Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials

Motivated by relevant physical applications, we study Schrodinger equations with state-dependent potentials. Existence, localization and multiplicity results are established for positive standing wave solutions in the case of oscillating potentials. To this aim, a localized Pucci-Serrin type critical point theorem is first obtained. Two examples are then given to illustrate the new theory.


Introduction. The nonlinear Schrödinger equation of general form
iu t + ∆u + g (|u|) u = 0 (1) where u is a complex-valued function of (x, t) ∈ R n × R, has applications in several domains of theoretical physics describing the propagation of waves due to dispersive effects (see, e.g., [1,23] and [36]). Such equations with state-dependent potential appear for instance in quantum mechanics and allow entanglement generation both for individual atoms and ensembles [6]. For mathematical aspects and methods concerning the nonlinear Schrödinger equation we refer to the monographs [5,9,12,13] and [36].
Of particular interest is the study of the so called standing wave solutions, i.e., solutions of the form where ω is a real number and v is a complex-valued function depending only on the space variables. Looking for such type of solutions, (1) reduces to the elliptic stationary equation There is a rich literature devoted to the study of nonlinear elliptic equations by means of many techniques and approaches, see among others [2,4,8,17,22,35,37].
In this paper, we shall investigate the existence, localization and multiplicity of positive standing waves solutions, i.e., solutions of the form (2) where v is a positive real-valued function. For some contributions on positive standing wave solutions of nonlinear Schrödinger equations, we refer to [3,10,11,14,21,38] and the references therein. Our results will take into consideration the oscillatory properties of the nonlinear state-dependent potential g in connection with the wave frequency ω. More exactly, we shall discuss the problem in a bounded open set Ω ⊂ R n . By a solution of (3) we mean a function v belonging to the positive cone K = H 1 0 (Ω; R + ) of the real Hilbert space H 1 0 (Ω) , and such that (v, w) H 1 0 (Ω) = (g (v) v − ωv, w) L 2 (Ω) , for all w ∈ H 1 0 (Ω) , where Our results make the connection between the oscillatory properties of the function g and the norm localization of a sequence (v k ) of solutions in distinct energy strips: r 01 < |v 1 | H 1 0 (Ω) < R 11 ≤ r 02 < |v 2 | H 1 0 (Ω) < R 12 ≤ ... The upper estimates |v k | H 1 0 (Ω) < R 1k , k = 1, 2, ... are obtained in a way similar to that from the a priori bounds technique [17]. On the contrary, getting lower estimates like r 0k < |v k | H 1 0 (Ω) is in general a difficult task for the solutions of PDEs, except for some special cases such as radial solutions (see, e.g., [16] and [18]). The lower estimations require weak Harnack type inequalities, which in case of PDEs have only a local expression in terms of the norm of solutions on a proper subdomain of Ω. Thus instead of the global lower estimate r 0k < |v k | H 1 0 (Ω) we may expect a local one, R 0k < |v k | H , where |.| H is a norm in a space H of functions defined on some Ω 0 ⊂ Ω. Such lower estimations are possible based on local (in a subdomain) weak Harnack inequalities which are known for several classes of PDEs (see, e.g., [17,30]).
As an example, such a space H may be L p (Ω 0 ) , with 1 ≤ p ≤ 2 * (n ≥ 3) endowed with its usual norm Clearly, in this case, one has the embedding H 1 0 (Ω) ⊂ L p (Ω 0 ) and there is c 0 > 0 such that Then, from the local estimate via (4), one can nevertheless obtain , namely a lower estimate in the energetic norm. Therefore, it suffices to localize solutions in a conical shell of a less regular geometry, namely in a set denoted by K R0R1 , such that The choice of a second space (H, |.| H ) is imposed by the Harnack type inequalities and is essential in order to guarantee the conditions on the boundary of the shell. The localization of the solutions in conical shells of type (5) can be done making use of the critical point theory developed by the second author in [25,26]. The paper is organized as follows. In Section 2 we give our main abstract result, namely a critical point theorem in a conical shell. This result of Pucci-Serrin type [28,29], immediately yields a three critical points principle in a conical shell. The difference from the original Pucci-Serrin theorem consists in the possibility to localize critical points. The iterative application of this new principle yields in a natural way multiplicity results. For a recent Pucci-Serrin type result in a ball, we refer to [27].
The results are then applied in Section 3, and yield the existence, localization and multiplicity of positive standing wave solutions for general nonlinear Schrödinger equations. We next give two examples. The first one deals with one dimensional waves. In this case we are able to make a complete study of the existence, localization and multiplicity of solutions, in connection with the oscillatory properties of the potential. The second example is concerning with higher dimensional radial waves in an annular domain. Here the theory applies to a nonautonomous problem, obtained after two appropriate changes of variable. We conclude by a concrete example of state-dependent potential, on which we illustrate the practical applicability of the three critical points principle.
2. Abstract theory. There exists a huge literature devoted to the existence and localization of positive solutions of various types of integral, ordinary differential and partial differential equations. One of the most common approaches is based on the Krasnoselskii's fixed point theorem in cones [19] which yields solutions in a conical shell assuming that some compression or expansion boundary conditions are satisfied. An analogue critical point theory was established in [25] and [26] as an extension of Schechter's bounded critical point theory [33,34]. The theory guarantees multiple positive solutions of a problem provided that suitable conditions are fulfilled for disjoint shells. Another tool for multiple solutions of nonlinear problems is the three critical points theorem of Pucci-Serrin [28,29]. This principle was successfully extended by Ricceri [31,32] (see also [7]) in the study of parametrized problems with applications to partial differential equations. For details and several applications, we also refer to the book [20].

A preliminary result.
For reader's convenience, we recall from [25] a saddle point theorem under a linking condition, which is the key ingredient in the proof of our main abstract result of this section, Theorem 2.7 below. We start by a short description of the functional setting and some definitions.
Let X be a Hilbert space with inner product and norm (., .) , | | |.| | |, H be a Banach space with norm . , H its dual with norm . H , and assume that there exists an injective continuous linear map I : X → H, by which we can identify X to the linear subspace I (X) of H, and any element u ∈ X to its image Iu ∈ H. Thus X ⊂ H holds with continuous embedding. Let c 0 be the best embedding constant with u ≤ c 0 | | |u| | | for all u ∈ X, i.e., c 0 = sup{ u : u ∈ X, | | |u| | | = 1}. It is assumed that the duality map P of H associated to some C 1 normalization function σ : R + → R + , i.e., σ (0) = 0, σ (τ ) > 0 for every τ > 0 and σ (τ ) → ∞ as τ → ∞, is a continuous single-valued map, namely P : H → H and By a wedge of X we understand a convex closed nonempty set K ⊂ X, K = {0} , with λu ∈ K for every u ∈ K and λ ≥ 0. If in addition K ∩ (−K) = {0} , then K is called a cone.
Definition 2.1. For every numbers R 0 , R 1 with 0 < R 0 < φ R 1 , we define the conical shell K R0R1 by being the connected component of the set One can easily see that from 0 < R 0 < φ R 1 , there is µ > 0 such that R 0 < µφ and | | |µφ| | | < R 1 . This shows that the set K R0R1 is nonempty.
Let L be the linear operator from X to X (the canonical isomorphism of X onto X ), given by (u, v) = Lu, v for all u, v ∈ X, and let J from X into X be the inverse of L. Then (Ju, v) = u, v for all u ∈ X , v ∈ X.
Let F : X → R be a C 1 functional and assume that the following conditions are satisfied: where ν 0 is a positive constant.

Remark 1.
It is worthy to mention that the invariance condition (6) is essential for all the results which follow. Notice that it trivially holds in the particular case K = X.
Remark 2. Note that (7) holds whenever JF maps bounded sets into bounded sets.
Definition 2.2. We say that the functional F satisfies the Modified Palais-Smale-Schechter (MPSS) condition in K R0R1 provided that any sequence (w k ) , w k ∈ K R0R1 , for which (F (w k )) converges and one of the following three conditions holds has a convergent subsequence.
Remark 3. In particular, F satisfies the (MPSS)-condition in any conical shell K R0R1 if the mappings N (u) := u − JF (u) and JP are completely continuous from X into X.
Definition 2.4. Let S ⊂ K R0R1 be a closed set, T ⊂ K R0R1 a submanifold with relative boundary ∂T, and assume that S and ∂T link with respect to Γ : For more information about linking we refer the reader to [34] and [35]. Referring to Definition 2.4 we denote We are now ready to state the announced saddle point theorem.
Theorem 2.5. Assume that conditions (6), (7) are satisfied. In addition assume that sup and there is a ρ > 0 with Then there exists a sequence (w k ) with w k ∈ K R0R1 such that and one of the conditions (8), (9), (10) holds. If in addition, the (MPSS)-condition holds in K R0R1 and F satisfies the compression boundary condition in K R0R1 , then there exists u ∈ K R0R1 with F (u) = 0 and F (u) = c.
We note that Theorem 2.5 was proved in [25] under the following additional assumption on the duality map P : (s): For each function u ∈ C 1 (R + , H) , the function P u (t) belongs to C 1 (R + , H ) and This condition was used in the proof of Theorem 2.5 only to guarantee that P u (t) , u (t) and d u (t) /dt have the same sign for every t with u (t) = 0. However this happens without assumption (s), in virture of the following result.
Lemma 2.6. If the duality map P of H associated to the normalization function σ is a continuous single-valued map, then for each Proof. From If h > 0, then and letting h ↓ 0 we obtain

Similarly from
Since lim inf is less or equal to lim sup, we may infer that the function u (t) is differentiable to the right at t and its right derivative is P u (t) , u (t) /σ ( u (t) ) .
Taking h < 0 we arrive to the conclusion that u (t) is differentiable to the left at t and its left derivative is also equal to P u (t) , u (t) /σ ( u (t) ) .

2.2.
A bounded Pucci-Serrin type theorem. In this section we state and prove a version of the Pucci-Serrin critical point theorem in a conical shell. This extension is based on second author's bounded critical point result, Theorem 2.5, and on a refined application of Ekeland's variational principle. Roughly speaking, this theorem guarantees that if a functional has two local minima in a conical shell, then a critical point exists in that shall, different from the two minima. This implies that if the functional has a finite or infinite number of local minima located in disjoint conical shells, then another sequence of critical points exists.
Theorem 2.7 (Pucci-Serrin type critical point theorem in a conical shell). Let F : X → R be a C 1 functional which satisfies the (MPSS)-condition in K R0R1 and such that (6) and (7) hold. Assume that u 1 , u 2 ∈ K R0R1 are distinct, and there exists ρ > 0 such that one of the following conditions is satisfied: Then F has a critical point u 3 such that Remark 4. In particular, if H = X and | | |.| | | = . , since R 0 < R 1 , there are no elements which simultaneously satisfy | | |u| | | = R 1 and u = R 0 . Hence in this case the conditions (h 1 ) and (h 2 ) trivially hold. The same is true for the similar condition (23) below.
We show now that condition (14) holds. Indeed, if (h 1 ) is true, then taking hence (14) holds. If (h 2 ) is true, then since c ≥ F (u 1 ) , we have c ≥ F (u) + ρ for every u ∈ K R0R1 with both | | |u| | | = R 1 and u = R 0 , so (14) holds. This, together with the (MPPS)-condition in K R0R1 gives the result. Case 2. Assume that there is no r ∈ (0, r 0 ) satisfying (17). Then, for every r ∈ (0, r 0 ), one has This equality implies that, for r ∈ (0, r 0 ) and every integer k ≥ 1, there exists v k ∈ K R0R1 such that Now fix r ∈ (0, r 0 ) and choose k large enough such that Since V ⊂ B r0 (u 1 ), then inf u∈V F (u) = c 1 . Hence, an application of the Ekeland variational principle [15] to F | V and v k gives the existence of w k ∈ V such that Let v = w k − tJF (w k ). We claim that v ∈ V for all t > 0 sufficiently small, and so it can be used in (iii). We first prove that v ∈ K R0R1 . Step and both w k and w k − JF (w k ) belong to K, we have that v ∈ K for all t ∈ (0, 1). Step we have | | |w k | | | < R and consequently Step and w k > R 0 , one also has v ≥ R 0 for t sufficiently small.
Thus v ∈ V for all t > 0 sufficiently small, as claimed. Therefore we may use this choice of v in (iii) and obtain for all t > 0 sufficiently small Dividing by t and letting t → 0 + , we deduce that This gives It follows that F (w k ) → 0 as k → ∞. Moreover, from (i), F (w k ) → c 1 as k → ∞. Now the existence of a critical point u 3 ∈ K R0R1 follows from the (MPSS)-condition, while the inequalities (20) give | | |u 3 − u 1 | | | = r, whence u 3 = u 1 and u 3 = u 2 .
The steps that we propose in order to use Theorem 2.7 are: • Choose u 1 as a global minimum of F in a conical shell K 1 ; • Choose u 2 as a global minimum of F in a conical shell K 2 , where K 1 ∩K 2 = ∅; • Take a conical shell K 3 including both K 1 and K 2 and apply Theorem 2.7 in K 3 . More precisely, let R 01 , R 11 , R 02 , R 12 , R 0 and R 1 be positive numbers such that Let K 1 := K R01R11 , K 2 := K R02R12 and K 3 := K R0R1 the connected components of the sets Theorem 2.8. Let F : X → R be a C 1 functional and assume that (6) holds. In addition assume that the boundary conditions (7) and the (MPSS)-condition are satisfied on K 3 . If for each j ∈ {1, 2} , u j is a global minimum of F in K j with u j > R 0j , | | |u j | | | < R 1j , and there exists ρ > 0 such that either (h 1 ) or (h 2 ) holds, then F has a critical point u 3 such that Proof. It is easy to see that u j , j = 1, 2 are local minima of F in K 3 with u j > R 0 and | | |u j | | | < R 1 . Hence, the assumptions of Theorem 2.7 are satisfied on K 3 and this gives the conclusion.
The following result from [25] gives us a way to obtain a global minimum in a conical shell. Theorem 2.9. Assume that conditions (6), (7) are satisfied. In addition assume that m := inf and there is a ρ > 0 with Then there exists a sequence (w k ) with w k ∈ K R0R1 such that and one of the conditions (8), (9), (10) holds. If in addition, the (MPSS)-condition holds in K R0R1 and F satisfies the compression boundary condition in K R0R1 , then there exists u ∈ K R0R1 with E (u) = 0 and E (u) = m.
The next remark gives us a way to obtain a three critical points result in a conical shell.
where we assume that Ω is a bounded open subset of R n , h is a continuous function from R + into R + and η ∈ L ∞ (Ω) with 0 < α ≤ η (x) ≤ β for a.e. x ∈ Ω. Notice that, for the particular choice η (x) ≡ 1 and h (s) = (g (s) − ω) s, (25) is exactly (3). Assume the growth property: This atypical growth condition, where a, b and p vary with r, will be used to obtain multiple solutions by the iterative application of Theorem 3.1 below.
For the application of Theorem 2.7, we give the functional setting (see Subsection 2.1) and we check the theorem hypotheses. Here we take X = H 1 0 (Ω) endowed with the inner product and norm , and H = L p (Ω 0 ) , with a fixed Ω 0 ⊂⊂ Ω and the usual norm In what follows, in order to conform ourselves to the abstract setting from the previous section, we shall associate to the functional spaces considered as X, H and to any other space Y the following notations for the norms: (X, | | |.| | |), (H, . ) and (Y, |.| Y ) , respectively. Thus, for the present application, the notation | | |.| | | stands for the norm |.| H 1 0 (Ω) , while the symbol . means the norm |.| L p (Ω0) . Also, in this case, Lv = −∆v and J = (−∆) −1 , Here F : H 1 0 (Ω) → R is given by In particular this shows that F maps bounded sets into bounded sets, so according to Remark 2, relation (7) holds. Also, we have The (MPSS)-condition. Since p < n/ (n − 2) < 2 * = 2n/ (n − 2) , the embedding of H 1 0 (Ω) into L p (Ω 0 ) is compact and so N and JP are completely continuous from H 1 0 (Ω) to itself, and the Remark 3 applies. Definition of K. The hypothesis p < n/ (n − 2) , allows us to make use of the weak Harnack inequality for nonnegative superharmonic functions (see [30]), which implies the existence of a constant M > 0 (depending on the choice of Ω 0 ) such If v ∈ K, then v ≥ 0 in Ω and since η ≥ 0 and h (R + ) ⊂ R + , one has that ηh (v) ≥ 0 in Ω. If w := J [ηh (v)] , then −∆w = ηh (v) ≥ 0, and, by the maximum principle, w is nonnegative. Then w satisfies (27), and consequently Choice of φ. In this case we take φ be the positive eigenfunction corresponding to the first eigenvalue λ 1 of the Dirichlet problem for −∆, i.e., ∆φ + λ 1 φ = 0 in Ω, φ = 0 on ∂Ω, with | | |φ| | | = 1. Thus, all the conical shells considered below intersect R + φ.
Theorem 3.1. Let h : R + → R + be continuous and satisfies (h). Assume that R 0 and R 1 are positive numbers such that R 0 < φ R 1 and for r = M R 0 the following conditions are satisfied: In addition assume that there exists ρ > 0 such that for every v ∈ K satisfying both | | |v| | | = R 1 and v = R 0 . Then (25) has a solution v which is the global minimum of F in K R0R1 and Proof. We apply Theorem 2.9. First we note that since R 0 φ/ φ belongs to K R0R1 , we have F (R 0 φ/ φ ) ≥ m and so (31) guarantees (22). Regarding to the previous considerations it remains only to prove the compression boundary conditions (11) and (12). We need these conditions even for λ = 0, in order to guarantee for the minimum the strict inequalities (32), as required by Theorem 2.7. Assume that (12) does not hold for every λ ≥ 0. Then JF (v) + λv = 0 for some v ∈ K R0R1 with | | | v| | | = R 1 and λ ≥ 0. It follows that in Ω 0 and for r = M R 0 , with the notations a = a (r) , b = b (r) and p = p (r) , we have Then relations (33) and (34) imply which contradicts (29). So (12) holds for λ ≥ 0. Next we show by contradiction that (11) Taking the norm in this inequality we obtain which contradicts (30). Hence Theorem 2.9 yields a critical point v ∈ K R0R1 which is a global minimum of F. Remark 6. We give two distinct sufficient conditions which guarantee (31).
(a): F is coercive in K and R 1 is sufficiently large. (b): R 1 is given and there exists r 0 > 0 such that where ϕ r0 is defined in the last line of relation (28). In this case ρ = ϕ r0 (R 1 )− F (R 0 φ/ φ ) .

Remark 7.
In the one-dimensional case, Ω is an interval I, and since H 1 0 (I) ⊂ C I , we have |v| C(I) ≤ c 0 | | |v| | | for all v ∈ H 1 0 (I) . Then instead of condition (29) one can use a more precise inequality, namely Indeed, as for the estimations (33), we have If in addition h is nondecreasing on R + , then (30) and (35) become These show the behavior of the nonlinearity h only at two points M R 0 and c 0 R 1 .
Indeed, the problem of finding positive standing wave solutions for (37) is exactly problem (25) with η (x) ≡ 1 and h (s) = (g (s) − ω) s. Obviously, in this case, the functions ϕ r and ψ r depend on ω, and so do the assumptions of Theorem 3.1. This shows the interdependence between the value of the frequency ω and the kinetic energy of the wave.
A three solution type result can be now obtained from Theorem 3.1, Theorem 2.8 and Remark 5.
Theorem 3.2. Let h : R + → R + be continuous and satisfy (h), and let F be coercive in K. Assume that R 01 , R 02 , R 11 and R 12 are positive numbers such that c 0 R 11 < R 02 and for each k ∈ {1, 2} , all the assumptions of Theorem 3.1 hold for R 0 = R 0k and R 1 = R 1k . Then problem (25) has three solutions v 1 , v 2 , v 3 such that v 1 and v 2 are global minima of F in K R01R11 and K R02R12 , respectively, and Proof. Theorem 3.1 guarantees the existence of the first two solutions v 1 and v 2 with the desired properties. Clearly v 1 , v 2 are local minima of F in any conical shell K R01R for every R ≥ R 12 . The conclusion will follow from Corollary 2.8 applied to K R01R once we have guaranteed the condition (h 1 ). To this aim it is sufficient to observe that Hence it is enough to chose R ≥ R 12 large enough that where ρ > 0 is arbitrary taken. This is possible owing to the coercivity of F.

Remark 9.
In virtue of (28), the functional F is coercive in K provided that p < 2.
In the next sections we illustrate the applicability of Theorem 3.2 to problem (3) in two particular situations: the one-dimensional case and the case of the ndimensional radial waves in an annulus.
3.1. One-dimensional waves. Without loss of generality, let Ω = I = (0, 1) . We first recall the following result from [24]. Lemma 3.3. Let h : R → R + be a continuous function with h (0) = 0 and nondecreasing on R + . Assume that there are two numbers 0 < R 0 < R 1 such that Then the problem Note that if the inequalities in (40) are strict, then the solution satisfies the strict inequalities R 0 < | | |v| | | < R 1 .
By a successive application of Lemma 3.5, we derive the following multiplicity result.
Proof. From Lemma 3.5 we obtain a solution z j in each shell K R0j R1j , for j = 1, 2, ..., k. The additional solution z k+1 is obtained if we apply the three critical points theorem 2.7 to the shell K R 0k R , where R ≥ R 1k is large enough that conditions (h 1 ) and g (c 0 R) < ω + 1 βc 2 0 (b − a) hold, which is possible thanks to the sublinear growth of h (p < 2) .
Example. We apply Theorem 3.6 to problem (42) for k = 2, assuming that b − a = 1 and h is the increasing function satisfying the sublinear growth condition (48) where β is that from (43), γ > 0 is large enough and d is the unique solution bigger than 1 of the equation β −1 1 + (s − 1) 3 = As. Here A = 1/ (αM Jχ Ω0 ) . Note that s = 2 is the unique solution bigger than 1 of the equation 1 + (s − 1) 3 = s, and it can be seen that 2 < d.
Also, in this case c 0 = 1, and the second inequality from (46) reads as h (R 11 ) R 11 < 1 β , and due to the convexity of the function h on the interval [1,2] , it holds for every R 11 ∈ (1, 2) . Now we look for a radius R 11 such that (47) holds. Since R 01 ≤ 2, one has h (R 01 ) ≤ h (2) , and direct computation gives h (2) = β −1 23/12. Hence for (47) it suffices to have R 2 which in view of (49) is true provided that Clearly this happens if we choose R 11 < 2 close enough to 2. Therefore we found a first shell K R01R11 with the desired properties. A second shell of this type is obtained if we take a suitable R 02 > d/M (which exists for large γ) and a large enough R 12 (which is possible due to the coercivity of F ). Therefore, Theorem 3.6 applies and gives three positive solutions.