Normalized multi-bump solutions for saturable Schrödinger equations

1 Introduction and main results

This paper deals with the existence of solutions (v, λ) ∈ H¹(ℝ², ℝ) × ℝ to the following nonlinear eigenvalue problem with saturable nonlinearity

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 > 0. 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 L² norm prescribed and λ as a Lagrange multiplier. For small ε > 0 in (1.1), we will make a first attempt to study the existence and concentration behavior of multi-bump type solutions in H¹(ℝ²). 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(ε x) for small ε > 0

It is well known that equation (1.2) is the Euler-Lagrange equation of the following minimization problem subject to a L² constraint

where

Also observe that in this case the parameter λ ∈ ℝ depending on ε (so in what follows, we denote λ = λ_ε), 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 (1.2) in H¹(ℝ²) for small ε > 0, under the condition

(B1) I(x) satisfies

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 H¹(ℝ²). 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 difficulties which require new ideas and variational techniques.

Let k ≥ 2 be a fixed positive integer. We define

where

l = 1, 2, …, k, and O(2) is the group of orthogonal transformations in ℝ². 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 satisfied and I(x) = I(|x|) ∈ C(ℝ², ℝ) ∩ L^∞(ℝ², 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 S¹ = {x ∈ ℝ²| |x| = 1}, and there exist δ₀ > 0 and r₀ > 0 such that I(x) ≥ I_∞ + δ₀ for ∥x| − 1| ≤ r₀.

Then we have the following Theorem.

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 u_ε ∈ Hk1 (ℝ²) as ε ∈ (0, ε₀). Generally speaking, this conclusion is not necessarily true. For example, if only (B1) is satisfied, by Theorem 2.4 in [14], we see that u_ε ∈ Hk1 (ℝ²) 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(0) are sufficient 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 ℝ²; All dx in the integrals are omitted; L^p ≡ L^p(ℝ²)(1 ≤ p < + ∞) is the usual Lebesgue space with the norm ||u||pp = ∫_ℝ² |u|^p; H¹ ≡ H¹(ℝ²) denotes the uaual Sobolev space with the norm ∥u∥² = ∫_ℝ²(|∇ u|² + |u|²); o_n(1) (resp. o_ε(1)) will denote a generic infinitesimal as n → ∞ (resp. ε → 0⁺); → denotes the strong convergence and ⇀ the weak convergence.

2 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).

This follows from the proofs in [9], in which I(x) is fixed throughout the proof there. As we need to place a condition on I(0) as in Theorem 1.1, closer examination tells us that the proof of the above result works through if we fix the property of I in the neighborhood of the maximum points while allowing changes of I(0). We omit the details here.

Next in order to analyze the asymptotic behavior of the minimizers u_ε we prepare some estimates.

Now in the following, for given k ∈ ℕ⁺, we always fix

where Γ0′ (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| − 1| ≤ 2r₀, for each fixed Γ < Γ₀, there exists ε₀ = ε₀(Γ) > 0, such that m_Γ,k(ε) is always achieved by some u_ε ∈ Hk1 (ℝ²) for ε ∈ (0, ε₀). At the same time, by Theorem 2.1, we know that u_ε satisfies

where λ_ε is associated Lagrange multiplier.

Next, we start to study the qualitative properties for the minimizer u_ε of m_Γ,k(ε).

Let u_ε be the minimizer of m_Γ,k(ε) for ε ∈ (0, min{ε₁, ε₂}). Now we can obtain that there exists a sequence of points {y_n} ≡: {y_εn} in ℝ² such that most of the “mass” of un2 (x) ≡: uεn2 (x) is contained in a ball of fixed size centered at {y_n}. Here and below, we note that ε_n → 0 if and only if n → + ∞. At the same time, in the following we may assume that, up to a subsequence, u_n ⇀ v in Hk1 (ℝ²) as n → +∞.