Existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in ℝ³

Theorem 1.1

Let ( A 1 ) , ( A 2 ) , and ( g 1 ) – ( g 3 ) be satisfied. Then, for any δ > 0 , there is λ * > 0 such that problem (1.1) has at least one nontrivial solution u λ that satisfies the following properties:

1. 1 2 − 1 θ ∫ R 3 ( ∣ ∇ u λ ∣ 2 + A ( x ) u λ 2 ) d x ≤ δ λ 2 2 − 2 * ;

2. λ 4 − λ 2 * ∫ R 3 ∣ u λ ∣ 2 * d x + λ θ 4 − 1 ∫ R 3 G ( u λ ) d x ≤ δ λ 2 2 − 2 * .

Theorem 1.2

Let ( A 1 ) , ( A 2 ) , and ( g 1 ) – ( g 3 ) be satisfied and g ( t ) is odd in t. Then, for any m * ∈ N and σ > 0 , there is Λ m * , σ > 0 such that if λ ≥ Λ m * , σ , problem (1.1) has at least m * pairs of solutions u ± λ , m * , which satisfy the estimates ( i ) and ( i i ) in Theorem 1.1.

Remark 1.1

We should also point out that compared to previous papers, this article has several important characteristics. First, let us emphasize that the work presented in this article is completely different from previous papers, such as [21–25], they used decomposing the corresponding functional and truncating techniques to prove the compactness condition. However, in this article, we attempt to prove the compactness condition by using the concentration-compactness principle, which is entirely distinct from the prior methods. Second, the main difficulty of this article is caused by the nonlinear term, which has critical growth and can lead to the loss of compactness. Another difficulty we encounter is the lack of compactness due to unbounded domains and Poisson terms. In order to prove the symmetric mountain path theorem, we must establish some tools and technological achievements. In a sense, we have generalized and improved the previous results.

Notations 1.1

Now, let us explain some symbols of the article, while others will be provided when we use them:

• H 1 ( R 3 ) is the usual Sobolev space with norm ‖ ⋅ ‖ ;

• ∣ u ∣ p is the usual L p -norm ( ∫ R 3 ∣ u ∣ p d x ) 1 p ;

• for any number r > 0 , B r ( 0 ) denotes a sphere of radius r centered on 0 with respect to the norm topology in D 1 , 2 ( R 3 ) ;

• o n ( 1 ) represents a vanishing sequence;

• the arrows ⇀ , → denote weak convergence and strong convergence, respectively;

• C , C * , C i , i = 1 , 2 , … , denote generic constants, which may change from line to line.

Lemma 2.1

(see [1, Lemma 3.4]) For every u ∈ H 1 ( R 3 ) , we have the following properties:

1. for every y ∈ R 3 , Φ u ( ⋅ + y ) = Φ u ( ⋅ + y ) ;

2. Φ u ≥ 0 ;

3. for every s ∈ ( 3 , + ∞ ] , Φ u ∈ L s ( R 3 ) ∩ C 0 ( R 3 ) ;

4. for every s ∈ ( 3 2 , + ∞ ] , ∇ Φ u ∈ L s ( R 3 ) ∩ C 0 ( R 3 ) ;

5. ∣ Φ u ∣ 6 ≤ C ‖ u ‖ 2 for some constant C > 0 .

6. Moreover, if u is radial also Φ u is, and if u n ⇀ u in H rad 1 ( R 3 ) , then

7. Φ u n → Φ u in D ;

8. ∫ R 3 Φ u n u n 2 d x → ∫ R 3 Φ u u 2 d x ;

9. ∫ R 3 Φ u n u n v d x → ∫ R 3 Φ u u v d x for any v ∈ H 1 ( R 3 ) .

Lemma 3.1

Assume ( A 1 ) , ( A 2 ) , and ( g 1 ) – ( g 3 ) hold. For any finite-dimensional subspace F ⊂ E , there is e ∈ F such that the functional defined in (2.5) satisfies the following properties:

1. For each λ > 0 there is α λ > 0 and ρ λ > 0 such that I λ ( u ) > 0 for u ∈ B ρ λ \ { 0 } . Moreover, I λ ( u ) > α λ for all u ∈ W λ with ‖ u ‖ = ρ λ , where B ρ λ = { u ∈ W λ : ‖ u ‖ < ρ λ } .

2. There is e ∈ W λ with ‖ e ‖ λ > ρ λ such that I λ ( e ) < 0 for all λ > 0 .

Proof

(i) First, by conditions ( g 2 ) and ( g 3 ) , we obtain that for any κ > 0 there is C κ > 0 such that

So, for every u ∈ W λ , let κ ∈ ( 0 , 1 4 λ c 2 * 2 * ) , we have that

Let

Clearly, lim t → 0 + f ( t ) = 1 4 > 0 , because 2 * > 2 and q > 2 . So, we can choose ρ λ ≔ ‖ u ‖ small enough such that

So, we conclude

This completes the proof of Lemma 3.1 (i).

(ii) We note that conditions ( g 2 ) and ( g 3 ) mean that for every ϱ > 0 , there is C ϱ > 0 such that

Let u 0 ∈ W λ with ‖ u 0 ‖ = 1 , by Lemma 2.1- ( Φ 5 ) , we have

For all u ∈ W λ due to 2 * > 2 and p > 4 , we can choose t ≥ 1 large enough such that ‖ t u 0 ‖ > ρ λ and I λ ( t u 0 ) < 0 . It is easy to obtain this conclusion by letting e = t u 0 and this means that the proof of Lemma 3.1 (ii) is completed.□

Definition 3.1

Let J : E → R be a C 1 functional on a Banach space E .

(1) For c ∈ R , a sequence { u n } ⊂ E is called a Palais-Smale sequence at level c (in short ( PS ) c ) in E for J if J ( u n ) = c + o n ( 1 ) and J ′ ( u n ) → 0 in E * (dual of E ) as n → ∞ .

(2) We say J satisfies ( PS ) c condition if for any Palais-Smale sequence { u n } in E , J has a convergent subsequence.

Lemma 3.2

Suppose that ( A 1 ) , ( A 2 ) , and ( g 1 ) – ( g 3 ) hold. Let { u n } ⊂ W λ be any ( PS ) c sequence. The functional I λ satisfies ( PS ) c condition for all c ∈ ( 0 , α 0 λ 2 2 − 2 * ) , where α 0 = ( 1 2 − 1 θ ) S 2 * 2 * − 2 .

Proof

This proof is divided into four steps.

Step I. We claim that for all c ≥ 0 , { u n } is bounded in W λ .

In fact, let { u n } ⊂ W λ be a ( PS ) c sequence related to the functional I λ . By ( g 3 ) , one has

Together with 1 2 − 1 θ > 0 , we obtain that { u n } is bounded on W λ . This means that the proof of Step I is complete.

Step II. We claim w j = 0 for all j ∈ J .

In fact, from Step I, we know that { u n } is bounded in W λ . Hence, we can conclude that

and

Thus, we assume that

in the sense of measure. By the concentration compactness principle from Lions [28], there is at most countable sets of J , families of positive numbers { ω j : j ∈ J } , { ξ j : j ∈ J } , and sequences of point { z j } j ∈ J ⊂ R 3 such that

and

where δ z is Dirac mass of mass 1 concentrated at x ∈ R 3 . In addition, we can structure a smooth truncation function ϕ ε , j centered at z j so that

and

for any ε > 0 small. Note that ⟨ I λ ′ ( u n ) , u n ϕ ε , j ⟩ = o n ( 1 ) as n → ∞ . So, we obtain

With the help of the boundedness of { u n } in W λ , we have

On the other hand, by (3.3), (3.4), and the Hölder inequality we find that

and

Now, according to (3.6), (3.7), (3.8), and (3.9), we obtain λ ξ j ≥ ω j . Combining (3.5), we have

We claim that the first case is not valid. If not, then there is j 0 ∈ J such that ω j 0 ≥ S 2 * 2 * − 2 λ 2 2 − 2 * , we have

Due to θ > 2 , this gives a contradiction with c ∈ ( 0 , α 0 ) , and we obtain that