High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness

1 Introduction and main results

The main purpose of this article is to study high-energy positive solutions for the following coupled Schrödinger equations:

where Ω ⊂ R 4 is an unbounded exterior domain, ∂ Ω ≠ ∅ , R 4 \ Ω is bounded, V 1 , V 2 ∈ L 2 ( Ω ) ∩ L loc ∞ ( Ω ) are non-negative functions, and μ 1 , μ 2 , and β are the positive constants.

On the one hand, the motivation to studying the coupled equations (1.1) is inspired from solitary wave solutions of the following nonlinear Schrödinger system:

where Ω ⊂ R N ( N ≤ 4 ) is a smooth domain and i is the imaginary unit. Systems like (1.2), also known as Gross-Pitaevskii equations, have a very important physical background when N ≤ 3 . For instance, in condensed matter physics, system (1.2) arises in the Hartree-Fock theory for a double condensate, i.e., a binary mixture of Bose-Einstein condensates in two different hyperfine states [23]. System (1.2) also appears in nonlinear optics, when problem (1.2) models two optical waves of different frequencies co-propagating in a medium and interacting nonlinearly through the medium, or two polarization components of a wave interacting nonlinearly at some central frequency [1].

In order to find solitary wave solutions, i.e., solutions like

system (1.2) is reduced to the following elliptic system:

We note that a solution ( u , v ) of system (1.4) is said to be nontrivial if both u ≠ 0 and v ≠ 0 ; we call a solution ( u , v ) semi-trivial if ( u , v ) is of type ( u , 0 ) with u ≠ 0 or ( 0 , v ) with v ≠ 0 ; we call a solution ( u , v ) positive if u > 0 and v > 0 . From a physical viewpoint, only nontrivial solutions have physical meaning and are worth further investigation. In [1], various analytical and numerical results for system (1.4) were investigated by authors in the case N = 1 . As far as we know, Lin and Wei [28] first studied system (1.4) with general dimension ( N ≤ 3 ). We remark that if N ≤ 3 , the nonlinearity and the coupling terms in (1.4) are of subcritical growth with respect to Sobolev critical exponent. Since then, many authors have been interested in studying existence, multiplicity, and qualitative properties of solutions for system like (1.4). There are many interesting results in the case N ≤ 3 ; we just refer the readers to [2,3,6–8,14,19,21,22,28,29,33–40,45,46,49–52] and references therein.

When N = 4 , system (1.4) is a problem with Sobolev critical exponent, which can be seen as a critically coupled perturbed Brezis-Nirenberg problem in dimension 4. Different from N ≤ 3 , the discussion on system (1.4) with N = 4 becomes quite delicate and requires some new techniques and ideas due to the lack of compactness. So, compared with the case N ≤ 3 , the study of system (1.4) with N = 4 is not frequent in the literature. We refer to Chen and Zou [13] who proved that the critical system (1.4) has a positive ground-state solution for negative β , positive small β , and positive large β , respectively. If λ 1 = λ 2 , the authors obtained the uniqueness of positive ground-state solutions. Furthermore, they also studied the limit behavior of the ground-state solutions as β → − ∞ , and phase separation is expected. To the best of our knowledge, the results obtained in [13] are the first contribution to Schrödinger systems in the critical case. Soon afterward, it turned out that results for the critical system in the higher dimensions ( N ≥ 5 , see [15]) are quite different from those obtained if N = 4 . Chen and Zou [16] investigated the existence and symmetry properties of positive ground-state solutions for coupled nonlinear Schrödinger system with critical exponent and a Hardy potential on the whole R N . Note that the positive solutions studied in [13,15,16] are ground-state solutions. Some authors are interested in higher-energy positive solutions to systems like (1.4), cf. [17,25,31,41,43,44]. Some of these articles considered positive solutions to coupled Schrödinger equations with critical exponent when the domain is bounded and has nontrivial topology [41,43,44]. The corresponding results extended the celebrated work for the Coron and Bahri-Coron problems [5,20] to coupled Schrödinger equations. For systems of two coupled equations with critical exponent in a domain Ω ⊂ R N ( N ≥ 3 ) , by proving uniqueness of ground-state solutions and non-degeneracy of a manifold of synchronized type of positive solutions when Ω = R N , Peng et al. [41] succeeded to obtain a positive solution when Ω is bounded and has nontrivial topology. For systems of m -coupled equations with critical exponent in a bounded domain Ω ε ⊂ R N ( N = 3,4 ) , having k distinct small shrinking holes as the parameter ε → 0 , by a perturbation approach based on the Lyapunov-Schmidt finite-dimensional reduction, Pistoia and Soave [43] proved the existence of positive solutions of two different types: either each density concentrates around a different hole, or they have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points. For systems of m -coupled equations with critical exponent in a bounded symmetric domain Ω ε ⊂ R 4 with a small shrinking hole B ε ( ξ 0 ) , using Lyapunov-Schmidt finite-dimensional reduction, Pistoia et al. [44] constructed non-synchronized solution that looks like a fountain of positive bubbles concentrating around the same point ξ 0 as ε → 0 . In [17,25,31], higher-energy positive solutions to coupled system are considered on whole space R N . Specifically, Clapp and Pistoia [17] obtained that a weakly coupled elliptic system has infinitely many fully nontrivial solutions, which are higher-energy solutions and invariant with respect to some subgroup of O ( N + 1 ) . By proving a global compactness result for coupled Schrödinger system, Liu and Liu [31] studied positive solutions for the following system:

where W 1 , W 2 ∈ L 2 ( R 4 ) ∩ L loc ∞ ( R 4 ) are non-negative functions and μ 1 , μ 2 , and β are the positive constants. If β > max { μ 1 , μ 2 } and ∣ W 1 ∣ L 4 ( R 4 ) + ∣ W 2 ∣ L 4 ( R 4 ) > 0 , the authors obtained the existence of positive solutions to system (1.5) when ∣ W 1 ∣ L 4 ( R 4 ) and ∣ W 2 ∣ L 4 ( R 4 ) are suitably small. In fact, the result obtained in [31] generalized the well-known result in [10] due to Benci and Cerami on semilinear Schrödinger equations to the case of coupled nonlinear Schrödinger systems. Very recently, Guo et al. [25] considered system (1.5) with W j ( x ) = λ + W j 0 ( x ) and λ > 0 , W j 0 ∈ L 2 ( R 4 ) ∩ L loc ∞ ( R 4 ) , j = 1 , 2 , and obtained the multiplicity of positive solutions to system (1.5) under W j 0 , j = 1 , 2 and λ are suitable small. We remark that some authors are also interested in sign-changing solutions for elliptic systems with critical exponent (see [12,32,42] for example).

On the other hand, the topic of system (1.1) is closely related to the following classical problems in exterior domains:

where D ⊂ R N ( N ≥ 3 ) is an unbounded domain, ∂ D ≠ ∅ is bounded, 2 < p < 2 N N − 2 . In the classical study of Benci and Cerami [9], the authors showed that problem (1.6) does not have any ground-state solution. So, they have to search for positive solutions to (1.6) at high-energy levels. To achieve their goal, the authors first analyzed the behavior of Palais-Smale sequences and proved a precise estimate of the energy levels where the Palais-Smale condition fails, then using variational method and Brouwer degree theory succeeded to obtain that the problem (1.6) has at least one positive solution for λ sufficiently small or for R N \ D small enough. We point out that there are very few results for systems in exterior domain. Yu [54] investigated the relationship between the topology of the domain and multiplicity properties of solutions to the following elliptic system:

where Ω ⊂ R N ( N ≥ 3 ) is an exterior domain with bounded boundary, λ and μ are the real parameters with 0 < μ < 1 , ∂ Ω ≠ ∅ , R N \ Ω are bounded, and 1 < p < N + 2 N − 2 . To be more precise, the author obtained that there exists μ ∗ > 0 , such that for every 0 < μ < μ ∗ , there is λ ( μ ∗ ) > 0 such that for λ > λ ( μ ∗ ) , system (1.7) has at least 3 cat Ω ¯ [ Ω ¯ , R N \ B ρ ¯ ] pairs of nontrivial solutions, where ρ ¯ = inf { ρ : R N \ Ω ⊂ B ρ ( 0 ) } > 0 and cat Ω ¯ [ Ω ¯ , R N \ B ρ ¯ ] denotes the relative category of Ω ¯ with respect to R N \ B ρ ¯ in Ω ¯ . Recently, Liu and Liu [30] studied the existence and multiplicity of positive solutions of the elliptic system

where Ω ε is an exterior domain in R N ( 1 ≤ N ≤ 3 ) such that R N \ Ω ε is far away from the origin and contains a sufficiently large ball, V j , μ j , β ∈ C ( R N ) , j = 1 , 2 all tend to positive constants at infinity. Specifically, by considering the interaction of potentials and topology of the domain, the authors proved that system (1.8) has at least three positive solutions for ε > 0 small enough. The main results extended work by Cerami and Molle [11] from single elliptic equations to the coupled nonlinear Schrödinger system. Among other critical systems on R N or bounded domain with C 2 -boundary in R N , Clapp and Szulkin [18] considered the following subcritical m -coupled system:

where Ω is an exterior domain in R N ( N ≥ 3 ) , κ i , μ i > 0 , λ i j = λ j i < 0 , α i j , β i j > 1 , α i j = β j i , and α i j + β i j = p ∈ ( 2 , 2 N N − 2 ) . When the exterior domain Ω is invariant under the action of a closed subgroup G of the group O ( N ) of linear isometries of R N , by a simple variational setting, the authors succeeded to obtain an unbounded sequence of nontrivial solutions whose components are G -invariant.

To the best of our knowledge, there seems to be no existence result for elliptic system with critical exponent in exterior domains in the literature. We remark that, very recently, by establishing global compact lemma, combining variational method with Brouwer degree, Jia et al. [27] obtained a positive solution for a class of Kirchhoff-type nonlocal problem with critical exponent and nonconstant potential function in exterior domains when the hole suitable small. Motivated by the aforementioned works, we are concerned in this article with the existence of high-energy positive solutions to coupled Schrödinger equations (1.1) with critical exponent in exterior domains.

The main result, in the case of small perturbations from infinity of the indefinite potentials, establishes the following existence property of high-energy positive solutions.

Our article is organized as follows. In Section 2, we give some preliminary results and obtain that system does not have any ground-state solution. In Section 3, we establish a global compactness result. In Section 4, we provide some estimates that are required to prove Theorem 1.1. By using some auxiliary results obtained in Section 4, we give the proof of Theorem 1.1 in Section 5 of this article. We will use C , C i , i ∈ N , to denote positive constants that can differ from line to line.

2 Preliminary results

Throughout this article, without any loss of generality, we may assume that 0 ∈ R 4 \ Ω . The norm of u in L r ( Ω ) and L r ( R 4 ) are denoted by ∣ u ∣ r and ∣ u ∣ r , R 4 , 1 ≤ r < ∞ . Define

with the inner product and the norm

Let ℋ 0 ≔ D 0 1 , 2 ( Ω ) × D 0 1 , 2 ( Ω ) with the norm

Define

with the inner product and the norm

Let ℋ 1 ≔ D 1 , 2 ( R 4 ) × D 1 , 2 ( R 4 ) with the norm

For considering the positive solution to (1.1), we study the modified system

where u ± = max { ± u , 0 } and v ± = max { ± v , 0 } such that u = u + − u − and v = v + − v − . Since V 1 , V 2 ∈ L loc ∞ ( Ω ) are the nonnegative functions and μ 1 , μ 2 , and β are the positive constants, it follows from the strong maximum principle (see Theorem 8.19 in [24]) that if ( u , v ) is a nontrivial solution of system (2.1), then u ( x ) > 0 , v ( x ) > 0 for x ∈ Ω , which shows that ( u , v ) is a positive solution of system (1.1). Therefore, to obtain a positive solution of system (1.1), we only need to prove the existence of a nontrivial solution of system (2.1).

The energy functional associated with equation (2.1) is defined by

for ( u , v ) ∈ ℋ 0 . It is easy to see that solutions to system (2.1) correspond to critical points of the functional Φ ( u , v ) . Moreover, we have

for ( u , v ) , ( φ , ψ ) ∈ ℋ 0 .

Define the Nehari manifold as

and

Furthermore, we have the following results about N .

The limit problem of system (2.1) is the following system:

of which the associated functional by

Let

To avoid semitrivial solutions in searching for nontrivial solutions of system (2.1), we need to study the following single equation:

and the associated energy functional is given by

Let

The limit equations of (2.5) are the following equations:

and associated energy functional is given by

Let

For δ > 0 , z ∈ R 4 , let

According to results in [4,48], the set { Ψ δ , z ∣ δ > 0 , z ∈ R 4 } contains all positive solutions of

Moreover, we have ‖ Ψ δ , z ‖ R 4 2 = ∣ Ψ δ , z ∣ 4 , R 4 4 = S 2 , where S is the best Sobolev constant for the embedding D 1 , 2 ( R 4 ) ↪ L 4 ( R 4 ) , i.e.,

It is well-known that c j ∞ ∗ = I j ∞ ( μ j − 1 2 Ψ δ , z ) = 1 4 μ j − 1 S 2 .