CONSTRUCTION OF SOLUTIONS FOR SOME LOCALIZED NONLINEAR SCHR¨ODINGER EQUATIONS

. For an N -body system of linear Schr¨odinger equation with space dependent interaction between particles, one would expect that the correspond- ing one body equation, arising as a mean ﬁeld approximation, would have a space dependent nonlinearity. With such motivation we consider the follow- ing model of a nonlinear reduced Schr¨odinger equation with space dependent nonlinearity where V ( x ) = − χ [ − 1 , 1] ( x ) is minus the characteristic function of the interval [ − 1 , 1] and where h (cid:48) is any continuous strictly increasing function. In this article, for any negative value of λ we present the construction and analysis of the inﬁnitely many solutions of this equation, which are localized in space and hence correspond to bound-states of the associated time-dependent version of the equation.


1.
Introduction. Autonomous Nonlinear Schrödinger (NLS) equations appear in many fields of theoretical physics, in particular as effective models in the study of many body quantum systems [24], with the standard form: where f is a suitable real-valued function and Du denotes the spatial derivatives of the complex-valued solution u. The Gross-Pitaevskii equation is the archetype of such evolution equations with f given by f (x, u, Du) = V (x)u − |u| 2 u, where the potential V is a real-valued function. The specification of the function f leads to various models of NLS equations, some of which have been extensively studied in the literature like the semilinear case f (x, u, Du) = V (x)u − g(|u|)u, where V (the linear potential) and g are real-valued. For an overview of the subject we refer to e.g. [1,4,5,10,33,39,40].
In the case of semilinear NLS equations, as in the linear case, the bound states (or standing waves) are defined as solutions of the form e −iλt ϕ(x), where λ is a real number and ϕ is a non trivial solution of the corresponding stationary non-linear equation, −∆ϕ + (V (x) − λ)ϕ = g(|ϕ|)ϕ , which belongs to H 1 (R d ) and vanishes at infinity. The existence and properties of bound states in the semilinear case have been widely studied with a quickly growing literature. These studies have mainly tackled the existence problem, in perturbative regimes ( [43] and references therein), in semiclassical regimes, see e.g. [2,7,13,20,30,44] and for asymptotically linear perturbations, see e.g. [15,27,28]. Some of these approaches also include nonlinear terms with some spatial modulations and some additional regular linear potential. In absence of potentials, see also [12]. These results have also been extended to include singular potentials [21,17], asymptotically periodic settings [16,33,38] and magnetic fields [3,14,19]. In general, the construction of such bound states is not very explicit and there are few results on the control of the multiplicity of the associated "eigenvalue". For the dynamical consequences related to the existence of bound states, see e.g. [37,41,42,43]. An important effort has also been placed in the specific study of non-negative solutions (ground states), for results on existence, non-existence and regularity of ground states see e.g. [22,34] and references therein.
In this paper we consider the existence problem of bound states for semilinear NLS equations with a compactly supported non-linearity. Such a model may arise as a mean field approximation of a many body quantum system, with space dependent interaction between particles. To our knowledge, this situation has not been studied systematically and provides interesting and useful insights for more general cases, see e.g. [8,31]. By studying a one dimensional basic model we provide a construction of infinitely many bound states for any negative energy of the system (see Theorems 3.3 and 3.4). These bound states will be characterized by their oscillation properties. The situation contrasts abruptly with the linear case, where the spectrum is known to be bounded from below and it is discrete in any compact interval of negative energies (Weyl Theorem). Our approach relies on very precise manipulations of ODE techniques and adequately constructed barrier functions.
The paper is structured as follows. The model and basic setting is presented in section 2, the main results are in section 3. In section 4, we study some consequences of our results, considering the limiting case of a singular non-linearity (like in [8,31]) and using the solution of our stationary equation to describe soliton-type solutions for some non-autonomous NLS equations. The proofs of the main results are presented in section 5, which is divided in three subsections. Subsection 5.1 develops the technical results in ODEs that are required as the main tools, that are used in subsection 5.2, to present the proofs of the main theorems in subsection 3.1. While subsection 5.3 contains some technical observations. 2. Setting of the problem and notation. The following notation and hypotheses will be used throughout sections 3 and 4, and will be omitted in the statements of the results. Let Let h : R + → R be a continuous strictly increasing function satisfying h (0) ≥ 0 and lim x→+∞ h (x) = ∞.
We consider the equation By solutions of equation (2) we mean H 2 (R) solutions, which in this setting ends up being equivalent to C 1 (R) solutions. As shown in subsection 5.3 we only need to consider λ ∈ R and ϕ real valued. Furthermore, theorem 3.1 will imply λ < 0.
By letting h(x) = x 0 h (s)ds, h satisfies h(0) = 0 and the hypotheses on h are equivalent to assume h ∈ C 1 (R + ) is strictly increasing and strictly convex with unbounded derivative.

Main theorems.
In this subsection we present the main theorems describing the solutions of equation (2). The proofs of these results are presented later in subsection 5.2.
The potential V (x) vanishes outside the interval [−1, 1] and therefore the nonlinear part of equation (2) is restricted to the interval [−1, 1]. The first step in the analysis is the study of equation (2) in each component of R \ {−1, 1}, obtaining the following characterization.
and ϕ = 0 only at isolated points.
Theorem 3.1 directs our attention to solutions of equation (4) that only vanish at isolated points. The analysis of equation (4), of the form |u | = g(u), requires some technical considerations due to the right hand side not necessarily Lipschitz at g(u) = 0. All the technical aspects of this equation are addressed in subsection 5.1 and a summary of such results is the following.
ϕ has exactly m zeroes, ϕ = 0 only at isolated points.  (2) can also be adapted to study non-linear Schrödinger equations on the one dimensional torus (related aspects can be found in [32]). Namely, consider the equation In this setting, we have that: ϕ(x) → 0 when x → ±∞.
As mentioned before, cubic NLS equations are effective models in the analysis of many body quantum systems involving two-body interactions in the limit of large particles numbers, see e.g. [25,26] and references therein.
Higher order NLS equations also appear as effective models in the study of many body quantum systems with more complex interactions, see e.g. [11]. c) If h (x) = e x then equation (2) becomes Observe that in this case h (0) = 1 hence in theorem 3.3, m 0 = 0 only for λ < −1. This kind of nonlinearity appear in plasma physics [23]. We refer to [18] and discussion therein for related problems on bounded domains of R d .

4.1.
Convergence of the potential to a delta function. The results in sections 3 and 5 provide a very complete characterization of the solutions of equation (2). In this subsection we use those results to study in detail what happens to the positive solution of equation (2) when the potential V (x) approaches a delta function. Namely, for µ > 0 let which satisfies that lim µ→∞ W µ (x) = −δ(x) in the sense of distributions. Let λ < 0 and consider the equation As µ → ∞ equation (7) resembles the equation and we will establish (assuming h (0) = 0)) that the positive solution of equation (7) converges to the positive solution of equation (8). Let us start by describing the solution of equation (8). Proof. It is immediate that equation (8), in (−∞, 0) and in (0, ∞), implies that any positive solution v(x) has the form . Hence α = s and the lemma is proved.
And now let us establish the convergence of the solutions.

4.2.
Description of soliton solutions for the dynamic equation. Let us consider the dynamic nonlinear Schrödinger equation As mentioned in the introduction, such models may arise as a mean field approximation of a many body quantum system. More specifically, in Bose-Einstein condensates, the system is studied using a nonlinear equation with interaction between particles summarized by a nonlinear potential of the form 1 2 (|ψ| 2 * w)ψ, where * denotes the convolution and w is the pair potential, i.e. the model for interaction between particles (see e.g. [24]). When the interaction between particles in the system is a very short range interaction (contact interaction) that happens all over the space, the pair potential w is proportional to the delta function and the Gross-Pitaevskii equation is obtained. If the interaction between particles in the system is a short range interaction that additionally has a space and time dependency, one might expect the corresponding mean field equation to be of the form (12).
For some specific cases of V (x, t), an adequate ansatzs of solution allows equation (12) to be reduced to the equation Some examples of potentials V (x, t) where such reduction is possible are the following cases.
Assume that λ ∈ R and assume that ϕ(x) is a solution of equation (13). Then, in the following cases, equation (12) admits the corresponding ψ(x, t) as a solution: After a change of variable, this results also tackles potentials of the formṼ ( Proof. It is obtained by direct calculations. To exemplify the calculations we prove Using these in equation (12), and since V (x, t) = t n−2 V (x/t), we obtain, which reduces equation (12) into equation (13) as prescribed.
Corollary 4.4. If ϕ(x) is a solution of equation (2), provided in theorem 3.3, then ψ(x, t) constructed in lemma 4.3 is a soliton solution of equation (12) in the one dimensional case for the corresponding potential V (x, t).

5.
Technical results and main proofs.
5.1. Some technical ODE results. This subsection can be read independently from the rest of the article. In particular, in here we are not assuming the hypotheses of section 2.
In this subsection we study the technical aspects of autonomous equations of the form |u | = g(u) and some particular cases of it. Such equations can be analyzed with standard ODE results when g(u) = 0 and difficulties arise only when g(u) = 0. Given our final setting, we will restrict our attention to solutions satisfying u = 0 only at isolated points, hence we will proceed with the following strategy: construct local solutions of |u | = g(u) satisfying g(u) = 0 except at the end points, then paste these local solutions in an appropriate way.
We start this subsection with results that analyze these local solutions or building blocks. Let us recall a basic comparison result of ODEs that will be used repeatedly throughout the subsection.
Since we are working with autonomous equations, for simplicity and without loss of generality we will write the results with initial conditions at x = 0.
Standard ODE theory (Picard-Lindelöf Theorem) implies the existence of a maximal a > 0 such that equation (15) admits a unique solution u ∈ C 1 ([0, a)). Since g(y) > 0 then u(x) is strictly increasing and y 0 ≤ u(x) < c, ∀x ∈ [0, a). We now prove that a < ∞, in which case it follows that lim x→a − u(x) = c (otherwise the solution could be extended beyond a, contradicting the maximality of a). Let If additionally κ 2 > 0 is such that g(y) ≤ κ 2 (c−y), ∀y ∈ [y 0 , c], lemma 5.1 implies the bound a ≥ 2 (c − y 0 )/κ 2 in a similar way.
We now establish the continuous dependence of the solution in lemma 5.2 with respect to the parameters of the equation. There are two main difficulties that we need to address in this result and they lie at the right end of the domain interval. They are: 1) the right end point moves when changing the parameters of the equation, therefore changing the domain of the solution; and 2) at the right end point the ODE stops being locally Lipschitz. Let a > 0, u ∈ C 1 ([0, a]) and let b > 0, v ∈ C 1 ([0, b]) be the solutions (as in lemma 5.2) of Then there exists δ > 0 that only depends on κ 1 , (c − y 0 ), (d − y 1 ) and the Lipschitz constant of h(x), such that for any 0 < < 1 small enough, Proof. Our first step is the comparison of the solutions u(x) and v(x) for x ∈ The appropriate criteria for the choice of x is the following: where δ > 0 is a constant that will be prescribed later.
Since x < min{a, b}, the definition of x and the continuity of the functions imply that max{ g(u(x)), h(v(x))} ≤ −δ/ ln( ). With this we now proceed to b), the analysis in the interval [x, min{a, b}], where the functions are almost constant and close to their limiting values c and d respectively. The equation for u implies the condition on g implies κ 1 (c − u(x)) ≤ g(u(x)), while the choice of x gives g(u(x)) ≤ (−δ/ ln( )) 2 .
These three conditions together imply Similarly for v, we have Since |c − d| ≤ we conclude In particular In summary, for δ large enough and for > 0 small enough, the estimates in Our second step is to estimate |a − b|. Observe that u and v solve Lemma 5.2 implies 0 ≤ a−x ≤ 2 (c − u(x))/κ 1 . Since we also have κ 1 (c−u(x)) ≤ g(u(x)) ≤ (−δ/ ln( )) 2 , we conclude Similarly, we obtain And these bounds imply To finish the proof we review the dependence of δ on the parameters of the equation.
The argument above requires δ ≥ 2L min{a, b}, but lemma 5.2 implies min{a, b} ≤ 2 (d − y 1 )/κ 1 , hence δ can be chosen as only depending on κ 1 , (d − y 1 ) and L. The estimates obtained are the same as the one described in the statement of the lemma after absorbing some constants into δ. Lemma 5.2 and lemma 5.3 study solutions u(x) of equations of the form u = g(u) on intervals where u (x) = 0 only at the right end of the interval. We now study how to use these solutions as building blocks, pasting them at points where u = 0 and constructing solutions of the equation |u | = g(u) for larger domains. As we will see below, by allowing u = 0 only at isolated points this construction also solves the equation u = g (u)/2 if g is C 1 . Let a > 0 and u ∈ C 1 ([0, a]) be the solution in lemma 5.2 of the equation Proof. Our first step is to verify that v(x) is a solution of the equation (18). From its definition it is clear that  Now we proceed to study a particular form of the equation above, namely, we consider the equation |u | = f (u 2 ) and its solutions satisfying u = 0 only at isolated point. The results obtained above provide a very complete description of these solutions. This is summarized in the following theorem.
This u also solves the equation u = uf (u 2 ) and it depends continuously on the parameters of the equation in any bounded interval.
Proof. Let g : [0, c] → R + be defined as g(y) = f (y 2 ). Then Hence we are in the setting of the previous results and w solves w = f (w 2 ) in [0, a] as described in lemma 5.2. Because of the symmetries, v = f (v 2 ) in [−a, a] and we can apply lemma 5.4 on each end of the intervals 2an + [−a, a], concluding that the prescribed u is indeed the only solution of equation (20). The stated properties of u are direct consequence of lemma 5.2 and lemma 5.4 and u is 4a-periodic by construction.
In theorem 5.5 we have a very complete description of the solutions of |u | = f (u 2 ) satisfying u = 0 at isolated points. These solutions are periodic and the last part of this appendix is devoted to study how the period of these solutions depend on some very specific parameters of the equation when β − f is a convex function for some constant β > 0. The family of convex functions providing the structure for the analysis is the following.
Definition 5.6. We define the family of functions H as The next lemma remarks on some useful basic properties for h ∈ H and introduce the definition of some quantities that will be used later in the parameters of the equations or in the estimates.
Proof. Let 0 < β < γ. Let a = a(β), c = c(β), b = b(γ), d = d(γ) and let u ∈ Lemma 5.7 implies 0 < c < d and then another part of lemma 5.7 implies since h is convex and u is positive and strictly increasing Evaluating in x = a and some manipulation imply We conclude that lim β→0 + a = ∞.
Theorem 5.8 will be applied with a slightly different notation in the main part of the article. For simplicity we rewrite theorem 5.8 in such notation as a corollary. Then c(β) is continuous strictly increasing on β, a(β) is continuous strictly decreasing on β, lim β→∞ c = ∞ and lim β→∞ a = 0. If additionally h (0) < −λ then lim β→0 + a = +∞.
Proof. If h ∈ H then (λ + h) ∈ H for any λ ∈ R, hence the corollary is exactly theorem 5.8 and some parts of lemma 5.7.
And the last result in this appendix is a version of corollary 5.9 considering a specific initial condition.
In the proof of theorem 5.8 we already obtained hence all that we need to apply lemma 5.1 is that y 0 = r and y 1 = s satisfy d c y 0 ≤ y 1 . From the definitions, and from lemma 5.7 Applying lemma 5.1 as in the proof of theorem 5.8 implies a 0 (γ) < a 0 (β).

5.2.
Proofs of the main theorems. Now we proceed to use the results and tools from subsection 5.1 to present the proofs of the main theorems of this paper, which were introduced in section 3.

Real valued parameters.
In section 2 we mentioned that without loss of generality we can restrict our attention to λ ∈ R and solutions ϕ that were realvalued. In this section we justify such consideration. We let V (x) = −1 for |x| ≤ 1 and V (x) = 0 for |x| > 1. We assume h : R + → R is continuous strictly increasing, satisfying h (0) ≥ 0 and lim z→∞ h (z) = ∞. Let us study the equation Lemma 5.11. If there is ϕ : R → C, a non-trivial H 2 (R) solution of equation (24), then λ ∈ R.
Proof. Multiplying equation (24) by ϕ, integrating over R and integrating by parts we obtain