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On coupled systems of nonlinear Schrödinger and Choquard equations with distinct exponents

  • Dohoon Choi , Subong Lim and Jinmyoung Seok EMAIL logo
Published/Copyright: September 23, 2025
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Advanced Nonlinear Studies
From the journal Advanced Nonlinear Studies

Abstract

In this paper, we are interested in the existence of a positive solution of the two coupled system of nonlinear Schrödinger and Choquard equations. Our equations admit the case that the nonlinearity exponents of two components are different. This includes, in particular, the upper and lower critical cases. We first investigate the existence of ground states and provide a criterion for two-component positiveness of ground states. In case where the ground state is semi-trivial, we construct a higher energy positive solution of the mountain-pass type on the Nehari manifold.

Keywords: nonlinear Schrödinger; Choquard; ground state; variational method
2020 Mathematics Subject Classification: 35J20; 35J50; 35J61

1 Introduction

Let N ≥ 3, 0 ≤ α < N and N + α N ≤ p ≤ q ≤ N + α N − 2 . We are concerned with the system of equations

(1.1) − Δ u + u = I α ∗ 1 p | u | p + 1 q | v | q | u | p − 2 u , − Δ v + v = I α ∗ 1 p | u | p + 1 q | v | q | v | q − 2 v , lim | x | → ∞ u ( x ) = 0 , lim | x | → ∞ v ( x ) = 0 ,   in  R N .

Here, I α is Riesz potential given by

I α ( x ) : = Γ N − α 2 Γ α 2 π N / 2 2 α | x | N − α

and Γ denotes the Gamma function. For the case α = 0, we regard I 0 as δ 0 so that (1.1) is written as the nonlinear Schrödinger system

(1.2) − Δ u + u = 1 p | u | p + 1 q | v | q | u | p − 2 u , − Δ v + v = 1 p | u | p + 1 q | v | q | v | q − 2 v , lim | x | → ∞ u ( x ) = 0 , lim | x | → ∞ v ( x ) = 0 ,   in  R N

but in this case, we exclude the lower critical p = N + α N = 1 since the energy functional J 0 is not continuously differentiable due to the term ∫|u‖v| q  dx. In this paper, we regard the system (1.2) as just a special case of (1.1) with α = 0.

The system (1.2) appears in considering a solitary wave solution Φ(x, t) = e it u(x), Ψ(x, t) = e it v(x) of the coupled Schrödinger system

− i ∂ ∂ t Φ   = Δ Φ + 1 p | Φ | p + 1 q | Ψ | q | Φ | p − 2 Φ − i ∂ ∂ t Ψ   = Δ Ψ + 1 p | Φ | p + 1 q | Ψ | q | Ψ | q − 2 Ψ   in  R N ,

which arises from many physical models such as incoherent wave packets in Kerr medium in nonlinear optics [1], [2], [3] and in Bose Einstein condensates for multi-species condensates [4], [5].

Since the groundbreaking work by Maia-Montefusco-Pellacci [6], Ambrosetti-Colorado [7] and Sirakov [8], much attention is paid to the study of the ground and bound states of (1.2) for most standard case p = q = 2. For examples, we refer to refs. [9], [10], [11] and references therein. The identical exponent case p = q is considered and well studied in refs. [12], [13], [14], [15], [6]. However, to the best of the authors’ knowledge, there has been no literature dealing with the distinct exponent case p < q, on which the main interest of this paper lies.

On the other hand, by setting v = 0, the system (1.1) reduced to the so-called Choquard equation

(1.3) − Δ u + u = I α ∗ 1 p | u | p | u | p − 2 u , lim | x | → ∞ u ( x ) = 0 ,   in  R N ,

whose ground states are shown to exist for the range N + α N < p < N + α N − 2 in ref. [16]. When N = 3, α = 2 and p = 2, a solution of equation (1.3) provides a standing wave of the nonlinear Schrödinger-Newton equation, which describes through Hartree–Fock approximation, a dynamics of condensed states to a system of non-relativistic boson particles with two-body attractive Newtonian potential. For related works on the existence and properties of solutions of Choquard or Schrödinger-Newton equations, we refer to refs. [17], [18], [19], [20], [21], [22], [23], [16], [24], [25], [26].

In this paper, we investigate the existence of positive solutions of (1.1). The system (1.1) is the Euler–Lagrange equations of the functional

J α ( u , v ) : = 1 2 ∫ R N | ∇ u | 2 + | ∇ v | 2 + u 2 + v 2 d x − 1 2 ∫ R N I α ∗ 1 p | u | p + 1 q | v | q 1 p | u | p + 1 q | v | q d x

defined on the standard Sobolev space H : = H 1 ( R N ) × H 1 ( R N ) . By a solution of (1.1), we mean a critical point of J α in H, which actually coincides with a weak solution of (1.1). Since we are interested in finding a radial solution, we introduce the space H r 1 ( R N ) , the set of radial functions in H 1 ( R N ) . The space H r 1 ( R N ) × H r 1 ( R N ) is denoted by H r .

We say a solution (u 0, v 0) ∈ H r of (1.1) is a ground state in radial class if (u 0, v 0) ∈ H r is a nontrivial solution of (1.1) and J α (u 0, v 0) = J min, where

(1.4) J min : = inf J α ( u , v ) | ( u , v ) ∈ H r \ { ( 0,0 ) } , J α ′ ( u , v ) = 0 .

Using the rearrangement technique [27], it is not hard to see that a ground state in radial class is actually a ground state, namely, a minimizer of J α among every nontrivial critical point of J α in H. Thus, we may identify these two concepts of ground state and refer to a ground state in radial class just as a ground state throughout the paper.

Before stating our main results, we define some terminologies.

Definition 1.1.

A vector function (u, v) is said to be

  1. nontrivial if either u ≠ 0 or v ≠ 0;

  2. semi-trivial if it is nontrivial but either u = 0 or v = 0;

  3. vectorial if both of u and v are not zero;

  4. nonnegative if u ≥ 0 and v ≥ 0;

  5. positive if u > 0 and v > 0.

It is worth mentioning that if (u, v) is a nonnegative solution of (1.1) then each component of (u, v) is either strictly positive or identically zero. This comes from the fact that the solution (u, v) is given by

u = K ∗ 1 p | u | p + 1 q | v | q | u | p − 2 u , v = K ∗ 1 p | u | p + 1 q | v | q | v | q − 2 v ,

where K denotes the kernel of the inverse operator (−Δ + I)−1, which is strictly positive.

Our first concern is the existence and vectorial property of a ground state of (1.1). The following theorem states the existence of a nonnegative ground state.

Theorem 1.2

(Existence of ground states). There exists a nonnegative ground state (u 0, v 0) of (1.1) in H r if one of the following conditions holds.

  1. (Subcritical case) N ≥ 3, 0 ≤ α < N and N + α N < p ≤ q < N + α N − 2 ;

  2. (Upper critical case) N ≥ 5, 0 ≤ α < N and N + α N < p < q = N + α N − 2 ;

  3. (Lower and doubly critical case) N ≥ 5, 0 < α < N − 4 and N + α N = p < q ≤ N + α N − 2 ;

For the case α = 0 (nonlinear Schrödinger system) with p = q, the structure of a ground state of (1.1) is fully understood.

Theorem 1.3

(Liu-Wang [14], [15]). Suppose that 1 < p = q < N N − 2 . Let (u 0, v 0) be a nonnegative radial ground state of (1.1) with α = 0 and w be a positive radial ground state solution of the equation −Δw + w = |w|2p−2 w.

  1. If p < 2, then (u 0, v 0) = ((p/2)1/(2p−2) w, (p/2)1/(2p−2) w);

  2. If p > 2, then (u 0, v 0) is either (p 1/(2p−2) w, 0) or (0, p 1/(2p−2) w);

  3. If p = 2, then (u 0, v 0) is of the form ( 2 − t w , t w ) , 0 ≤ t ≤ 2 .

The proof of Theorem 1.3 is largely based on the work by Correia [12], which asserts that (u 0, v 0) = (a 1 w, a 2 w) for some constants a 1 and a 2. Applying the same argument, one has exactly the same result for the case α ≠ 0 as the following theorem states.

Theorem 1.4.

Suppose that N ≥ 3, 0 < α < N and N + α N < p = q < N + α N − 2 . Let (u 0, v 0) be a nonnegative ground state of (1.1) and w be a positive radial ground state solution of the equation − Δ w + w = I α ∗ | w | p | w | p − 2 w . Then there holds the same statements with (i)–(iii) of Theorem 1.3.

Our next result shows that the vectorial or semi-trivial property of a ground state is still generalized to the case p ≠ q although the argument by Correia does not apply for this case.

Theorem 1.5

(Classification of ground states). Let N ≥ 3, 0 ≤ α < N.

  1. If N + α N ≤ p < q < 2 and (α, p) ≠ (0, 1), then any ground state is vectorial, provided it exists.

  2. If 2 ≤ p < q ≤ N + α N − 2 , then any ground state is semi-trivial, provided it exists.

Remark 1.6.

In particular, Theorem 1.5 says a ground state of (1.1) is positive if N + α N − 2 < 2 , which is equivalent to N ≥ 5 and 0 ≤ α < N − 4. Thus the ground state constructed in Theorem 1.2.(iii) is always positive.

The final result of this paper is to construct a positive vectorial solution when 2 < p < q < N + α N − 2 in which case, the ground state becomes semi-trivial. We however need an additional nondegeneracy condition denoted by (ND). We say a tuple (α, s) satisfies (ND) if there exists a unique positive radial solution u 0 ∈ H r 1 ( R N ) of the equation

(1.5) − Δ u + u = I α ∗ | u | s s | u | s − 2 u

such that the linearized equation of (1.5) at u 0

(1.6) − Δ ϕ + ϕ = I α ∗ u 0 s − 1 ϕ u 0 s − 1 + I α ∗ u 0 s s ( s − 1 ) u 0 s − 2 ϕ

admits only the trivial solution in H r 1 ( R N ) .

Theorem 1.7.

Let N ≥ 3, 0 ≤ α < N and 2 < p < q < N + α N − 2 . Suppose (ND) holds true for the tuples (α, p) and (α, q). Then there exists a positive solution of (1.1).

Remark 1.8.

In case 2 < p = q < N + α N − 2 , a positive solution of (1.1) simply comes from identifying u and v.

It is not completely known whether (ND) holds true for any tuple (α, s) in the whole range α ∈ [0, N) and N + α N < s < N + α N − 2 . The following proposition illustrates some examples of (α, s) for which (ND) is proven to hold.

Proposition 1.9.

The value (α, s) satisfying (ND) includes the following cases:

  1. [28] N ≥ 3, α = 0 and 1 < s < N N − 2 (Nonlinear Schrödinger case);

  2. [29] N = 3, 1 < s < N N − 2 and α sufficiently close to 0;

  3. [29] N ≥ 3, 2 < s < 2 N N − 2 and α sufficiently close to N;

  4. [30], [31] N = 3, α = s = 2 (Schrödinger-Newton case);

  5. [32] N = 3, α = 2 and s > 2 sufficiently close to 2.

The paper is organized as follows. In Section 2, we prepare some basic inequalities and convergence results useful for our analysis. We prove Theorem 1.2 in Section 3, Theorems 1.4 and 1.5 in Section 4, and Theorem 1.7 in Section 5.

2 Preliminaries

In this section, we collect some basic inequalities and convergence results for proofs of the main theorems. The first one is the well-known HLS (Hardy-Littlewood-Sobolev) inequality.

Proposition 2.1

(HLS inequality [33], [27]). Let p, r > 1 and α ∈ (0, N) be such that

1 p + 1 r = 1 + α N .

Then, there exists C(N, α, p) > 0 depending only on N, α, p such that for any f ∈ L p ( R N ) and g ∈ L r ( R N )

∫ R N ∫ R N f ( x ) g ( x ) | x − y | N − α d x d y ≤ C ( N , α , p ) ‖ f ‖ L p ( R N ) ‖ g ‖ L r ( R N ) .

In particular, the HLS inequality implies that

(2.1) S 1 ∫ R N I α ∗ | u | N + α N | u | N + α N d x N N + α ≤ ∫ R N u 2 d x ,

(2.2) S 2 ∫ R N I α ∗ | u | N + α N − 2 | u | N + α N − 2 d x N − 2 N + α ≤ ∫ R N | ∇ u | 2 d x ,

where S 1 and S 2 denote the best constants depending only on N and α.

The HLS inequality is the dual form of the Riesz potential estimate below [33], [27].

Proposition 2.2

(Riesz potential estimate). Let 1 ≤ r < s < ∞ and 0 < α < N be such that

1 r − 1 s = α N .

Then, for any f ∈ L r ( R N ) one has

1 | ⋅ | N − α ∗ f L s ( R N ) ≤ K ( N , α , r ) ‖ f ‖ L r ( R N ) .

By the Riesz potential estimate, we see that for any f ∈ L 2 N N + α ( R N ) ,

∫ R N ( I α ∗ f ) f d x < ∞ .

The following proposition says ∫ R N ( I α ∗ f ) g d x < ∞ whenever f , g ∈ L 2 N N + α ( R N ) .

Proposition 2.3.

Let f , g ∈ L 2 N N + α ( R N ) . Then for any α ∈ (0, N), there holds

∫ R N ( I α ∗ f ) g d x ≤ ∫ R N ( I α ∗ | f | ) | f | d x ∫ R N ( I α ∗ | g | ) | g | d x .

Moreover, if the equality holds, then |f| and |g| are co-linear, i.e., there exists a constant λ ∈ R such that |f| = λ|g|.

Proof.

Recalling the fact that the Riesz potential I α is the kernel of ( − Δ ) − α 2 , integrating by parts and the Hölder inequality give

∫ R N ( I α ∗ f ) g d x ≤ ∫ R N ( I α ∗ | f | ) | g | d x = ∫ R N I α 2 ∗ | f | I α 2 ∗ | g | d x ≤ ∫ R N I α 2 ∗ | f | 2 d x ∫ R N I α 2 ∗ | g | 2 d x = ∫ R N ( I α ∗ | f | ) | f | d x ∫ R N ( I α ∗ | g | ) | g | d x .

If the equality holds, we see from the Hölder inequality that there exists λ ∈ R such that I α/2∗|f| = λI α/2∗|g|. Denote h := |f| − λ|g|. Then for every ϕ ∈ C c ∞ ( R N ) ,

0 = ∫ R N I α / 2 ∗ h ( − Δ ) α / 4 ϕ d x = ∫ R N h ( I α / 2 ∗ ( − Δ ) α / 4 ϕ ) d x = ∫ R N h ϕ d x .

This shows h = 0. □

The following convergences are proved in the previous works [29], [34].

Proposition 2.4.

Let N + α N ≤ p ≤ q ≤ N + α N − 2 and α ∈ (0, N). Suppose that { u j } , { v j } ⊂ H r 1 ( R N ) are sequences converging weakly to some u 0 , v 0 ∈ H r 1 ( R N ) , respectively, in H 1 ( R N ) as j → ∞.

  1. Unless p = q = N + α N or p = q = N + α N − 2 ,

    ∫ R N 1 | ⋅ | N − α ∗ | u j | p | v j | q d x → ∫ R N 1 | ⋅ | N − α ∗ | u 0 | p | v 0 | q d x .

  2. For any ϕ ∈ H 1 ( R N ) ,

    ∫ R N 1 | ⋅ | N − α ∗ | u j | p | v j | q − 2 v j ϕ d x → ∫ R N 1 | ⋅ | N − α ∗ | u 0 | p | v 0 | q − 2 v 0 ϕ d x .

The Brezis-Lieb lemma for the Riesz potential plays crucial role for our analysis. We refer to ref. [16] for the proof.

Proposition 2.5.

Let α ∈ (0, N) and N + α N ≤ p ≤ N + α N − 2 . If {u j } is a bounded sequence in L 2 N p N + α ( R N ) such that u j → u almost everywhere as j → ∞ for some function u, then u ∈ L 2 N p N + α ( R N ) and

lim j → ∞ ∫ R N I α ∗ | u j | p | u j | p d x − ∫ R N I α ∗ | u j − u | p | u j − u | p d x = ∫ R N I α ∗ | u | p | u | p d x .

3 Existence of ground states

3.1 Variational setup and subcritical case

We denote the standard H 1 norm by ‖⋅‖, i.e.,

‖ u ‖ 2 = ∫ R N | ∇ u | 2 + u 2 d x .

Then the norm on H is given by

‖ ( u , v ) ‖ H ≔ ‖ u ‖ 2 + ‖ v ‖ 2 for  ( u , v ) ∈ H .

We say a sequence {(u j , v j )} ⊂ H r is a (PS) sequence of J α | H r at level c if

J α ′ ( u j , v j ) → 0    in  H r   and   J α ( u j , v j ) → c   as  j → ∞ .

We define the mountain pass energy level c 0 by

c 0 ≔ inf γ ∈ Γ max t ∈ [ 0,1 ] J α γ ( t ) ,

where Γ ≔ { γ ∈ C ( [ 0,1 ] , H r ) | γ ( 0 ) = 0 , J α γ ( 1 ) < 0 } and another two levels

c 1 ≔ inf ( u , v ) ∈ H r ( u , v ) ≠ ( 0,0 ) max t ≥ 0 J α ( t u , t v ) , c 2 ≔ inf ( u , v ) ∈ N J α ( u , v ) ,

where N denotes the so-called Nehari manifold given by

N ≔ ( u , v ) ∈ H r | J α ′ ( u , v ) ( u , v ) = 0 , ( u , v ) ≠ ( 0,0 ) .

We also recall that J min denotes the ground energy level given in (1.4).

It is standard to prove the following two propositions. See, for examples, [35], [36].

Proposition 3.1.

Let α ∈ [0, N), N + α N ≤ p ≤ q ≤ N + α N − 2 and (α, p) ≠ (0, 1).

  1. There exists a (PS) sequence of J α | H r at level c 0.

  2. There holds 0 < c 0 = c 1 = c 2.

  3. If c 0 is a critical value of J α | H r , then c 0 is a critical value of J α and c 0 = J min.

  4. If J min is a critical value of J α | H r , then J min is a critical value of J α and c 0 = J min.

Proposition 3.2.

If (u 0, v 0) is a ground state, then (|u 0|, |v 0|) is also a ground state.

By Propositions 3.1 and 3.2, it suffices to show the relative compactness of a (PS) sequence of J α | H r at level c 0 for proving Theorem 1.2. For subcritical case N + α N < p ≤ q < N + α N − 2 , this compactness is easily obtained.

Proposition 3.3.

Let N + α N < p ≤ q < N + α N − 2 and α ∈ [0, N). Assume that {(u j , v j )} ⊂ H r is a (PS) sequence of J α | H r at level c 0. Then, it is relatively compact in H r .

Proof.

We only prove the proposition for the case α ≠ 0. The simpler case α = 0 can be covered by the same argument. We first show that {(u j , v j )} is bounded in H. Since {(u j , v j )} is a (PS) sequence at level c, we have

c + o ( 1 ) = J α ( u j , v j ) = 1 2 ‖ ( u j , v j ) ‖ H 2 − 1 2 ∫ R N I α ∗ 1 p | u j | p + 1 q | v j | q 1 p | u j | p + 1 q | v j | q d x

and

o ( 1 ) ‖ ( u j , v j ) ‖ H = J α ′ ( u j , v j ) ( u j , v j ) = ‖ ( u j , v j ) ‖ H 2 − ∫ R N I α ∗ 1 p | u j | p + 1 q | v j | q | u j | p + | v j | q d x .

Then, we have

1 2 ‖ ( u j , v j ) ‖ H 2 ≤ c + o ( 1 ) + 1 2 p ∫ R N I α ∗ 1 p | u j | p + 1 q | v j | q | u j | p + | v j | q d x = c + o ( 1 ) + 1 2 p ( ‖ ( u j , v j ) ‖ H 2 + o ( 1 ) ‖ ( u j , v j ) ‖ H .

Since p > 1, this shows that ‖(u j , v j )‖ H is bounded. Therefore, up to a subsequence, {(u j , v j )} weakly converges to some (u 0, v 0) ∈ H r .

By (ii) of Proposition 2.4, (u 0, v 0) is a weak solution of (1.1). From this, we have

0 = J α ′ ( u 0 , v 0 ) ( u 0 , v 0 ) = ‖ ( u 0 , v 0 ) ‖ H 2 − ∫ R N I α ∗ 1 p | u 0 | p + 1 q | v 0 | q | u 0 | p + | v 0 | q d x .

Define (w j , t j ) ≔ (u j − u 0, v j − v 0). Then, we have

(3.1) ‖ ( w j , t j ) ‖ H 2 = ‖ ( u j , v j ) ‖ H 2 − ‖ ( u 0 , v 0 ) ‖ H 2 + o ( 1 ) = ∫ R N I α ∗ 1 p | u j | p + 1 q | v j | q | u j | p + | v j | q d x + o ( 1 ) ‖ ( u j , v j ) ‖ H − ∫ R N I α ∗ 1 p | u 0 | p + 1 q | v 0 | q | u 0 | p + | v 0 | q d x = o ( 1 ) .

We used the fact that (u 0, v 0) is a weak solution of (1.1) in the second equality, and the last equality follows from (i) of Proposition 2.4. □

We note that Theorem 1.2.(i) is immediately deduced from Proposition 3.3.

3.2 Critical case

The aim of this section is to prove the existence of a nonnegative ground state of (1.1) when

(3.2) N ≥ 5  and  0 < α < N − 4   if  p = N + α N < q ≤ N + α N − 2   , N ≥ 5  and  0 ≤ α < N   if  N + α N < p < q = N + α N − 2 .

The following proposition characterizes the energy levels at which a (PS) sequence of J α | H r converges.

Proposition 3.4.

Let N + α N ≤ p ≤ q ≤ N + α N − 2 . Assume that {(u j , v j )} ⊂ H r is a (PS) sequence of J α | H r at level c. Then, it is relatively compact in H r when

c < min 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1 , 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1   if  p = N + α N , q = N + α N − 2 ,  and  α ∈ ( 0 , N − 4 ) , c < 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1   if  p = N + α N < q < N + α N − 2  and  α ∈ ( 0 , N − 4 ) , c < 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1   if  N + α N < p < q = N + α N − 2  and  α ∈ [ 0 , N ) .

Proof.

We only prove the case where p = N + α N , q = N + α N − 2 , and α ∈ (0, N − 4). Other cases can be proved more easily by following the same argument. Suppose that p = N + α N and q = N + α N − 2 . By the same argument in the proof of Proposition 3.3, ‖(u j , v j )‖ H is bounded. Therefore, up to a subsequence, {(u j , v j )} weakly converges to some (u 0, v 0) ∈ H r .

Since (u 0, v 0) is a solution of (1.1), we have

J α ( u 0 , v 0 ) = 1 2 ‖ ( u 0 , v 0 ) ‖ H 2 − 1 2 ∫ R N I α ∗ 1 p | u 0 | p + 1 q | v 0 | q 1 p | u 0 | p + 1 q | v 0 | q d x ≥ 1 2 ‖ ( u 0 , v 0 ) ‖ H 2 − 1 2 p ∫ R N I α ∗ 1 p | u 0 | p + 1 q | v 0 | q | u 0 | p + | v 0 | q d x = 1 2 − 1 2 p ‖ ( u 0 , v 0 ) ‖ H 2 .

Since p ≥ 1, this shows that J α (u 0, v 0) ≥ 0.

Define (w j , t j ) ≔ (u j − u 0, v j − v 0). Then, we have

(3.3) ‖ ( w j , t j ) ‖ H 2 = ‖ ( u j , v j ) ‖ H 2 − ‖ ( u 0 , v 0 ) ‖ H 2 + o ( 1 ) = ∫ R N I α ∗ 1 p | u j | p + 1 q | v j | q | u j | p + | v j | q d x + o ( 1 ) ‖ ( u j , v j ) ‖ H − ∫ R N I α ∗ 1 p | u 0 | p + 1 q | v 0 | q | u 0 | p + | v 0 | q d x = 1 p ∫ R N I α ∗ | w j | p | w j | p d x + 1 q ∫ R N I α ∗ | t j | q | t j | q d x + o ( 1 ) .

The last equality follows from (1) of Propositions 2.4 and 2.5. If we combine (2.1) and (2.2) with (3.3), then we have

(3.4) S 1 ∫ R N I α ∗ | w j | p | w j | p d x 1 p + S 2 ∫ R N I α ∗ | t j | q | t j | q d x 1 q ≤ ∫ R N | w j | 2 d x + ∫ R N | ∇ t j | 2 d x ≤ 1 p ∫ R N I α ∗ | w j | p | w j | p d x + 1 q ∫ R N I α ∗ | t j | q | t j | q d x + o ( 1 ) .

We define

x ≔ lim sup j → ∞ ∫ R N I α ∗ | w j | p | w j | p d x , y ≔ lim sup j → ∞ ∫ R N I α ∗ | t j | q | t j | q d x .

They are finite since ‖(w j , t j )‖ H is bounded. If we pass to a limit j → ∞ in (3.4), then we have

S 1 x 1 p + S 2 y 1 q ≤ 1 p x + 1 q y .

We claim that x = y = 0. If it is shown, then the equality (3.3) says that ‖(w j , t j )‖ H → 0 as j → ∞ up to a subsequence. Therefore, (u j , v j ) → (u 0, v 0) in H r as j → ∞ up to a subsequence.

Now, we show the claim holds. Suppose that x ≠ 0 or y ≠ 0. Then, either x ≥ ( p S 1 ) N + α α or y ≥ ( q S 2 ) N + α 2 + α . We compute

(3.5) J α ( u j , v j ) = 1 2 ‖ ( u j , v j ) ‖ H 2 − 1 2 ∫ R N I α ∗ 1 p | u j | p + 1 q | v j | q 1 p | u j | p + 1 q | v j | q d x = 1 2 ‖ ( u 0 , v 0 ) ‖ H 2 + 1 2 ‖ ( w j , t j ) ‖ H 2 − ∫ R N I α ∗ 1 p | u 0 | p 1 q | v 0 | q d x − 1 2 ∫ R N I α ∗ 1 p | u 0 | p 1 p | u 0 | p d x − 1 2 ∫ R N I α ∗ 1 p | w j | p 1 p | w j | p d x − 1 2 ∫ R N I α ∗ 1 q | v 0 | q 1 q | v 0 | q d x − 1 2 ∫ R N I α ∗ 1 q | t j | q 1 q | t j | q d x + o ( 1 ) = J α ( u 0 , v 0 ) + 1 2 ‖ ( w j , t j ) ‖ H 2 − 1 2 p 2 ∫ R N I α ∗ | w j | p | w j | p d x − 1 2 q 2 ∫ R N I α ∗ | t j | q | t j | q d x + o ( 1 ) = J α ( u 0 , v 0 ) + 1 2 p − 1 2 p 2 ∫ R N I α ∗ | w j | p | w j | p d x + 1 2 q − 1 2 q 2 ∫ R N I α ∗ | t j | q | t j | q d x + o ( 1 ) .

We used (1) of Propositions 2.4 and 2.5 in the second equality, and the last equality followed from the equality (3.3). Since J α (u 0, v 0) ≥ 0, if we pass to a limit j → ∞ in (3.5), then we see that either c ≥ 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1 or c ≥ 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1 . But this contradicts with the assumption of the proposition. So, the claim is proved. □

Completion of proof of Theorem 1.2.

By Proposition 3.1, there is a (PS) sequence { ( u j , v j ) } ⊂ H r 1 of J α | H r 1 at level c 0. We claim that c 0 satisfies

c 0 < min 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1 , 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1   if  p = N + α N , q = N + α N − 2 ,  and  α ∈ ( 0 , N − 4 ) , c 0 < 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1   if  p = N + α N < q < N + α N − 2  and  α ∈ ( 0 , N − 4 ) , c 0 < 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1   if  N + α N < p < q = N + α N − 2  and  α ∈ [ 0 , N ) .

If this is shown, we get a ground state by Proposition 3.1.

Now, we show the claim holds. Suppose that p = N + α N and α ∈ (0, N − 4). For a positive number λ > 0, we define

μ λ ( x ) ≔ A λ N 2 ( λ 2 + | x | 2 ) N 2 .

If N ≥ 3, one has μ λ ∈ H r 1 ( R N ) for every λ and they are the extremal functions of the inequality (2.1). The constant A is chosen to satisfy

∫ R N | μ 1 | 2 d x = ∫ R N I α ∗ | μ 1 | p | μ 1 | p d x .

This implies that ∫ R N | μ 1 | 2 d x = S 1 p p − 1 . Let β be a fixed number and ψ 1 ∈ H r 1 ( R N ) . For a positive number λ > 0, we define

ψ λ ( x ) ≔ λ β ψ 1 x λ .

Then, ψ λ ∈ H r 1 ( R N ) for every λ > 0.

For a fixed positive number λ, let t λ > 0 be such that

J α t λ μ λ , ψ λ = max t > 0 J α t μ λ , ψ λ .

Let t ̃ λ be the number that satisfies J α t ̃ λ μ λ , ψ λ < 0 . We have seen that t ̃ λ > t λ . We define γ 1 ( t ) ≔ t t ̃ λ ( μ λ , ψ λ ) . Then, we see that

c 0 ≤ max t ∈ [ 0,1 ] J α γ 1 ( t ) = J α t λ μ λ , ψ λ .

We compute

(3.6) 0 = d d t t = t λ J α t μ λ , ψ λ = t λ ∫ R N | ∇ μ λ | 2 d x + t λ ∫ R N | μ λ | 2 d x + t λ ∫ R N | ∇ ψ λ | 2 d x + t λ ∫ R N | ψ λ | 2 d x − t λ 2 p − 1 p ∫ R N I α ∗ | μ λ | p | μ λ | p d x − t λ 2 q − 1 q ∫ R N I α ∗ | ψ λ | q | ψ λ | q d x − ( p + q ) t λ p + q − 1 p q ∫ R N I α ∗ | μ λ | p | ψ λ | q d x = t λ λ − 2 ∫ R N | ∇ μ 1 | 2 d x + t λ ∫ R N | μ 1 | 2 d x + t λ λ 2 β − 2 + N ∫ R N | ∇ ψ 1 | 2 d x + t λ λ 2 β + N ∫ R N | ψ 1 | 2 d x − t λ 2 p − 1 p ∫ R N I α ∗ | μ 1 | p | μ 1 | p d x − t λ 2 q − 1 λ 2 β q + N + α q ∫ R N I α ∗ | ψ 1 | q | ψ 1 | q d x − t λ p + q − 1 λ − N 2 p + β q + N + α p q ∫ R N I α ∗ | μ 1 | p | ψ 1 | q d x .

We define t ∞ ≔  lim sup λ→∞ t λ . Then, if β satisfies

(3.7) β < − N 2 ,

then we can see that t ∞ < ∞ since 2β − 2 + N, 2β + N < 0.

If we assume (3.7) and pass to a limit λ → ∞ in (3.6), then we obtain

0 = t ∞ ∫ R N | μ 1 | 2 d x − t ∞ 2 p − 1 p ∫ R N I α ∗ | μ 1 | p | μ 1 | p d x = t ∞ − t ∞ 2 p − 1 p ∫ R N | μ 1 | 2 d x

since 2 β q + N + α , − N 2 p + β q + N + α < 0 . This implies that t ∞ = p 1 2 p − 2 .

We compute

J α t λ μ λ , ψ λ = t λ 2 λ − 2 2 ∫ R N | ∇ μ 1 | 2 d x + t λ 2 2 ∫ R N | μ 1 | 2 d x + t λ 2 λ 2 β + N − 2 2 ∫ R N | ∇ ψ 1 | 2 d x + t λ 2 λ 2 β + N 2 ∫ R N | ψ 1 | 2 d x − t λ 2 p 2 p 2 ∫ R N I α ∗ | μ 1 | p | μ 1 | p d x − t λ 2 q λ 2 β q + N + α 2 q 2 ∫ R N I α ∗ | ψ 1 | q | ψ 1 | q d x − t λ p + q λ − N 2 p + β q + N + α p q ∫ R N I α ∗ | μ 1 | p | ψ 1 | q d x ≤ t λ 2 2 − t λ 2 p 2 p 2 ∫ R N | μ 1 | 2 d x + t λ 2 λ − 2 2 ∫ R N | ∇ μ 1 | 2 d x + t λ 2 λ 2 β + N − 2 2 ∫ R N | ∇ ψ 1 | 2 d x + t λ 2 λ 2 β + N 2 ∫ R N | ψ 1 | 2 d x − t λ p + q λ − N 2 p + β q + N + α p q ∫ R N I α ∗ | μ 1 | p | ψ 1 | q d x .

Note that the function g ( t ) ≔ t 2 2 − t 2 p 2 p 2 attains its maximum at t = t ∞. This shows that

t λ 2 2 − t λ 2 p 2 p 2 ∫ R N | μ 1 | 2 d x ≤ 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1 .

Since 4 + α < N, if β satisfies

(3.8) β > − ( N + α + 4 ) ( N − 2 ) 2 ( N + α ) ,

(3.9) β < − ( N − 2 ) ( N − α ) 2 ( N − α − 4 ) ,

then, we have

− 2 < − N 2 p + β q + N + α , 2 β + N < − N 2 p + β q + N + α .

Therefore, if β satisfies (3.7)– (3.9), then for sufficiently large λ > 0, we have

J α ( t λ ( μ λ , ψ λ ) ) < 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1 .

Since we have

− ( N + α + 4 ) ( N − 2 ) 2 ( N + α ) < − N 2 , − ( N + α + 4 ) ( N − 2 ) 2 ( N + α ) < − ( N − 2 ) ( N − α ) 2 ( N − α − 4 ) ,

we can find a number β satisfying (3.7)– (3.9). This shows that

c 0 < 1 2 1 − 1 p p 1 p − 1 S 1 p p − 1 .

Suppose that q = N + α N − 2 . We still give a proof only for the case α ≠ 0 since the case α = 0 is easier to deal with. For a positive number λ > 0, we define

ν λ ( x ) ≔ B λ N − 2 2 ( λ 2 + | x | 2 ) N − 2 2 .

If N ≥ 5, one has ν λ ∈ H r 1 ( R N ) for every λ and they are the extremal functions of the inequality (2.2). The constant B is chosen to satisfy

∫ R N | ∇ ν 1 | 2 d x = ∫ R N I α ∗ | ν 1 | q | ν 1 | q d x .

This implies that ∫ R N | ∇ ν 1 | 2 d x = S 2 q q − 1 . For a fixed positive number λ, let s λ > 0 be such that

J α s λ ψ λ , ν λ = max t > 0 J α t ψ λ , ν λ .

Let s ̃ λ be the number that satisfies J α s ̃ λ ψ λ , ν λ < 0 . We have seen that s ̃ λ > s λ . We define γ 2 ( t ) ≔ t s ̃ λ ( ψ λ , ν λ ) . Then, we see that

c 0 ≤ max t ∈ [ 0,1 ] J α γ 2 ( t ) = J α s λ ψ λ , ν λ .

We compute

(3.10) 0 = d d t t = s λ J α t ψ λ , ν λ = s λ ∫ R N | ∇ ψ λ | 2 d x + s λ ∫ R N | ψ λ | 2 d x + s λ ∫ R N | ∇ ν λ | 2 d x + s λ ∫ R N | ν λ | 2 d x − s λ 2 p − 1 p ∫ R N I α ∗ | ψ λ | p | ψ λ | p d x − s λ 2 q − 1 q ∫ R N I α ∗ | ν λ | q | ν λ | q d x − ( p + q ) s λ p + q − 1 p q ∫ R N I α ∗ | ψ λ | p | ν λ | q d x = s λ λ 2 β − 2 + N ∫ R N | ∇ ψ 1 | 2 d x + s λ λ 2 β + N ∫ R N | ψ 1 | 2 d x + s λ ∫ R N | ∇ ν 1 | 2 d x + s λ λ 2 ∫ R N | ν 1 | 2 d x − s λ 2 p − 1 λ 2 β p + N + α p ∫ R N I α ∗ | ψ 1 | p | ψ 1 | p d x − s λ 2 q − 1 q ∫ R N I α ∗ | ν 1 | q | ν 1 | q d x − s λ p + q − 1 λ β p − N − 2 2 q + N + α p q ∫ R N I α ∗ | ψ 1 | p | ν 1 | q d x .

We define s 0 ≔ lim sup λ→0 t λ . Then, if β satisfies

(3.11) β > − N − 2 2 ,

then we can see that s 0 < ∞ since 2β − 2 + N, 2β + N > 0.

If we assume (3.11) and pass to a limit λ → 0 in (3.10), then we obtain

0 = s 0 ∫ R N | ∇ ν 1 | 2 d x − s 0 2 q − 1 q ∫ R N I α ∗ | ν 1 | p | ν 1 | p d x = s 0 − s 0 2 q − 1 p ∫ R N | ∇ ν 1 | 2 d x

since 2 β q + N + α , − N 2 p + β q + N + α < 0 . This implies that s 0 = q 1 2 q − 2 .

We compute

J α s λ ψ λ , ν λ = s λ 2 λ 2 β + N − 2 2 ∫ R N | ∇ ψ 1 | 2 d x + s λ 2 λ 2 β + N 2 ∫ R N | ψ 1 | 2 d x + s λ 2 2 ∫ R N | ∇ ν 1 | 2 d x + s λ 2 λ 2 2 ∫ R N | ν 1 | 2 d x − s λ 2 p λ 2 β p + N + α 2 p 2 ∫ R N I α ∗ | ψ 1 | p | ψ 1 | p d x − s λ 2 q 2 q 2 ∫ R N I α ∗ | ν 1 | q | ν 1 | q d x − s λ p + q λ β p − N − 2 2 q + N + α p q ∫ R N I α ∗ | ψ 1 | p | ν 1 | q d x ≤ s λ 2 2 − t λ 2 q 2 q 2 ∫ R N | ∇ ν 1 | 2 d x + s λ 2 λ 2 β + N − 2 2 ∫ R N | ∇ ψ 1 | 2 d x + s λ 2 λ 2 β + N 2 ∫ R N | ψ 1 | 2 d x + s λ 2 λ 2 2 ∫ R N | ν 1 | 2 d x − s λ p + q λ β p − N − 2 2 q + N + α p q ∫ R N I α ∗ | ψ 1 | p | ν 1 | q d x .

Note that we have

s λ 2 2 − s λ 2 q 2 q 2 ∫ R N | ∇ ν 1 | 2 d x ≤ 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1 .

If β satisfies

(3.12) β > − N ( N − α − 4 ) 2 ( N − α ) ,

(3.13) β < − N ( N − α − 4 ) 2 ( N + α ) ,

then we have

β p − N − 2 2 q + N + α < 2 β + N − 2 , β p − N − 2 2 q + N + α < 2 .

Therefore, if β satisfies (3.11)– (3.13), then for sufficiently small λ > 0 we have

J α ( s λ ( ψ λ , ν λ ) ) < 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1 .

Since we have

− N − 2 2 < − N ( N − α − 4 ) 2 ( N + α ) , − N ( N − α − 4 ) 2 ( N − α ) < − N ( N − α − 4 ) 2 ( N + α ) ,

we can find a number β satisfying (3.11)– (3.13). This shows that

c 0 < 1 2 1 − 1 q q 1 q − 1 S 2 q q − 1 .

□

4 Classification of ground states

In this section we prove Theorem 1.5. Recall that our basic assumption for N, α, p, q is N ≥ 3, 0 ≤ α < N and N + α N ≤ p ≤ q ≤ N + α N − 2 excluding the case α = 0 and p = 1.

4.1 N + α N ≤ p < q < 2 case

In this case we need to prove any ground state is vectorial if it exists. We show that if a semi-trivial solution (u 0, 0) or (0, v 0) exists then it cannot be a minimizer of J α on N . Then it is not a ground state by Proposition 3.1. We prove this only for (u 0, 0) because the situation is exactly the same for (0, v 0).

Let (u 0, 0) be a nontrivial nonnegative semi-trivial solution of (1.1). Note that

(4.1) ‖ u 0 ‖ 2 = ∫ R N I α ∗ 1 p u 0 p u 0 p d x .

Therefore, we have

J α ′ ( ( 1 + t ) u 0 , s u 0 ) ( ( 1 + t ) u 0 , s u 0 ) = ( ( 1 + t ) 2 + s 2 ) ‖ u 0 ‖ 2 − ( 1 + t ) 2 p p ∫ R N I α ∗ u 0 p u 0 p d x − s 2 q q ∫ R N I α ∗ u 0 q u 0 q d x − 1 p + 1 q ( 1 + t ) p s q ∫ R N I α ∗ u 0 p u 0 q d x = ( ( 1 + t ) 2 + s 2 − ( 1 + t ) 2 p ) ‖ u 0 ‖ 2 − s 2 q q ∫ R N I α ∗ u 0 q u 0 q d x − 1 p + 1 q ( 1 + t ) p s q ∫ R N I α ∗ u 0 p u 0 q d x .

Let f be the two variable function defined by

f ( t , s ) : = J α ′ ( ( 1 + t ) u 0 , s u 0 ) ( ( 1 + t ) u 0 , s u 0 ) .

It is clear that f is C 1. Since (u 0, 0) is a solution of (1.1), f(0, 0) = 0. Also one has ∂ t f(0, 0) = 2(1 − p)‖u 0‖2 ≠ 0 so we can invoke the implicit function theorem to assert that there exists a neighborhood of (0, 0) such that t is a unique C 1 function of s and f(t(s), s) = 0 in this neighborhood. In particular t(s) → 0 as s → 0. Then observe that

J α ( ( 1 + t ) u 0 , s u 0 ) = ( 1 + t ) 2 + s 2 2 ‖ u 0 ‖ 2 − ( 1 + t ) 2 p 2 p 2 ∫ R N I α ∗ u 0 p u 0 p d x − s 2 q 2 q 2 ∫ R N I α ∗ u 0 q u 0 q d x − 1 p q ( 1 + t ) p s q ∫ R N I α ∗ u 0 p u 0 q d x = 1 2 ‖ u 0 ‖ 2 − 1 2 p 2 ∫ R N I α ∗ u 0 p u 0 p d x + 2 t + t 2 + s 2 2 ‖ u 0 ‖ 2 − ( 1 + t ) 2 p − 1 2 p 2 ∫ R N I α ∗ u 0 p u 0 p d x − s 2 q 2 q 2 ∫ R N I α ∗ u 0 q u 0 q d x − 1 p q ( 1 + t ) p s q ∫ R N I α ∗ u 0 p u 0 q d x = J ( u 0 , 0 ) + 2 t + t 2 2 ‖ u 0 ‖ 2 − ( 1 + t ) 2 p − 1 2 p 2 ∫ R N I α ∗ u 0 p u 0 p d x ︸ ( A ) + s 2 2 ‖ u 0 ‖ 2 − s 2 q 2 q 2 ∫ R N I α ∗ u 0 q u 0 q d x − 1 p q ( 1 + t ) p s q ∫ R N I α ∗ u 0 p u 0 q d x ︸ ( B ) .

Since q < 2 and lim s→0 t(s) = 0, we easily see that (B) < 0 for sufficiently small s > 0. For (A), we use the equality (4.1) and the Taylor expansion to see

( A ) = − ( 1 + t ) 2 p − 1 − 2 p t − p t 2 2 p ‖ u 0 ‖ 2 = − 2 p ( 2 p − 1 ) t 2 / 2 − p t 2 + o ( t 2 ) 2 p ‖ u 0 ‖ 2 = − 2 p ( p − 1 ) t 2 + o ( t 2 ) 2 p ‖ u 0 ‖ 2 < 0

for sufficiently small t > 0. Therefore we conclude that

J α 1 + t ( s ) u 0 , s u 0 < J ( u 0 , 0 )

for small s > 0 and consequently for small t > 0. Since J α ′ 1 + t ( s ) u 0 , s u 0 1 + t ( s ) u 0 , s u 0 = 0 , (u 0, 0) cannot be a minimizer of J α on N . This completes the proof.

4.2 2 ≤ p < q ≤ N + α N − 2 case

In this case, we prove that if a nonnegative ground state of (1.1) exists then it must be semi-trivial. We define the Morse index of a solution (u, v) of (1.1) as the maximal dimension of subspace H ⊂ H such that J α ′ ′ ( u , v ) [ ( ϕ , ψ ) , ( ϕ , ψ ) ] < 0 for every ( ϕ , ψ ) ∈ H \ { ( 0,0 ) } .

The linearized operator of the system (1.1) at a nonnegative solution (u, v) is given by

(4.2) L ( ϕ , ψ ) = − Δ ϕ + ϕ − I α ∗ ( u p − 1 ϕ + v q − 1 ψ ) u p − 1 − ( p − 1 ) I α ∗ 1 p u p + 1 q v q u p − 2 ϕ − Δ ψ + ψ − I α ∗ ( u p − 1 ϕ + v q − 1 ψ ) v q − 1 − ( q − 1 ) I α ∗ 1 p u p + 1 q v q v q − 2 ψ .

We note that

(4.3) J α ′ ′ ( u , v ) [ ( ϕ , ψ ) , ( ϕ , ψ ) ] = ⟨ L ( ϕ , ψ ) , ( ϕ , ψ ) ⟩ L 2 .

Lemma 4.1.

Let (u, v) be a positive solution of (1.1). Then the Morse index of (u, v) is at least two.

Proof.

Let V : = { ( s u , t v ) | s , t ∈ R } . Then

J α ′ ′ ( u , v ) [ ( s u , t v ) , ( s u , t v ) ] = s 2 ‖ u ‖ 2 − ∫ I α ∗ u p u p − ( p − 1 ) ∫ I α ∗ 1 p u p + 1 q v q u p + t 2 ‖ v ‖ 2 − ∫ I α ∗ v q v q − ( q − 1 ) ∫ I α ∗ 1 p u p + 1 q v q v q − 2 s t ∫ I α ∗ u p v q .

By multiplying the first and second equation of (1.1) by respectively u and v and integrating by part, we get

‖ u ‖ 2 = ∫ I α ∗ 1 p u p + 1 q v q u p d x , ‖ v ‖ 2 = ∫ I α ∗ 1 p u p + 1 q v q v q d x .

Then we see that

J α ′ ′ ( u , v ) [ ( s u , t v ) , ( s u , t v ) ] = − s 2 ( p − 2 ) ∫ I α ∗ 1 p u p + 1 q v q u p − t 2 ( q − 2 ) ∫ I α ∗ 1 p u p + 1 q v q v q − s 2 ∫ I α ∗ u p u p + 2 s t ∫ I α ∗ u p v q + t 2 ∫ I α ∗ v q v q ︸ ( A ) .

We claim that (A) ≥ 0 for any s , t ∈ R . Using Proposition 2.3, we actually see that

( A ) ≥ ∫ ( I α ∗ ( | s | u p ) ) ( | s | u p ) − 2 ∫ ( I α ∗ ( | s | u p ) ) ( | t | v q ) + ∫ ( I α ∗ ( | t | v q ) ) ( | t | v q ) ≥ 0 .

This shows that J″(u, v)[(ϕ, ψ), (ϕ, ψ)] < 0 for every (ϕ, ψ) ∈ V \ {(0, 0)}. Thus the proof is done because dim(V) = 2. □

Since a ground state of (1.1) is a minimizer of J α on the Nehari manifold, it has Morse index one. Thus we immediately get the desired result from Lemma 4.1.

4.3 N + α N < p = q < N + α N − 2 case

We use the following lemma which characterizes the ground states of the (1.1) when p = q.

Lemma 4.2.

Suppose that N + α N < p = q < N + α N − 2 and α ∈ [0, N). Then, a ground state of (1.1) is of the form (au 0, bu 0), where a , b ∈ R and u 0 is a nontrivial solution of the equation

(4.4) − Δ u + u = I α ∗ | u | p | u | p − 2 u .

Proof.

Let X ≔ H r 1 ( R N ) and M = 2. We define operators

I 1 ( u ) ≔ 1 2 ∫ R N ( | ∇ u | 2 + u 2 ) d x , J 1 ( u ) ≔ ∫ R N I α ∗ | u | p | u | p d x , C ( u , v ) ≔ ∫ R N I α ∗ | u | p | v | p d x

for u, v ∈ X. We check the conditions (H1), (H2), and (H3) in ref. [12], Section 3] for I 1, J 1, and C. It is easy to see that I 1 and J 1 are homogeneous of degrees 2 and 2p, respectively, and that J 1(u) > 0 if u ≠ 0. By the definition of C(u, v), we have C(ηu, ξu) = η p ξ p J 1(u) for all η, ξ > and all u ∈ X. By Proposition 2.3, we have

C ( u , v ) ≤ ∫ R N I α ∗ | u | p | u | p d x 1 2 ∫ R N I α ∗ | v | p | v | p d x 1 2 = J 1 ( u ) 1 2 J 1 ( v ) 1 2 .

Therefore, operators I 1, J 1 and C satisfy the conditions (H1), (H2), and (H3) in ref. [12], Section 3]. By Lemma 7 in ref. [12], we get the desired result. □

By Lemma 4.2, a ground state of (1.1) can be written as (au 0, bu 0) for some a , b ∈ R , where u 0 is a nontrivial solution of (4.4).

First, suppose that p ≠ 2 and that (au 0, bu 0) is a vector ground state of (1.1). Since (au 0, bu 0) is a solution of (1.1), we have

− Δ u 0 + u 0 = I α ∗ a p + b p p | u 0 | p a p − 2 | u 0 | p − 2 u 0 , − Δ u 0 + u 0 = I α ∗ a p + b p p | u 0 | p b p − 2 | u 0 | p − 2 u 0 .

Comparing each right-hand side in the above equations, we can see that a and b should be the same if both of a and b are not zero. Moreover, since u 0 is a solution of (4.4), we see that a = p 2 1 2 p − 2 . In the same way, we see that if (au 0, bu 0) is a semi-trivial ground state of (1.1), then it should be p 1 2 p − 2 u 0 , 0 or 0 , p 1 2 p − 2 u 0 .

Now, we compare the values of J α at p 2 1 2 p − 2 u 0 , p 2 1 2 p − 2 u 0 and p 1 2 p − 2 u 0 , 0 to determine which one is a ground state of (1.1). For this, we compute the difference. Since u 0 is a solution of (4.4), we have ∫ R N | ∇ u 0 | 2 + u 0 2 d x = ∫ R N I α ∗ | u 0 | p | u 0 | p d x . Therefore, we have

J α p 2 1 2 p − 2 u 0 , p 2 1 2 p − 2 u 0 − J α p 1 2 p − 2 u 0 , 0 = 1 2 1 p − 1 − 1 2 1 − 1 p p 1 p − 1 A ,

where A = ∫ R N | ∇ u 0 | 2 + u 0 2 d x . Since p > 1, we see that 1 − 1 p p 1 p − 1 A > 0 . Therefore, if p > 2, then there is no vector ground state of (1.1), and if p < 2, then there is no semi-trivial ground state of (1.1).

If p = 2, then it is easy to see that (au 0, bu 0) is a ground state of (1.1) if a 2 + b 2 = 2.

5 Existence of positive vector solutions

In this section, we prove Theorem 1.7, which asserts the existence of a positive solution when α ∈ [0, N) and 2 < p < q < N + α N − 2 with the condition (ND) stated in Section 1. Our strategy follows the basic idea by Ambrosetti and Colorado [7]. We prove that two semi-trivial solutions (u 0, 0) and (0, v 0) are strict local minimums of J α on the Nehari manifold N . Then the well-known mountain pass lemma says there exists a third critical point of J α , which is different from these two semi-trivial solutions.

We recall a tuple (α, s) satisfies (ND) if there exists a unique positive radial solution u 0 ∈ H r 1 ( R N ) of the equation

(5.1) − Δ u + u = I α ∗ | u | s s | u | s − 2 u

such that the linearized equation of (5.1) at u 0

(5.2) − Δ ϕ + ϕ = I α ∗ u 0 s − 1 ϕ u 0 s − 1 + I α ∗ u 0 s s ( s − 1 ) u 0 s − 2 ϕ

admits only the trivial solution in H r 1 ( R N ) .

Lemma 5.1.

Let N ≥ 3, α ∈ [0, N), N + α N < s < N + α N − 2 and (ND) hold for (α, s). Then there exists a positive constant c > 0 such that

‖ ϕ ‖ 2 − ∫ I α ∗ u 0 s − 1 ϕ u 0 s − 1 ϕ d x − s − 1 s ∫ I α ∗ u 0 s u 0 s − 2 ϕ 2 d x ≥ c ‖ ϕ ‖ 2

for each ϕ ⊥ u 0 in H r 1 ( R N ) . Here u 0 denotes the unique positive radial solution of (5.1).

Proof.

We refer to Lemma 5.2 in ref. [37], in which our assertion is proved for the case α = 0. The exactly same arguments apply to derive the same conclusion. We omit the proof. □

In the rest of this section, we denote by u 0 and v 0 the unique positive radial solutions of (5.1) respectively with s = p and s = q when (ND) is assumed for (α, p) and (α, q).

Lemma 5.2.

Let N ≥ 3, 2 < p ≤ q < N + α N − 2 and (ND) hold for (α, p) and (α, q). Then the semi-trivial solutions (u 0, 0) and (0, v 0) of (1.1) are strict local minimums of J α on the Nehari manifold N .

Proof.

We note that for proving the lemma, it suffices to show the coerciveness of semi-trivial solutions (u 0, 0) and (0, v 0) on N , i.e., there exist constants c 1, c 2 > 0 such that

J α ′ ′ ( u 0 , 0 ) [ ( ϕ , ψ ) , ( ϕ , ψ ) ] ≥ c 1 ( ‖ ϕ ‖ 2 + | ψ ‖ 2 ) , J α ′ ′ ( 0 , v 0 ) [ ( ϕ , ψ ) , ( ϕ , ψ ) ] ≥ c 2 ( ‖ ϕ ‖ 2 + | ψ ‖ 2 )

for any ( ϕ , ψ ) ∈ T ( u 0 , 0 ) N and ( ϕ , ψ ) ∈ T ( 0 , v 0 ) N respectively.

By (4.3), we compute

J α ′ ′ ( u 0 , 0 ) [ ( ϕ , ψ ) , ( ϕ , ψ ) ] = ‖ ϕ ‖ 2 + ‖ ψ ‖ 2 − ∫ I α ∗ u 0 p − 1 ϕ u 0 p − 1 ϕ d x − p − 1 p ∫ I α ∗ u 0 p u 0 p − 2 ϕ 2 d x , J α ′ ′ ( 0 , v 0 ) [ ( ϕ , ψ ) , ( ϕ , ψ ) ] = ‖ ϕ ‖ 2 + ‖ ψ ‖ 2 − ∫ I α ∗ v 0 q − 1 ψ v 0 q − 1 ψ d x − q − 1 q ∫ I α ∗ v 0 q v 0 q − 2 ψ 2 d x .

Now, we claim that ( ϕ , ψ ) ∈ T ( u 0 , 0 ) N if and only if ϕ ⊥ u 0 in H r 1 ( R N ) . We define N ̃ ( u , v ) : = ⟨ J α ′ ( u , v ) , ( u , v ) ⟩ L 2 so that N = { ( u , v ) ∈ H r | N ̃ ( u , v ) = 0 , ( u , v ) ≠ ( 0,0 ) } . Then one has by definition, ( ϕ , ψ ) ∈ T ( u 0 , 0 ) N if and only if ⟨ N ̃ ′ ( u 0 , 0 ) , ( ϕ , ψ ) ⟩ L 2 = 0 . A direct computation shows

⟨ N ′ ( u 0 , 0 ) , ( ϕ , ψ ) ⟩ L 2 = 2 ∫ ∇ u 0 ⋅ ∇ ϕ + u 0 ϕ d x − 2 ∫ ( I α ∗ u 0 p ) u 0 p − 1 ϕ d x = 2 ∫ ∇ u 0 ⋅ ∇ ϕ + u 0 ϕ d x − 2 p ∫ ∇ u 0 ⋅ ∇ ϕ + u 0 ϕ d x = 2 ( 1 − p ) ∫ ∇ u 0 ⋅ ∇ ϕ + u 0 ϕ d x

from which the claim is proved. Similarly we see that ( ϕ , ψ ) ∈ T ( 0 , v 0 ) N if and only if ψ ⊥ v 0 in H r 1 ( R N ) . Therefore, we can derive the coerciveness of (u 0, 0) and (0, v 0) on N from Lemma 5.1. □

Proof of Theorem 1.7.

We first introduce the functional

J + ( u , v ) ≔ 1 2 ∫ R N | ∇ u | 2 + | ∇ v | 2 + u 2 + v 2 d x − 1 2 ∫ R N I α ∗ 1 p u + p + 1 q v + q 1 p u + p + 1 q v + q d x .

Since p, q > 2, J + is C 2. Also, it is easy to see that a critical point of J + is a nonnegative critical point of J α and vice versa. Since u 0 and v 0 are positive, Lemma 5.2 then still holds true for J + and N + : = ( u , v ) ∈ H r \ { ( 0,0 ) } | ⟨ J + ′ ( u , v ) , ( u , v ) ⟩ L 2 = 0 so that there exists a mountain pass type critical point (u ∗, v ∗) by minimaxing J + among the curves lying on the constraint N + and connecting two strict local minimum points (u 0, 0) and (0, v 0). See [7] for a reference. As a result, (u ∗, v ∗) is neither (u 0, 0) nor (0, v 0). Since N + is a natural manifold, (u ∗, v ∗) is a unconstrained critical point of J +, which is a nonnegative, nontrivial and radial solution of (1.1). By Proposition 3.2, u ∗ and v ∗ are either identically zero or strictly positive. Thus the uniqueness assumption in (ND) shows u ∗ and v ∗ cannot be zero. This completes the proof. □

Corresponding author: Jinmyoung Seok, Department of Mathematics Education, Seoul National University, Seoul 08826, Republic of Korea, E-mail: 

Funding source: Ministry of Science and ICT

Award Identifier / Grant number: NRF- 2020R1C1C1A01006415

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Conflict of interest: The authors state no conflict of interest.

  5. Research funding: J. Seok is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF- 2020R1C1C1A01006415).

  6. Data availability: Not applicable.

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Received: 2024-06-09
Accepted: 2025-08-26
Published Online: 2025-09-23

© 2025 the author(s), published by De Gruyter, Berlin/Boston

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https://doi.org/10.1515/ans-2023-0199
Keywords for this article
nonlinear Schrödinger; Choquard; ground state; variational method
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