Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation

In this paper, we consider \begin{document}$ L^2 $\end{document} constrained minimization problem for a modified Gross-Pitaevskii equation with higher order interactions in \begin{document}$ \mathbb{R}^2 $\end{document} . By using an auxiliary functional and some detailed energy estimates, the blow-up behavior of ground state for the modified Gross-Pitaevskii equation was obtained under different parameter regimes. Our conclusion extends some results of [ 3 ,Theorem 3.4].

1. Introduction. In this paper, we study the following modified Gross-Pitaevskii equation with higher order interactions in two dimensions where Ψ : R × R N → C, V : R N → R is a given potential. Equation (1) is used to describe the wave functions of Bose-Einstein condensations with higher order interactions, as well as other nonlinear topics. a and δ are two dimensionless constants for describing the contact interaction and higher order interaction strengths [4,17,18]. Standing waves of (1) are solutions of the form Ψ(t, x) = e −iµt u(x), where µ ∈ R is a fixed parameter. Taking it into equation (1), we see that u(x) satisfies − u + V (x)u − δ |u| 2 u = µu + a|u| 2 u, x ∈ R N . ( To prove the existence of solutions of (2), one approach is to treat equation (2) directly, or transform it into a semilinear elliptic equation by introducing a technique of changing variables, and then apply the mountain pass theorem and other variational methods, see [5,14,15] and the references therein. Another approach is to take µ ∈ R as an unknown Lagrange multiplier, then solutions of (2) can be solved by studying the following minimization problem [3,6,11,12,17,21] e a (δ) = inf u∈A E δ a (u),

XIAOYU ZENG AND YIMIN ZHANG
where E δ a (u) = The space X and H are defined by A minimizer of problem (3) is usually called a ground state of (2). When δ = 0, problem (3) was studied in [2,8,9,10,19,20] provided V (x) satisfies the following assumptions Particularly, when N = 2, problem (3) is mass critical and attracted much attention.
It was proved in [2,8] that there exists a * > 0, such that problem (3) has a minimizer if and only if a < a * . Moreover, the concentration and symmetry breaking of minimizers were investigated in [8,10] when a a * . Here, a * = |Q| 2 L 2 with Q = Q(|x|) being the unique positive solution of the equation [13] − When δ = 0, Colin, Jeanjean and Squassina [6] investigated the existence and stabilities of minimizers of (3) with V (x) = 0. For V (x) satisfies the condition (4), Bao, Cai and Ruan proved in [3] that (3) can be attained for all a when N = 1, 2, 3. We obtained a similar result results in [21] and proved that (3) has minimizers for more general nonlinear term for any δ > 0. In [3], the authors also studied the asymptotic behaviors of minimizers as δ → 0 + when V (x) is of the form By using the homogeneous of V (x) and applying some rescaling arguments, they obtained the following results.
Theorem A (Theorem 3.4 of [3]). Let V (x) be given by (6) and φ δ g be a nonnegative minimizer of (3). When N = 2 and a > a * , letφ is a nonnegative minimizer of the problem (7).
This theorem tells us that the limit behavior of minimizers depends on the value of the parameter a. Its proof mainly involves some technical rescaling arguments. Especially, the proofs rely on the homogeneity of V (x). In this paper, we intend to extend the results of Theorem A to some general potentials which are not homogeneous, such as the form of (4), and give refined energy estimates for e a (δ). We also want to study the asymptotic behaviors of minimizers as δ → 0 + for the case of N = 2 and a = a * , which is not involved in [3]. We note that, since the potential V (x) is not homogeneous, some of the arguments of [3] are no longer applicable for our case. To solve our problem, we need introduce some new ideas and derive detailed energy estimates of e a (δ). We first introduce the following auxiliary functional and the minimization problem Remark 1. We note that if a > a * , then problem (7) has at least one minimizer. Indeed, letū τ = τ |Q| L 2 Q(τ x), τ > 0, one can easily check thatū τ ∈Ā and it follows from (21) that Thus, we then conclude that (7) has at least one minimizer by applying the argument of [6, section 4].
Our following theorem addresses some refined blow-up behaviors of minimizers of (3) as δ → 0 + for the case of N = 2.
be a nonnegative minimizer of e a (δ) and z δ be one maximum point of u δ . Then, Moreover, where and where w 0 = w 0 (|x|) is a nonnegative minimizer of problem (7). (4), it is well know (see [2] for instance)that the embedding from H into L p (R 2 ) is compact with 2 ≤ p < ∞. Hence, from the proof of Theorem 1.4 of [6] or Theorem 2.1 of [3], one can easy to prove that there is a minimizer for problem (3).
When a > a * , the arguments of [3] only give that e a (δ) ≤ 1 δ d a + C as δ → 0 + , this is not enough to ensure that the blow-up point of u δ must be one minimum point of V (x). For the case of a = a * , we see that the blow-up of minimizers also happens as δ → 0 + , but the limit of minimizers is quite different from the case of a > a * . It also deserves to point out here that, (8) and (10) only indicate that the minimizers blow up around one minimal point of V (x), however, it does not give any information about the blow-up rate of ε δ . We think the main reason is that the asymptotic expansion of V (x) near its minimal points is unknown previously. To calculate the precise blow-up rate of the minimizers, motivated by [9,10] we now assume that V (x) has exactly l ∈ N + different minimal points, i.e., and there exist r 0 , p i , γ i > 0 such that Let (15) and denote the flattest minimal point of V (x) by Then, we have the following theorem which gives the blow-up rate and locates the blow-up point of minimizers. (4), (13) and (14). Let a = a * , x 0 be the blow-up point given in Theorem 1.1, and ε δ be given by (10), then x 0 ∈ Z. Moreover, we have and e a * (δ) = where From the above theorem we see that the minimizers concentrate at one of the flattest minimal points of V (x) as δ → 0 + . Moreover, (10) and (17) indicate that the blow-up rate of R 2 |∇u δ | 2 dx is the order of δ − 2 p+4 . (18) also gives the precisely energy estimates for e a (δ).
In this paper, we always denote |u| L p the L p -norm of a function u. C > 0 denotes some constant which may be different in different place.
2. Proofs of Theorems 1.1 and 1.2. We first recall the following Gagliardo-Nirenberg inequality [1] where the "=" holds when u = Q(x). Moreover, it follows from (5) and (20) that From Proposition 4.1 of [7], we also have We next estimate the energy of e a (δ) as δ → 0 + in the following lemma.

Lemma 2.2.
For any δ > 0 and a > 0, let and defineē Then, δē a (δ) = d a . Moreover, u δ is a minimizer ofē a (δ) if and only if δ Proof. For any u ∈Ā, set w δ = δ This also indicates that u δ is a minimizer ofē a (δ) if and only if and δ Moreover, if u δ (x) is a minimizer of e a (δ), then there holds that Proof. Let w 0 (x) = w 0 (|x|) be a nonnegative radial minimizer of d a . Then there exists µ 0 ∈ R such that Moreover, it follows from (33) and Remark 1 that Similar to the proof of Lemma 5.10 in [16] (see also [3, Theorem 2.2]), one can deduce from (33) that Let ϕ(x) be the cut-off function given by (25), and still denoteū δ := δ − 1 2 u δ (δ − 1 2 x) a nonnegative minimizer ofē a (δ) by Lemma 2.2. For any x 0 ∈ R 2 , we set where A δ ≥ 1 such that R 2ũ 2 δ dx ≡ 1 and A δ → 1 as δ → 0 + . Using the exponential decay of w 0 in (34), we have Similarly, one can also prove that Therefore, choosing x 0 ∈ R 2 such that V (x 0 ) = 0, we then deduce from the above estimates that This implies (31) by applying Lemma 2.2. Meanwhile, if u δ is a nonnegative minimizer of e a (δ), then We therefore obtain (32) and the proof of the lemma is finished.
Based on the above lemmas, we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. (I) Let u δ be a nonnegative minimizer of e a (δ), then there exists µ ∈ R such that u δ satisfies From (20)and (23) we have and We next prove that On the contrary, if {|∇u δ | 2 L 2 } is bounded, it then follows from (37) that {u δ } is bounded in H. Applying Lemma 2.1 of [9], there exists u 0 ∈ H such that u δ → u 0 in L p (R 2 ) for any p ∈ [2, +∞) as δ → 0 + .
Since x = 0 is the unique maximum point of Q(x), and each w δ (x) obtains its maximum point at x = 0, it then yields from (43) that This gives (9). Thus, d a ≤ F a (w δ ) + o(1)δ ≤ d a + o(1). This indicates that {w δ } is a minimizing sequence of d a . Therefore, it follows from [6, section 4] that there exists x δ ∈ R 2 such that, up to a subsequence, where w 0 is a minimizer of d a . Moreover, it follows from (32) that This together with (49) indicates that z δ := δ 1 2 x δ → x 0 as δ → 0 + , where x 0 ∈ R 2 satisfying V (x 0 ) = 0. Repeating the arguments of Part (I), one can further prove that (8) and (12).
Based on Theorem 1.1 and the assumptions (13) and (14), we now give the proof of Theorem 1.2.