On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity

Theorem 1.1

Suppose that Conditions ( V ) and ( M ) are satisfied. Then there exists λ ∗ > 0 such that if λ > λ ∗ > 0 , then there exists at least one solution u λ of Problem (1.1) and u λ → 0 as λ → ∞ .

Theorem 1.2

Suppose that Conditions ( V ) and ( M ) are satisfied. Then, for any m ∈ N , there exists λ m ∗ > 0 such that if λ > λ m ∗ , then Problem (1.1) has at least m pairs of solutions u λ , i , u λ , − i , i = 1 , 2 , … , m and u λ , ± i → 0 as λ → ∞ .

Proposition 2.1

(Hardy-Littlewood-Sobolev inequality [38, Theorem 4.3]) Let 1 < t , r < ∞ , and 0 < μ < N with 1 r + 1 t + μ N = 2 , u ∈ L t ( R N ) , and v ∈ L r ( R N ) . Then, there exists a sharp constant C ( N , μ , t , r ) > 0 , independent of u and v, such that

Lemma 2.1

(Diamagnetic inequality [1, Lemma 3.1, Remark 3.2]) For every u ∈ W A s , p ( R N , C ) , we obtain ∣ u ∣ ∈ W s , p ( R N ) . More precisely, we have [ ∣ u ∣ ] s ≤ [ u ] s , A .

Lemma 3.1

(See Xiang and Zhang [39]) Assume that there exist bounded non-negative measures ω , ζ , and ν on R N , and at most countable set { x i } i ∈ I ∈ Ω \ { 0 } such that

Then, there exist a countable sequence of points { x i } ⊂ R N and families of positive numbers { ν i : i ∈ I } , { ζ i : i ∈ I } , and { ω i : i ∈ I } such that

where I is at most countable. Furthermore, we have

where is the Dirac mass of mass 1 concentrated at { x i } ⊂ R N .

Lemma 3.2

(See Xiang and Zhang [39]) Let { u n } n ⊂ W s , p ( R N ) be a bounded sequence such that

and define

Then, the quantities ω ∞ , ζ ∞ , and ν ∞ are well defined and satisfy

In addition, the following inequality holds

Lemma 3.3

Assume that Conditions ( V ) and ( M ) hold. Then, any ( PS ) c sequence { u n } n for J λ is bounded in E and c ≥ 0 .

Proof

Suppose that { u n } n ⊂ E is a ( P S ) sequence for J λ . Then, we have

and

It follows from (3.6), (3.7), and ( M ) that

This fact implies that { u n } n is bounded in E . We also obtain c ≥ 0 from (3.8).□

Lemma 3.4

Assume that Conditions ( V ) and ( M ) hold. Then, J λ ( u ) satisfies ( PS ) c condition, for all s p < N < s p 2 and

Proof

Let { u n } n be a ( PS ) c sequence for J λ . Then, by Lemma 3.3, we know that the sequence { u n } n is bounded in E . Moreover, we know that there exists a subsequence, still denoted by { u n } , such that u n ⇀ u weakly in E . Moreover, we have

Now, by the concentration-compactness principle, we may assume that there exist bounded non-negative measures ω , ζ , and ν on R N , and an at most countable set { x i } i ∈ I ∈ Ω \ { 0 } such that

and

Now, there exists a countable sequence of points { x i } ⊂ R N and families of positive numbers { ν i : i ∈ I } , { ζ i : i ∈ I } , and { ω i : i ∈ I } such that

We can also obtain

In the sequel, we shall prove that

Suppose, to the contrary, that I ≠ ∅ . Then, we can define a smooth cut-off function such that ϕ ∈ C 0 ∞ ( R N ) and 0 ≤ ϕ ≤ 1 ; ϕ ≡ 1 in B ( x i , ε ) , ϕ ( x ) = 0 in R N \ B ( x i , 2 ε ) . Let ε > 0 and ϕ ε i = ϕ x − x i ε , where i ∈ I . It is not difficult to see that { u n ϕ ε i } n is bounded in E . Then, ⟨ J ′ ( u n ) , u n ϕ ε i ⟩ → 0 , which implies

where

By the Hölder inequality, we know that