Normalized multi-bump solutions for saturable Schrödinger equations

where ε is a small parameter (related to the Planck constant), Γ is a coupling constant, and I(x), the density function, is a bounded continuous function. This model describes paraxial counter-propagating beams in isotropic local media (e.g., [1–5]). An interesting issue concerning (1.1) is the existence of semiclassical states, which concerns the study of (1.1) for small ε > 0. From the physics point of view, semiclassical states describe a sort of transition from quantummechanics to classical mechanics as the parameter ε goes to zero. In (1.1), one can either consider the parameter λ ∈ R to be given, or to be an unknown of the problem. In this paper, we study the latter case, i.e., we look for normalized solutions with the L2 norm prescribed and λ as a Lagrange multiplier. For small ε > 0 in (1.1), we will make a rst attempt to study the existence and concentration behavior of multi-bump type solutions in H1(R2). We refer [6–8] for results on the problems of saturable nonlinearity without constraints and references therein such as existence and concentration property of solutions.


Introduction and main results
This paper deals with the existence of solutions (v, λ) ∈ H (R , R) × R to the following nonlinear eigenvalue problem with saturable nonlinearity −∆v + Γ I(εx) + v where ε is a small parameter (related to the Planck constant), Γ is a coupling constant, and I(x), the density function, is a bounded continuous function. This model describes paraxial counter-propagating beams in isotropic local media (e.g., [1][2][3][4][5]). An interesting issue concerning (1.1) is the existence of semiclassical states, which concerns the study of (1.1) for small ε > . From the physics point of view, semiclassical states describe a sort of transition from quantum mechanics to classical mechanics as the parameter ε goes to zero. In (1.1), one can either consider the parameter λ ∈ R to be given, or to be an unknown of the problem. In this paper, we study the latter case, i.e., we look for normalized solutions with the L norm prescribed and λ as a Lagrange multiplier. For small ε > in (1.1), we will make a rst attempt to study the existence and concentration behavior of multi-bump type solutions in H (R ). We refer [6][7][8] for results on the problems of saturable nonlinearity without constraints and references therein such as existence and concentration property of solutions.
The main goal of this paper is to establish the existence and concentration behavior of multi-bump solutions with a localizing potential I(εx) for small ε > −∆u + Γ I(εx) + u + I(εx) + u u = λu, for x ∈ R . (1.2) It is well known that equation (1.2) is the Euler-Lagrange equation of the following minimization problem subject to a L constraint mε(Γ, I) = inf{Eε(ρ) | ρ ∈ H (R ), where Also observe that in this case the parameter λ ∈ R depending on ε (so in what follows, we denote λ = λε), comes from problem (1.3) and can be interpreted as a Lagrange multiplier. Among all possible standing waves for equation (1.2), typically the most relevant are ground state solutions. Recently, in [9], by a global minimization method, we have obtained the existence and concentration behavior of positive normalized ground state solutions of equation ( We remark that in most cases the global minimizers are not necessarily multi-bump solutions, and that when I(x) is radially symmetric the global minimizers may be radially symmetric functions. In this paper we investigate conditions on I(x) = I(|x|) under which the minimizers are non-radial and multi-bump type solutions. In order to solve this problem, we introduce a local minimization procedure and work on a subspace of H (R ). The main ideas come from the methods introduced in [10,11] of the second author. This local minimization procedure has been successfully used to treat nonlinear Dirichlet problems [10,12] and nonlinear Neumann problems [13]. The advantage of this method is that we can get qualitative properties of the solutions constructed such as the concentration behavior and the shape of solutions with a discrete number of bumps. However this type of method and results have not been studied before for normalized solutions and there are new di culties which require new ideas and variational techniques.
Let k ≥ be a xed positive integer. We de ne . . , k, and O( ) is the group of orthogonal transformations in R . It is easy to see that G k is a cyclic group of order k. In order to get multi-bump type solutions, we consider the following minimization problem If (B1) is satis ed and I(x) = I(|x|) ∈ C(R , R) ∩ L ∞ (R , R) is radially symmetric, using a similar procedure as in the proof of Theorem 2.1 in [9], we may deduce the existence result of a minimizer for the above minimization problem m Γ,k (ε). But to show the minimizers are non-radial and of multi-bump type we would need additional conditions on the density function I(x).
By (B1) we deduce that the maximum value of I(x) must be obtained on a bounded closed set. We suppose that (B2) I(x) = I(|x|) is radial and achieves its unique maximum on S = {x ∈ R | |x| = }, and there exist δ > and r > such that I(x) ≥ I∞ + δ for ||x| − | ≤ r .
Then we have the following Theorem. Theorem 1.1. Assume that I(x) satis es (B1)-(B2). For any integer k ≥ xed, there exists Γ = Γ (I∞, k, δ , r ) (but independent of ε > , I and I( )), for each xed Γ < Γ , there exist ε = ε (Γ) > and α = α (Γ) > such that if < ε < ε and < + I( ) < α , m Γ,k (ε) has a minimizer solution uε ∈ H k (R ) satisfying (ii) uε is of k−bump type in the sense that uε has exactly k maximum points which form a G k −orbit G k (yε) for some yε ∈ R satisfying |εyε| → and up to subsequences The existence of a minimizer follows from the work of [14] (will be stated in Theorem 2.1), and we mainly concern whether there are multi-bumps for the local minimizers uε ∈ H k (R ) as ε ∈ ( , ε ). Generally speaking, this conclusion is not necessarily true. For example, if only (B1) is satis ed, by Theorem 2.4 in [14], we see that uε ∈ H k (R ) may be a radially symmetric solution and has only one bump centered at the origin. Therefore, in order to construct multi-bump solutions, we need to impose some additional conditions on I(x). We prove that (B2) and the condition on I( ) are su cient to assure the minimizers are of multi-bump type solutions. This paper is structured as follows. In Section 2 we will present and show some useful lemmas which are useful for the proof of Theorem 1.1. Afterwards, in Section 3 we will give the proof of Theorem 1.1.

Notation.
Throughout this paper, we denote by C a positive constant, which may vary from line to line; all integrals are taken over R ; All dx in the integrals are omitted; L p ≡ L p (R )( ≤ p < +∞) is the usual Lebesgue space with the norm ||u|| p p = R |u| p ; H ≡ H (R ) denotes the uaual Sobolev space with the norm ||u|| = R (|∇u| + |u| ); on( ) (resp. oε( )) will denote a generic in nitesimal as n → ∞ (resp. ε → + ); → denotes the strong convergence and the weak convergence.

Some technical results
In this section, we will establish several lemmas, which will be useful to prove Theorem 1.1 in next Section. First using a similar procedure as in the proof of Theorem 2.1 in [9], we may deduce the existence result of a minimizer for m Γ,k (ε). Theorem 2.1. Suppose that I(x) = I(|x|) satis es (B1)-(B2). Then for given integer k ≥ , there exists Γ = Γ (I∞, k) < (independent of ε > , I and I( )), for each xed Γ < Γ , there exists ε = ε (Γ) > such that for all ε ∈ ( , ε ), the minimization problem possesses a solution uε, which solves equation (1.2) for some λ < .
This follows from the proofs in [9], in which I(x) is xed throughout the proof there. As we need to place a condition on I( ) as in Theorem 1.1, closer examination tells us that the proof of the above result works through if we x the property of I in the neighborhood of the maximum points while allowing changes of I( ). We omit the details here. Next in order to analyze the asymptotic behavior of the minimizers uε we prepare some estimates.
Lemma 2.1. For given integer k ≥ , there exist Γ = Γ (I , k) < and < a = a(I , k) < k , such that for each xed Γ < Γ , is achieved by u k which is radially symmetric. Moreover, In particular, we have Proof. Using the same arguments as Theorem 2.1 in [14], we know that there exists Γ = Γ (I , k) < , such that for each xed Γ < Γ the minimization problem m( k , I ) is attained by u k . In addition, by Theorem 2.3 in [14], u k is a radially symmetric function. Moreover, we have and this implies On the other hand, we may nd a sequence of functions un such that ||un|| = k and an = R [u n − ln( + u n +I )] → k as n → ∞. In fact, this can be done by choosing v(x) ∈ C ∞ (B ( )) such that R v = k , and setting un(x) = nv(nx), then Here we used the fact for b > xed, Hence, there is a function u with ||u || = k and Therefore, there exists Γ = Γ (I , k), such that we can get the desired results for Lemma 2.1. Now in the following, for given k ∈ N + , we always x where Γ (I∞, k) is given in Theorem 2.1. We remark that by Theorem 2.1 and (B2), we know that no matter how I(x) changes outside the neighborhood of |x| − ≤ r , for each xed Γ < Γ , there exists ε = ε (Γ) > , such that m Γ,k (ε) is always achieved by some uε ∈ H k (R ) for ε ∈ ( , ε ). At the same time, by Theorem 2.1, we know that uε satis es where λε is associated Lagrange multiplier. Next, we start to study the qualitative properties for the minimizer uε of m Γ,k (ε).

Lemma 2.2. It holds that
Proof. Taking e ∈ S and the G k −orbit of e containing exactly k points, {g i e | i = , , . . . , k}, and de ning where w(x) ∈ H (R ) is the minimizer of m( k , I ) so w(x) → as |x| → ∞. Moreover, by Lemma 2.1, w is radially symmetric. Since Setting then ||Vε(x)|| = and Vε(x) ∈ H k (R ).
Proof. By Lemma 2.2, we know that there exists < ε ≤ ε , such that for ε ∈ ( , ε ), we have Therefore, by (2.2), we have Here we have used R u ε = and On the other hand, according to (2.3), we obtain Therefore, we have nished the proof of Lemma 2.3.

Lemma 2.4.
There exists < ε ≤ ε , such that for ε ∈ ( , ε ), the minimizer uε of m Γ,k (ε) satis es Here we have used the fact that R u ε (x) = . Thus, by Lemma 2.1, we have Let uε be the minimizer of m Γ,k (ε) for ε ∈ ( , min{ε , ε }). Now we can obtain that there exists a sequence of points {yn} ≡: {yε n } in R such that most of the "mass" of u n (x) ≡: u εn (x) is contained in a ball of xed size centered at {yn}. Here and below, we note that εn → if and only if n → +∞. At the same time, in the following we may assume that, up to a subsequence, un v in H k (R ) as n → +∞. Proof. We will do it by a contradiction argument. If not, for any R > , there exists a sequence un ≡: uε n such that lim n→+∞ sup y∈R B R (y) u n (x) = .
Then by Lions's Lemma (e.g., [15]) one has ||un||p → for any p > . By the fact that t − ln( + t ) ≤ Ct for some C > , we have Proof. Without loss of generality, we may assume that yn ≡ . By Lemma 2.5, we know that there exist positive constants R and β such that Since un v in H k (R ) as n → +∞, by (2.4), we have Thus, v ≠ and un → v in L loc (R ) as n → +∞. Therefore, un(x) v(x) in H (R ) and un → v in L (B R ( )). Hence, up to a subsequence, we may assume that un(x) → v(x) a.e. in B R ( ). In view of (2.5), v ≠ in B R ( ). By (2.5) there exists σ > such that where and µ(Ω) > .
To prove (2.17), we need the following lemma.
where y i n = g i yn , g i ∈ G k , i = , , . . . , k and g k yn = yn . Proof. We only consider k ≥ (for k = , this is the case of Lemma 3.1 in [9]). Applying the concentration compactness principle [17,18], we get three possibilities: vanishing, dichotomy and compactness. Vanishing can be ruled out by using Lemma 2.5. If compactness happens, there exists a subsequence of {un(x)} (still denoted by {un(x)}), and {yn} such that for any γ > there exists R = R(γ) > with the property that Now we can also get a contradiction as follows. Firstly, we claim that there exists R > such that {yn} in (2.27) satisfying |yn| ≤ R .
(2.28) If (2.28) is not true, then for a subsequence |yn| → +∞ as n → +∞. By the symmetry of R and un(g − x) = un(x) with g ∈ G k , we have This is a contradiction with R u n (x) = . On the other hand, by Proposition 2.1, we know that {yn} in (2.27) satisfying |yn| → +∞ as n → +∞, which produces a contradiction with (2.28). Therefore, compactness does not happen.
With vanishing and compactness both being ruled out, we obtain dichotomy of the sequence. Now by Proposition 2.1, we note that the orbit of {yn} under the action of G k contains exactly k points: y n , y n , . . . , y k− n , y k n = yn and the distance between any two of these k points tends to in nity as n → ∞. By the symmetry of the domain R and the fact that un(x) are G−invariant, one obtains for any xed n su ciently large, Since un(g i x) = un(x), i = , , . . . , k, we have Now we claim that for all γ > , there exists R = R(γ) > , such that as n → +∞. If not, we assume that there exists α satisfying < α < k , such that for all γ > , there exists as n → +∞. Then A := kα < . We will derive a contradiction as follows. Since |y i n | → +∞, i = , , . . . , k, as n → +∞ and |y i n − y j n | → +∞, i ≠ j as n → +∞, we have as n → +∞. Then, we get Therefore, by (2.38) and Brezis-Lieb Lemma [19], we have as n → ∞, using Lemma 5.2 in [14] we can nd δ > such that (2.40) as n → ∞.

Proof of Theorem 1.1
In this section, we will give the proof for the conclusions (i) and (ii) of Theorem 1.1 as ε → + . To be more speci c, the conclusions of (i) in Theorem 1.1 will be proved by Lemma 3.1, and the conclusions of (ii) in Theorem 1.1 will be proved by Lemma 3.2 and Lemma 3.3. On the other hand, by Lemma 2.9, for each γ > , there exists R = R(γ) > , such that Let η = η(t) be a smooth nonincreasing function on [ , +∞) such that η(t) = , for t ∈ [ , ], η(t) = , for t ≥ , and |η (t)| ≤ . Setting then w n,i (x) ∈ H (R ). By choosing R large enough (for xed γ > ), we may assume Therefore, we have Letting εn → and γ → , we have a contradiction to (3.1). ≥ km( k , I ), a contradiction. Here we have used w n,i w i in H (B R ( )), strongly in L (B R ( )) and and for each γ > , there exists R = R(γ) > , such that B R ( ) w i ≥ k − γ, i = , , . . . , k.
Therefore, there exists a subsequence εn such that x i n ≡ εn y i n → x i , w n,i w i ≥ in H (R ) and a.e. in R , where x i = g i x , g i ∈ G k , i = , , . . . , k and g k x = x . Applying the elliptic estimates theory to (2.31), we have w n,i → w i in C loc (R ) and −∆w i + Γ I(x i ) + w i + I(x i ) + w i w i = λ w i , x ∈ R , i = , , . . . , k, here λ < derives from Lemma 2.3 which implies λε n → λ as εn → + (up to a subsequence).
Since w n,i (x) = un(x + y i n ), i = , , . . . , k, and as n → +∞, w n,i (x) → w i (x) in L (B R ( )), by Lemma 3.1, Lemma 2.9, and the weakly lower semi-continuity of norm, we have  At last, combining Lemma 3.2 and the proof process of Lemma 3.2, we can obtain