Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System

1 Introduction and Main Result

In this paper, we consider the nonlinear Schrödinger–Newton type system

where ϵ>0 and Q⁢(x) is a bounded positive continuous function on ℝ3. This system describes the interaction of a particle with its own gravitational field. In Quantum Mechanics, the behavior of a single particle of mass m>0 can be described by the linear Schrödinger equation

where ℏ is the Planck constant and Φ⁢(x):ℝ3→ℝ is the time independent potential of the particle at the position x∈ℝ3.

When there are many particles one can try to simulate the effects of the mutual interactions by introducing a nonlinear term. Then we are led to a nonlinear equation of the following form:

where 1<p<5. If we suppose that the particle moves in its own gravitational field, where the field is generated by the particles probability density via the classical Newton field equation, then the potential Φ is given (up to constants) by

i.e., Φ⁢(x) is the solution of the Poisson equation

Moreover, if we look for standing waves, namely, waves of the form

then the system that we study is given by

For simplicity of notations, taking ℏ22⁢m=ϵ2 and ω⁢ℏ=1 into (1.3), we get (1.1).

The form of system (1.3) is very similar to the following standard Schrödinger–Poisson system:

where ℏ,i,m and ψ are the same as that of (1.2), E is a real number and ϵ>0. System (1.4) appears in quantum mechanics models (see, e.g., [5, 7, 21]) and in semiconductor theory (see [4, 22, 25]). There are various results about (1.4) under various assumptions of V⁢(x),K⁢(x) and f⁢(x,ψ); one can refer to[1, 2, 4, 9, 8, 10, 11, 12, 13, 14, 16, 15, 17, 20, 23, 27, 28, 33, 37] etc. All the papers referred above used the variational methods to study (1.4). Moreover, we want to point out that in [20] and [19], solutions of (1.4) are constructed by applying the finite-dimensional reduction method under some suitable conditions of V⁢(x),K⁢(x) and f⁢(x,ψ).

When there is no nonlinear term on the right-hand side of the first equation of (1.1), i.e., the classic Schrödinger–Newton system, for the more general Schrödinger–Newton system

there are also some results, and one can refer to [18, 24, 29, 30, 32, 34, 36] etc. In [34], when K⁢(x)≡1 in (1.5), Wei and Winter show that if for some positive integer k, the points Pi∈ℝ3, i=1,2,…,k, with Pi≠Pj for i≠j, are all local minimums or local maximums or non-degenerate critical points of V⁢(x), then for ϵ small enough, there exist solutions of (1.5) with k bumps which concentrate at Pi. They also prove that, given a local maximum point P0 of V⁢(x), there exists a solution with k bumps which all concentrate at P0 and whose distances to P0 are at least O⁢(ϵ13). Very recently, Luo, Peng and the second author, using a local Pohozaev type of identity, blow-up analysis and the maximum principle, proved the uniqueness of positive solutions for (1.5) concentrating at the non-degenerate critical points of V⁢(x) for ϵ small enough.

Compared with (1.4), to the best knowledge of us, there are very few results about (1.1). In [31], Vaira proved the existence of positive ground states for (1.5) with more general nonlinearity and potentials. Moreover, in [32], Vaira proved the ground state solution of (1.1) with the potential Q⁢(x)≡constant is unique up to translations and non-degenerate.

Applying the non-degenerate result of the ground state solution of the limit equation for (1.1) in [32], motivated by [20, 19, 26], we want to apply the finite-dimensional reduction method to study the existence of positive multi-peak solutions for (1.1). Our aim here is to show that corresponding to each strict local minimum point x0 of Q⁢(x) in ℝ3 and for each positive integer k, (1.1) has a positive solution with k-peaks concentrating near x0, provided ϵ>0 is sufficiently small, that is, a solution with k-maximum points converging to x0 while vanishing as ϵ→0 everywhere else in ℝ3.

Throughout this paper, we assume that Q⁢(x) satisfies the following conditions:

1. Q has a strict local minimum at some point x0∈ℝ3, that is, for some δ>0, Q⁢(x)>Q⁢(x0) for all 0<|x-x0|<δ.

2. There are constants C,θ>0 such that |Q⁢(x)-Q⁢(z)|≤C⁢|x-z|θ for all |x-x0|≤δ,|z-x0|≤δ.

In the sequel, denote by H1⁢(ℝ3) the standard Sobolev space endowed with the standard norm ∥⋅∥ and inner product 〈⋅,⋅〉. Let |⋅|p be the usual norm of Lp⁢(ℝ3). Define D1,2⁢(ℝ3) to be the completion of C0∞⁢(ℝ3) with respect to the Dirichlet norm ∥u∥D=(∫|∇u|2dx)12. ∫f⁢(x) denotes the Lebesgue integral of f⁢(x) in ℝ3, i.e., ∫ℝ3f⁢(x)⁢𝑑x.

We look for solutions (u,Φ)∈H1⁢(ℝ3)×D1,2⁢(ℝ3). Now we reduce (1.1) to a single equation with a non-local term. Noting that u∈Lq⁢(ℝ3) for all q∈[2,6], we have u2∈L65⁢(ℝ3) for all u∈H1⁢(ℝ3) and

By Riesz theorem, there exists a unique Φu∈D1,2⁢(ℝ3) such that

It follows that Φu satisfies the Poisson equation

and

Moreover, substituting (1.7) into (1.6) and taking v=ϕu in (1.6), we have

Substituting Φu into (1.1), we obtain

that is,

Now we consider the functional Iϵ:H1⁢(ℝ3)→ℝ given by

where (u)+ denotes the positive part of u⁢(x).

Denote by w⁢(x) the unique radially positive and non-degenerate solution of the following problem (see [32]):

Noting that

it follows that Iϵ is a well-defined C1 functional. If u∈H1⁢(ℝ3) is a critical point of it, then the pair (u,Φu) is a classical solution of (1.1).

The main result of this paper is the following.

We mainly use the finite-dimensional reduction method to prove our result as in [6, 20, 19, 26, 34]. Since the nonlocal term appears, we have to overcome much difficulties in the reduction process which involves some technical and careful computations. To our knowledge, all previous results on this problem are restricted to solutions with at most one positive peak near x0, where x0 is assumed to be a non-degenerate critical point of Q⁢(x), a strict local maximum point, or some other class of “topologically nontrivial critical points”. Moreover, it is the first time to study (1.1) by applying the finite-dimensional reduction method.

We want to point out that the reduction procedure has been modified here to allow for the degeneracy of the critical point of Q⁢(x). For the classical Schrödinger equation in [26], Noussair and Yan give a example that when x0 is a local maximum point of Q⁢(x), one cannot expect, in general, to have a positive solution with more than one peak concentrating at x0.

The paper is organized as follows. In Section 2, we do the finite reduction. In Section 3, we prove our main result and give some remarks. We put all the technical estimates and some known results to Appendices A–C.

2 The Finite-Dimensional Reduction

Denote

and

The norm induced by the product 〈⋅,⋅〉ϵ is

Let

We use ∥⋅∥ to denote the usual norm in H1⁢(ℝ3), that is to say

Without loss of generality, we assume that x0=0 and Q⁢(0)=1.

Let Br={x∈ℝ3,|x|<r}, and let B¯r be its closure. For δ,R>0, and any positive integer k=1,2,…, we define

Denote

and

Set

Then, by direct computations, we have

where we have used the fact that wxj,ϵ is the unique radial positive solution (up to translations) of

In order to find a critical point for Jϵ⁢(𝐱,φ), we need to discuss each term in the expansion (2.1).

where Rϵ,𝐱(i)⁢(φ) denotes the i-th derivative of Rϵ,𝐱⁢(φ).