Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity

Theorem 1.1

Under assumptions ( V 1 ) – ( V 2 ) , p > 8 , Problem (1.1) has a ground state solution u 0 ′ = g 0 ( v 0 ′ ) .

Lemma 2.1

[24–26]  g 0 ( t 0 ) , g 0 ′ ( t 0 ) satisfies the following properties:

1. ∣ g 0 ( t 0 ) t 0 ∣ ≤ 1 for all t 0 ∈ R ;

2. ∣ g 0 ( t 0 ) ∣ ∣ t 0 ∣ ≤ 2 1 / 4 for all t 0 ∈ R ;

3. lim t → + ∞ g 0 ( t 0 ) t 0 = 2 1 / 4 ;

4. g 0 ( t 0 ) / 2 ≤ t 0 g 0 ′ ( t 0 ) ≤ g 0 ( t 0 ) for all t 0 ≥ 0 ;

5. ∣ g 0 ( t 0 ) g 0 ′ ( t 0 ) ∣ ≤ 1 / 2 for all t 0 ∈ R ;

6. there exists a constant C > 0 such that ∣ g 0 ( t 0 ) ∣ ≥ C ∣ t 0 ∣ for ∣ t 0 ∣ ≥ 1 and ∣ g 0 ( t 0 ) ∣ ≥ C ∣ t 0 ∣ 1 / 2 for ∣ t 0 ∣ ≤ 1 .

Proposition 2.2

J ∈ C 1 ( X , R ) , the critical point of J is radial solution of (1.2).

Lemma 2.3

[7] If { v n } ⇀ v in X as n → + ∞ , then for any ψ ∈ X

Proposition 2.4

[2] For v ∈ X , there holds

Theorem 3.1

[27] Set ( E , ‖ ⋅ ‖ ) be a Banach space, T 0 ⊂ R + be a real interval. Consider a family Ψ η of C 1 -functional on E

where ℬ ( v ) is non-negative and when ‖ v ‖ → + ∞ , either A ( v ) → + ∞ or ℬ ( v ) → + ∞ . Assume that there exist two points v 1 and v 2 which hold

where Γ λ = { γ ¯ ∈ C ( [ 0 , 1 ] , E ) : γ ¯ ( 0 ) = v 1 , γ ¯ ( 1 ) = v 2 } . Then, for a.e. λ ∈ T 0 , there exist a sequence { v n } ⊂ E such that

1. { v n } is bounded in E ;

2. lim n → + ∞ Ψ λ ( v n ) = c λ ;

3. lim n → + ∞ Ψ λ ′ ( v n ) = 0 in the dual space E ∗ of E .

Lemma 3.2

Under the assumption ( V 1 ) , then there holds:

1. For λ ∈ T 0 , there exists v ∈ X \ { 0 } such that J λ ( v ) < 0 ;

2. For λ ∈ T 0 , max { J λ ( 0 ) , J λ ( v ) } < inf γ ¯ ∈ Γ λ max t ∈ [ 0 , 1 ] J λ ( γ ¯ ( t ) ) = c λ , where Γ λ = { γ ¯ ∈ C ( [ 0 , 1 ] , X ) : γ ¯ ( 0 ) = 0 , γ ¯ ( 1 ) = v } .

Proof

(i) Set v ∈ X \ { 0 } be fixed, we infer that

Arguing as in [28,29], we consider ξ ∈ C 0 ∞ ( R 2 ) , which satisfies 0 ≤ ξ ( x 0 ) ≤ 1 , ξ ( x 0 ) = 0 for ∣ x 0 ∣ ≥ 2 and ξ ( x 0 ) = 1 for ∣ x 0 ∣ ≤ 1 . By ( g 4 ), we can deduce that g 0 ( t 0 ξ ( x 0 ) ) ≥ g 0 ( t 0 ) ξ ( x 0 ) for t 0 ≥ 0 . Thus, from ( g 1 ), one has

for all t 0 > 0 . By p > 8 and ( g 3 ), we deduce that lim t 0 → + ∞ J λ ( t 0 ξ ) = − ∞ ; therefore, there exists a t ˆ 0 > 0 such that J λ ( t ˆ 0 ξ ) ≤ 0 . For all λ ∈ T 0 , taking a function v = t ˆ 0 ξ , one obtains J λ ( v ) < 0 .

(ii) By Chen and Wu [28] and Fang and Szulkin [30], there exists ρ ′ > 0 such that C ‖ v ‖ 2 ≤ ∫ R 2 ( ∣ ∇ v ∣ 2 + V ( ∣ x ∣ ) g 0 2 ( v ) ) for all ‖ v ‖ ≤ ρ ′ . From ( g 2 ), Hardy-Littlewood-Sobolev inequality in [1], and Sobolev imbedding inequality, one obtains

whenever ‖ v ‖ ≤ ρ ′ . Since p > 8 , we obtain J λ ( v ) > 0 if ‖ v ‖ is small enough. Hence, J λ ( 0 ) is strict local minimum, c λ > 0 .□

Lemma 3.3

Let λ ∈ [ η , 1 ] be fixed. Suppose that { w n } ⊂ X is a sequence obtained earlier. Then there exists w λ ∈ X \ { 0 } , such that J λ ′ ( w λ ) = 0 and J λ ( w λ ) = c λ .

Proof

By Theorem 3.1 and Lemma 3.2, we have ‖ w n ‖ ≤ C , then, up to a subsequence, there exists w λ ∈ X \ { 0 } such that w n ⇀ w λ in X , w n → w λ in L s ( R 2 ) ( s > 2 ) and w n → w λ almost everywhere in R 2 . Using Lebesgue dominated convergence Theorem, J λ ′ ( w λ ) = 0 . Similar to [29–32], we obtain

Furthermore, using Hardy-Littlewood-Sobolev inequality, the Hölder inequality and ( g 2 ) and ( g 5 ), one obtains

Similarly, we can obtain the following:

Thus, it follows from (2.2), (3.1)–(3.3), and Lemma 2.3 that