In this paper, we study the Schrödinger-Poisson system

in the dimension two, where ψ : ℝ 2 × ℝ → ℂ is the time-dependent wave function, a ∈ ℝ + and p > 4 . We seek for the standing wave solution of (1.1) of the form

The function u then satisfies

Solving the second equation in (1.2) via the Newtonian potential

problem (1.2) can be reduced to the single equation

The study of the three-dimensional Schrödinger-Poisson system (1.2), also known as the Schrödinger-Newton system, has a long history. With appropriate positive coupling constants, it first emerged in 1954 in the context of the quantum mechanics of a polaron at rest [1]. Subsequently it was employed in [2] to model an electron trapped in its own hole in 1976, and later in 1996, it appeared in the framework of self-gravitating matter [3]. In particular, the existence of a unique ground state of (1.2) in the three-dimensional case was established in [2] by solving a suitable minimization problem; this ground state is positive, spherically symmetric, an d strictly decreasing. Since then, the system has been extensively studied for N ≥ 3 . Variational methods have been employed to derive existence and multiplicity results for standing wave solutions [1]-[7], as well as for standing wave solutions with prescribed L 2 -norm [8]-[11].

Compared with the higher-dimensional counterparts, problem (1.3) exhibits several new features, which originate from the Newtonian potential N ( x ) . Moreover, this potential is sign-changing, and has singularities both at the origin and at infinity, which introduce additional difficulties in the analysis. One such difficulty is that the energy functional

is not well-defined on H 1 ( ℝ 2 ) as noted in [12]. To overcome this difficulty, Stubbe [12] introduced a new Hilbert space

endowed with the norm

It can be verified that the functional E p ( u ) is C 1 on the space X .

Using rearrangement theory, Stubbe [12] proved the existence and uniqueness of ground state solutions for (1.3). This work opened up a new approach to the study of two-dimensional Schrödinger-Poisson equations and has inspired many subsequent investigations, see [13]-[18] and the references therein. In particular, in [19] the authors employed a closed subgroup of the O ( 2 ) to prove that the Cerami sequence of the associated invariant functional has a convergent subsequence. For both critical and subcritical problems, various existence results have been obtained for (1.3) under different potential functions.

The existence and concentration of normalized solutions of (1.3) were investigated in [14][16][18] for mass-subcritical and mass-critical cases. In particular, for the mass-subcritical case p ∈ ( 2 , 4 ) , Cingolani and Jeanjean [14] established existence and multiplicity results for normalized solutions of (1.3).

Physically, p = 4 is the critical point separating different regimes. In particular, in [20] the authors studied Bose-Einstein condensates with attractive interactions in two dimensions, described by the Gross-Pitaevskii energy functional. They considered the case where the condensate is confined to a trapping potential V ( x ) , including the harmonic potential V ( x ) = | x | 2 . In the mass critical case p = 4 , it was shown in [20] that there exists a critical particle number a * > 0 for collapse of the Bose-Einstein condensate, even when 2 < p < 4 . They proved that the condensate survives by showing that the corresponding minimization problem is attained for 0 < a < a * . However, for the supercritical case, that is p > 4 , the minimization problem admits no minimizer for any a > 0 . Furthermore, in [21], the authors studied the evolution of the condensate as p → 4 − , and proved that the minimizers blow up in this limit.

In the present paper, we study the existence, asymptotic behavior and stability of solutions to problem (1.3) under the supercritical condition p > 4 , subject to the normalization constraint

Such solutions are called normalized solutions. The variational framework on the space X provides a fundamental tool for the study of problem (1.3). However, the analysis still requires many new ideas and delicate estimates because the norm on X is not translation-invariant, and the quadratic part of E p is not coercive.

Unlike the mass-critical exponent p = 4 , the case p > 4 is a mass-supercritical. The key difference is that the corresponding energy functional E p , when constrained on the sphere

is neither bounded above nor below, where

To overcome this difficulty, we consider the local minimization problem

where k > 0 to be determined and

We note that the space ℋ is a closed subspace of X , since the inequality

holds. Therefore, we can define a norm

on ℋ , and then the functional E p is well-defined on ℋ .

To state our results, we recall that the positive ground state solution φ p ( x ) of the scalar field equation

is unique and radially symmetric [22][23]. It satisfies the Pohozaev identity

see for instance [22][24].

Let

Theorem 1.1Suppose that 0 < a < a * .There exists a δ > 0 such that,for any p ∈ ( 4 , 4 + δ ) ,the minimization problem ℳ p , τ p 2 has a minimizer u p ∈ S ( 1 ) ∩ A τ p 2 ,where

and A τ p 2 denotes the interior of A τ p 2 .

The threshold constant a * in Theorem 1.1 is given by

where w is the ground positive solution of the equation

It is well-known (see [21]-[23]) that w is unique and radially symmetric. Moreover, | w | and | ∇ w | decay exponentially, in the sense that

By the Pohozaev identity, we have

Next, in contrast of [24], we investigate the asymptotic behavior of the minimizer u p of ℳ p k , τ p 2 as p → 4 + . The asymptotic behavior for p → 4 + reveals a universal attraction toward the critical state. Indeed, even though the system starts from the supercritical side, its profile near the critical point is captured by the solution at p = 4 .

Theorem 1.2For any sequence { p k } with p k → 4 + as k → ∞ ,there exists a corresponding subsequence of minimizers { u p k } of ℳ p k , τ p k 2 (still denoted as { u p k } )such that u p k → u 0 strongly in ℋ ,and u 0 is a global minimizer of the problem ℳ 0 : = inf u ∈ S ( 1 ) E 4 ( u ) ,that is, E 4 ( u 0 ) = ℳ 0 .

Finally, we study the stability of the minimizer u p .

We first establish the well-posedness of the time-dependent equation

with the initial data ψ 0 ∈ ℋ , that is,

Theorem1.3Suppose ψ 0 ∈ ℋ and p ∈ ( 4 , 4 + δ ) , 0 < δ < 1 .Then the Cauchy problem(1.15)-(1.16) admits a global-in-time solution ψ ( x , t ) . Moreover, ψ ( x , t ) satisfies the conservation laws of energy and mass,namely,

where the corresponding energy and mass are

and

respectively.

Based on Theorem 1.3, we have the following stability result. Let M p denote the set of local minimizers u p ∈ A τ p 2 of the minimization problem ℳ p , τ p 2 .

Theorem1.4Suppose that 0 < a < a * and p ∈ ( 4 , 4 + δ ) with δ ∈ ( 0 , 1 ) .For any u ∈ M p ,the normalized standing-wave solutions ψ ( x , t ) = e − i μ t u ( x ) of(1.3)is stable.More precisely,for every ε > 0 ,there exists η > 0 such that for any φ ∈ ℋ satisfying

we have

where Ψ is the unique solution of (1.15) with initial data φ .

The rest of this paper is organized as follows. In Section 2, we collect some preliminary facts that will be used throughout. In Section 3, we prove that ℳ p , τ p 2 is attained. The asymptotic behaviour of the minimizer and the stability result are established in Sections 4 and 5, respectively.

In this section, we collect some auxiliary facts that will be used in the sequel.

Let φ p and w be ground state solutions of (1.12) and (1.7) respectively. By Lemma 2.1 in [25], we have

and

It is known from [26] that a p * gives the best constant in the Gagliardo-Nirenberg inequality:

We now define on ℋ the following symmetric bilinear forms

and

Then we have the relation

Using the elementary inequality 0 < ln ( 1 + r ) < r for r > 0 , we obtain

Consequently, by the Gagliardo-Nirenberg inequality (2.3) and the Hardy-Littlewood-Sobolev inequality in dimension two [27]:

we have

Therefore,

Similarly,

The following results can be verified as in Lemma 2.2 of [19].

Lemma2.1The following statements hold:

1. The embedding ℋ ↪ L q ( ℝ 2 ) is compact, where q ∈ [ 2 , + ∞ ) .

2. The functionals B 0 ( u , u ) , B 1 ( u , u ) and B 2 ( u , u ) are of class C 1 on ℋ .

3. B 0 ( u , u ) is weakly lower semi-continuous on ℋ .

We also have the following compact result.

Lemma2.2Suppose V ∈ L l o c ∞ ( ℝ 2 ) and lim | x | → ∞ V ( x ) = ∞ .Then for any q ∈ [ 2 , ∞ ) ,the embedding ℋ ↪ L q ( ℝ 2 ) is compact.

In this section, we prove Theorem 1.1. The proof consists of two parts. First, we show ℳ p , k is attained by some u ∈ A k , then we prove u ∈ A k for some k > 0 .

Proof of Theorem1.1 Fix k > 0 . Let { u n } be a minimizing sequence of ℳ p , k . Since A k is a bounded closed set in ℋ , { u n } is bounded. Hence, by Lemma 2.2 we may assume that

This yields

Since u n → u in L 2 ( ℝ 2 ) , there exists v ∈ L 2 ( ℝ 2 ) such that

for any n [22]. Therefore,

and

By the Lebesgue dominated convergence theorem, we deduce that

and

Consequently,

Combining (3.2) and (3.3), we obtain

Thus u is a minimizer of ℳ p , k .

In order to prove that u is a weak solution of (1.3), it is necessary to show u ∈ A k . To this end, we shall choose k > 0 appropriately.

For any u ∈ S ( 1 ) , from (2.3) and (2.11) we deduce

Consider the function

It attains its maximum at

Let

Then

Choose θ = p − 4 4 ( p − 2 ) , then

We now show that u ∈ A τ p 2 .

Let φ ∈ C 0 ∞ ( ℝ 2 ) be such that φ ≥ 0 and ‖ φ ‖ 2 2 = 1 . Choose t p > 0 such that

satisfying

which implies φ t p ∈ A t p 2 . Thus,

This implies that t p → ∞ as p → 4 + , since τ p → + ∞ .

Set M : = sup x ∈ supp φ | x | 2 . By Lebesgue dominated convergence theorem,

as p → 4 + . Moreover, since

by (2.12) and (3.7) we obtain

In view of (3.7),

Consequently,

for p > 4 sufficiently close to 4. Hence we obtain

The proof is complete. □

In this section, we investigate the asymptotic behavior of the minimizers u p of ℳ p , τ p 2 as p → 4 + .

By Proposition 2.3 in [19], critical points u ∈ ℋ of E p on the L 2 -sphere are strong solutions of (1.3) in W 2 , q ( ℝ 2 ) for all q ≥ 1 , and, in the present setting, they are classical solutions in C 2 ( ℝ 2 ) . Moreover, it is shown in the same proposition that for every α > 0 ,

Consequently, u satisfies the Pohozaev identity:

Lemma4.1Any sequence of minimizers { u p } of ℳ p , τ p 2 is uniformlybounded in ℋ for p > 4 sufficiently close to4.

Proof. Let φ ∈ C c ∞ ( ℝ 2 ) be such that

Since τ p 2 given in (1.10) tends to infinity as p → 4 + , we have φ ∈ A τ p 2 for p > 4 sufficiently close to 4. Therefore,

By the Gagliardo-Nirenberg inequality (2.3),

which implies, since a p * → a * as p → 4 + , that

for some constant C > 0 independent of p .

Moreover, the estimate

combined with (4.4) yields

with C > 0 independent of p .

We now claim that

with C > 0 independent of p . Indeed, by the Pohozaev identity (4.1) we deduce, for p > 4 close to 4, that

Note that B 1 ( u , u ) ≥ 0 . From (2.7) and (2.11), we have

From (4.2) and (4.6), we have

Therefore,

which proves the claim (4.7). Combining (4.7) with the Pohozaev identity (4.1), we obtain

Hence,

The proof of Lemma 4.1 is complete. □

Proof of Theorem1.4. Let { u p } be a sequence of minimizers of ℳ p , τ p 2 . By Lemma 4.1, ‖ u p ‖ ℋ ≤ C . In view of Lemma 2.2, there exists a sequence p k , p k → 4 + as k → ∞ such that the corresponding sequence of minimizers { u p k } of ℳ p k , τ p k 2 satisfies

for q ∈ [ 2 , + ∞ ) as k → ∞ . The Lagrange multiplier λ k satisfies

which implies that { λ k } is bounded. Thus, up to a subsequence, we may assume λ k → λ 0 .

Furthermore, for any φ ∈ C 0 ∞ ( ℝ 2 ) , we write

In view of the weak convergence,

The the mean value theorem and the L p interpolation inequality imply