Ground states of Schrödinger systems with the Chern-Simons gauge fields

Theorem 1.1

Assume that one of the following conditions holds

1. p ∈ ( 2 , 3 ] and b ∈ ( 0 , b δ ) sufficiently small,

2. p ∈ ( 3 , 3 + 6 ) and b > max { b 1 , b 2 } ,

3. p ∈ [ 3 + 6 , ∞ ) and b > max { b 3 , b 4 } , then system (1.1) admits a positive vector ground-state.

Theorem 1.2

There exists b ˜ > 0 sufficiently small and ω ˜ > 0 sufficiently large such that b ∈ ( 0 , b ˜ ) and ω > ω ˜ , then the system (1.1) has only trivial solution if p ∈ ( 1 , 2 ] .

Lemma 2.1

[6] Suppose that a sequence { u n } converges weakly to a function u in H r 1 ( R 2 ) as n → ∞ . Then, for each φ ∈ H r 1 ( R 2 ) , B ( u n ) , B ′ ( u n ) φ , and B ′ ( u n ) u n converge up to a subsequence to B ( u ) , B ′ ( u ) φ , and B ′ ( u ) u , respectively, as n → ∞ .

Lemma 2.2

[6] For u ∈ H r 1 ( R 2 ) , the following inequality holds

Furthermore, the equality is attained by a continuum of functions

Lemma 2.3

If ( u , v ) is a solution of (1.1) then it satisfies the Pohozaev identity:

Proof

We adopt some ideas in [6]. Assume that ( u , v ) ∈ E is a weak solution for Problem (1.1). Similar to [4] and [6], we know that ∫ ∣ x ∣ ∞ h ( s ) s u 2 ( s ) d s , ∫ ∣ x ∣ ∞ g ( s ) s v 2 ( s ) d s , h 2 ( ∣ x ∣ ) ∣ x ∣ 2 , g 2 ( ∣ x ∣ ) ∣ x ∣ 2 ∈ L ∞ ( R 2 ) . Thus, the standard elliptic estimates [11] imply that u , v ∈ C loc 1 , γ ( R 2 ) for some γ > 0 . Then, we obtain ∫ ∣ x ∣ ∞ h ( s ) s u 2 ( s ) d s , ∫ ∣ x ∣ ∞ g ( s ) s v 2 ( s ) d s , h 2 ( ∣ x ∣ ) ∣ x ∣ 2 , g 2 ( ∣ x ∣ ) ∣ x ∣ 2 ∈ C ( R 2 ) . Since u , v ∈ H r 1 ( R 2 ) , we deduce that u , v ∈ C 2 ( R 2 ) . Then, multiplying the first equation in (1.1) by x ⋅ ∇ u and integrating by parts on a ball B R = { x ∈ R 2 : ∣ x ∣ < R } , then

Thus, we deduce that

In a similar way,

can be obtained. Then, summing up (2.2) and (2.3), we obtain

Arguing as in [6, Proposition 2.3.], there exists a suitable sequence R n → ∞ on which the right-hand side of the aforementioned equation tends to zero. Passing to the limit, we obtain the identity. This completes the proof.□

Proof of Theorem 1.2

Let ( u , v ) be a solution of (1.1). By Lemma 2.2, one can obtain

Denote

There exists b ˜ > 0 small enough such that

Then, we have

There exists a ω ˜ > 0 such that the function t ↦ ω t 2 + 1 2 t 4 − ( 1 + b ) t 2 p is nonnegative and strictly increases as ω > ω ˜ , b ∈ ( 0 , b ˜ ) . Hence, ( u , v ) must be identically ( 0 , 0 ) . This completes the proof.□

Lemma 3.1

(Mountain-Pass geometry) Assume b > 0 , then the functional I satisfies the following conditions:

1. There exists a positive constant r > 0 such that I ( u , v ) > 0 for ‖ ( u , v ) ‖ E = r ;

2. There exists ( e 1 , e 2 ) ∈ E with ‖ ( e 1 , e 2 ) ‖ E > r such that I ( e 1 , e 2 ) < 0 .

Proof

Since

and by the Sobolev embedding theorem, there exists a positive constant C such that

Hence, there exists r > 0 such that

For any ( u , v ) ∈ E \ { ( 0 , 0 ) } and t > 0 ,

which implies that I ( t u , t v ) → − ∞ as t → + ∞ . This completes the proof.□

Lemma 3.2

For every ( u , v ) ∈ E \ { ( 0 , 0 ) } , there exists a unique t ¯ u v > 0 such that t ¯ u v ( u , v ) ∈ N . The maximum of I ( t u , t v ) for t ≥ 0 is achieved at t = t ¯ u v .

Proof

∀ ( u , v ) ∈ E \ { ( 0 , 0 ) } and t > 0 , let

By Lemma 3.1, we deduce that there exists t ¯ = t ¯ u v > 0 such that

Moreover,

As p > 3 , thus g ′ ( t ) is strictly decreasing, the point t = t ¯ u v is the unique value of t > 0 at which t ¯ u v ( u , v ) ∈ N . This completes the proof.□

Lemma 3.3

c N = c .

Proof

Define

By Lemma 3.2, we can obtain c N = c 1 . Since I ( t u , t v ) < 0 for any t large, it follows that c ≤ c 1 . Since every γ ∈ Γ intersects N , c ≥ c N . This completes the proof.□

Lemma 3.4

Assume that one of the following conditions holds

1. p ∈ ( 3 , 3 + 6 ) and b > max { b 1 , b 2 } ,

2. p ∈ [ 3 + 6 , ∞ ) and b > max { b 3 , b 4 } , then c N < min { E 1 , E ω } , where E 1 is a ground-state energy of problem (3.1) with ω = 1 .

Proof

Denote

For fixed ( u , v ) ∈ E \ { ( 0 , 0 ) } , we have