponentially decaying together with its first derivatives at infinity. Unluckily, we know (see [17]) that single-mode optical fibers are not really "single-mode", but actually bimodal due to the presence of birefringence. This birefringence can deeply influence the way an optical evolves during the propagation along the fiber. Indeed, it can occur that the linear birefringence makes a pulse split in two, while nonlinear birefringent traps them together against splitting. The evolution of two orthogonal pulse envelopes in birefringent optical fibers is governed (see [25,26]) by the nonlinear Schrödinger system i ∂ψ ∂t + ∂ 2 ψ ∂x 2 + |ψ| 2 ψ + β|φ| 2 ψ = 0, where β is a positive constant depending on the anisotropy of the fibers. System (2) is also important for industrial applications in fiber communications systems [13] and all-optical switching devices, see [15]. If one looks for standing wave solutions of (2), namely solutions of the form Other physical phenomena, such as Kerr-like photorefractive media in optics, are also described by (3), see [1,8]. As a word of caution, (3) possesses the "simple" solutions of the form (u, 0) and (0, v), where u and v solve (1).
Problem (3), in a more general situation and also in higher dimension, has been studied in [9,10], where smooth ground state solutions (u, v) = (0, 0) are found by concentration compactness arguments. Later on, Ambrosetti et al. in [2], Maia et al. in [23] and Sirakov in [29] deal with problem and, among other results, they prove the existence of ground state solutions of the type (u, v), with u, v > 0, for β > 0 sufficiently big. Similar problems have been treated also in [14,18,20,30]. Some results in the singularly perturbed case can be found in [19,27,28], while the orbital stability and blow-up proprieties have been studied in [12,24]. Although the interest lies in solutions with both non-trivial components, solutions of (4) are somehow related to solutions of the single nonlinear Schrödinger equation (1). The nonlinearity g(u) = u 3 is typical in physical models, but much more general Schrödinger equation of the form −∆u + au = g(u), in R 3 , still have at least a ground state solution under general assumptions on the nonlinearity g which, for example, do not require any Ambrosetti-Rabinowitz growth condition. We recall that a function f : R → [0, +∞) satisfies the Ambrosetti-Rabinowitz growth condition if there exists some µ > 2 such that This condition essentially states that the function log (F (t)t −µ ) is monotone increasing for t ≥ 0, and is trivially satisfied by any power f (u) = u r with r > 1. The crucial feature is, to summarize, that ground states are necessarily radially symmetric with respect to some point, and this knowledge recovers some compactness. We refer to the celebrated papers [5,11] for a deep study of these scalar-field equations (see also [4,16]).
Motivated by these remarks, we want to find ground state solutions for the system where β ∈ R and f, g ∈ C(R 3 , R) satisfy the following assumptions: These assumptions are very weak, and it is easy to construct function f and g that match them but do not match the Ambrosetti-Rabinowitz condition. Without condition (5), one cannot perform a standard minimization over the Nehari manifold (see below for the definition of the functional I), as done in [2,29]. We recall that (5) is also used to prove the boundedness of minimizing sequences constrained to N , so its failure can cause troubles even at this stage. Our existence results not only cover more general systems than those in [2,29], but also give a different existence proof when f and g coincide with pure powers. Remark 1.2. For a single Schrödinger equation, even weaker assumptions can be requested, see [5]. Unluckily, the idea of locating solutions by means of the maximum principle does not seem to work for systems. System (6) has a variational structure, in particular solutions of (6) can be found as critical points of the functional I : where we have set We will call ground state solution any couple (u, v) = (0, 0) which solves (6) and minimizes the functional I among all possible nontrivial solutions. Thus we have to overcome the strong lack of compactness under our weak assumptions on f and g, and also to exclude "simple" solutions with a null component. To fix terminology, we introduce the following definition.
Scalar solutions for problem (6) exist by the results of [5]. Indeed, since f satisfies (f1-3), there exists a (least-energy) solution u 0 ∈ H 1 (R 3 ) for the single Schrödinger equation It can be checked immediately that the couples (u 0 , 0) and (0, v 0 ) are non-trivial solutions of (6). As a first step, we will prove that for any β ∈ R the problem (6) admits a ground state.
Then we will prove that vector solutions exist whenever the coupling parameter β is sufficiently large.
Theorem 1.5. Let f and g satisfy (f1-3) and (g1-3). Then there exists β 0 > 0 such that, for any β > β 0 , there there exists a vector solution of (6), which is a ground state solution. Moreover this solution is radially symmetric.
The main result of this paper is Theorem 1.5: up to our knowledge, indeed, this is the first vector solution existence result for problem (6). As already said in Remark 1.1, without condition (5), we cannot perform a standard minimization over the Nehari manifold and so our proof which is based on a constrained minimization over the Pohozaev manifold. The existence of vector solutions for any β is not known. The existence result of Theorem 1.4, instead, is already known and it is proved in [6]. For the reader's sake, here we give a different proof based on the constrained minimization over the Pohozaev manifold.
We denote with B r the ball of radius r centered in the origin. • We denote with · the norm of H 1 (R 3 ).
• With C i and c i , we denote generic positive constants, which may also vary from line to line.
By [3, Lemma 3.6] and repeating the arguments of [5], it is easy to see that each solution of (6), (u, v) ∈ H, satisfies the following Pohozaev identity: Therefore each non-trivial solution of (6) belongs to P, where We call P the Pohozaev manifold associated to (6). We collect its main properties of the set P in the next Proposition: the proof is easy and left to the reader. Then (2) P is a C 1 -manifold of codimension one.
Lemma 2.4. For any β ∈ R, P is a natural constraint for the functional I.
Proof. First we show that the manifold is nondegenerate in the following sense: By contradiction, suppose that (u, v) ∈ P and J ′ (u, v) = 0, namely (u, v) is a solution of the equation As a consequence, (u, v) satisfies the Pohozaev identity referred to (19), that is Since (u, v) ∈ P, by (20) we get 2 R 3 |∇u| 2 + |∇v| 2 = 0 and we conclude that u = v = 0: we get a contradiction since (u, v) ∈ P. Now we pass to prove that P is a natural constraint for I. Suppose that (u, v) ∈ P is a critical point of the functional I |P . Then, by Proposition 2.1, there exists µ ∈ R such that As a consequence, (u, v) satisfies the following Pohozaev identity which, since J(u, v) = 0, can be written Since either u = 0 or v = 0 we deduce that µ = 0, and we conclude.
By means of the previous lemma, we are reduced to look for a minimizer of I restricted to P. By the well known properties of the Schwarz symmetrization, we are allowed to work on the functional space H r as shown by the following Lemma 2.5. For any β > 0 and for any (u, v) ∈ P, there exists (ū,v) ∈ P ∩ H r such that I(ū,v) ≤ I(u, v).
Proof. Let (u, v) ∈ P and set u * , v * ∈ H 1 r (R 3 ) their respective symmetrized functions. We have Hence, there existst ∈ (0, 1] such that (ū,v) := (u * (·/t), v * (·/t)) ∈ P ∩ H r and Proposition 2.6. For any β > 0, the value m is achieved as a minimum by I on P by Proof. For any (u, v) ∈ P we have Let {(u n , v n )} n ⊂ P be such that I(u n , v n ) → m. By Lemma 2.5, we can assume that {(u n , v n )} n ⊂ P ∩ H r . By (21), we infer that {u n } n , {v n } n are bounded in D 1,2 (R 3 ). Let ε > 0 be given, and let C ε > 0 be the positive constant as in (9)- (16). We observe that, for any β ∈ R, there exists a positive constant C > 0 such that for all x, y ∈ R: Hence, since {(u n , v n )} n ⊂ P, by (12) and (16), we get Since {u n } n , {v n } n are bounded in D 1,2 (R 3 ) and D 1,2 (R 3 ) ֒→ L 6 (R 3 ), then {(u n , v n )} n is bounded in H.
Since we are dealing with radially symmetric functions, without loss of generality, we can assume that ξ n = 0, for all n ≥ 1.
By (22), we can argue that either u = 0 or v = 0 and, moreover, since {(u n , v n )} n ⊂ P, passing to the limit, we have By (23), it is easy to see that there existst ∈ (0, 1] such that (ū,v) = (u(·/t), v(·/t)) ∈ P ∩ H r . By the weak lower semicontinuity, we get hence (ū,v) is a minimum of I restricted on P and so, by Lemma 2.4, it is a (radially symmetric) ground state solution for the problem (6).
Finally, let us prove a lemma which will be a key point in the proof of Theorem 1.5 Lemma 2.7. Let u 0 , v 0 ∈ H 1 (R 3 ) be two non-trivial solutions respectively of (7) and (8).
3. Proofs of Theorem 1.4 and 1.5 Proof of Theorem 1.4. If we set Let {(u n , v n )} n ⊂ S be such that I(u n , v n ) → b. By (21), we infer that {u n } n , {v n } n are bounded in D 1,2 (R 3 ). Repeating the arguments of the proof of Proposition 2.6, we can argue that {(u n , v n )} n is bounded in H r . By Lemma 2.3 we know that {(u n , v n )} n does not vanish, namely there exist C, r > 0, {ξ n } n ⊂ R 3 such that (26) Br(ξn) u 2 n + v 2 n ≥ C, for all n ≥ 1.
Due to the invariance by translations, without loss of generality, we can assume that ξ n = 0 for every n.
Since {(u n , v n )} n is bounded in H, there exist u, v ∈ H 1 (R 3 ) such that, up to a subsequence, v n → v in L s loc (R 3 ), 1 ≤ s < 6. By (26), we can argue that either u = 0 or v = 0 and then it is easy to see that (u, v) ∈ S. By the weak lower semicontinuity, we get b ≤ I(u, v) = 1 3 R 3 |∇u| 2 + |∇v| 2 ≤ lim inf n→+∞ 1 3 R 3 |∇u n | 2 + |∇v n | 2 = lim inf n→+∞ I(u n , v n ) = b, hence (u, v) is a ground state for the problem (6). If β > 0, by Proposition 2.6, we can argue that there exists a ground state (u, v) which belongs to H r .
Note added in proof. After the manuscript had been submitted, the preprint [7] was pointed out to us. It is shown, in particular, that the radial symmetry of solutions to our system can be proved a priori. More precisely, every ground-state solution is necessarily invariant under rotations.