Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations

Lemma 1.1

(Hardy-Littlewood-Sobolev inequality [24]) Let α ∈ ( 0 , N ) and s ∈ ( 1 , N / α ) . Then for any Ψ ∈ L s ( R N ) , I α ∗ Ψ ∈ L N s N − α s ( R N ) , and

Remark 1.1

In this article, we use the Hardy-Littlewood-Sobolev inequality with the form

In fact, the following property of I α ensures that the aforementioned special form is true:

where s , t ∈ ( 0 , N ) , s + t < N , see [32].

Theorem 1.1

Suppose that V satisfies ( V ) . Then, for ε > 0 , problem (1.1) has at least a positive ground state solution ψ ε ( ε x ) in H ε ( R N ) under one of the following cases:

1. 3 ⩽ p < 2 ∗ − 1 , for any λ > 0 ;

2. α + 6 α + 2 < p < 3 , for λ > 0 small.

1. Let η ε ∈ R N be a global maximum point of this positive ground state solution ψ ε ( ε x ) to (1.1), then

   lim ε → 0 V ( η ε ) = V 0 .

   Meanwhile, for all ε n → 0 , ψ ε n ( ε n x + η ε n ) converges to a positive ground state solution of the following equation:

   (1.12) − Δ u + V 0 u + ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u , i n R N , u ∈ H 1 ( R N ) .

2. There exist C , κ > 0 independent of η ε satisfying

   ∣ ψ ε ( x ) ∣ ⩽ C e − κ ε ∣ x − η ε ∣ , ∀ x ∈ R N .

Theorem 1.2

Suppose that 3 ⩽ p < 2 ∗ − 1 , λ > 0 , and V satisfies ( V ) . Then for any δ > 0 , there exists ε δ > 0 such that problem (1.1) has at least  cat Θ δ ( Θ ) positive solutions in H ε ( R N ) for all ε ∈ ( 0 , ε δ ) . Also, the statements ( c ) and ( d ) of Theorem 1.1  hold true for these positive solutions.

Remark 1.2

If V ( x ) ≡ V 0 , then it is not difficult to deduce the following Pohozaev identity:

see, e.g., [30]. From (1.13), it follows that there is only trivial solution for problem (1.1) with ε = 1 , N < 4 + α and p = 2 ∗ − 1 . As for some recent results involving semiclassical ground state solutions in the critical case, for example, we refer to [13,25] and the references therein.

Remark 1.3

In [21], we obtained the existence, multiplicity, and concentration behavior of positive solutions for the following Schrödinger-Poisson problem in R 2 :

where f ∈ C ( R , R ) satisfies critical exponential growth and V ∞ < ∞ . Compared with the work in [21], we consider (1.1) in dimension R N ( N ⩾ 2 ) , and we introduce some new techniques and subtle analyses in the present article. For detailed description, we consider: (i) the case V ∞ = ∞ , this shows that we cannot obtain the positive solution of (1.1) by maximum principle; hence, the Harnack’s inequality enters into our consideration; (ii) for the Schrödinger-Poisson-Slater problem (1.1), we need a new method to prove that the solution of (1.1) is nonnegative; and (iii) we find a more straightforward approach to certify the existence of ground state solutions for the limit equation.

Lemma 2.1

The following statements hold:

1. There exists ρ > 0 such that I V 0 ∣ S ρ ( u ) > 0 , ∀ u ∈ S ρ = { u ∈ H ε ( R N ) : ‖ u ‖ V 0 = ρ } .

2. There is e ∈ H ε ( R N ) with ‖ e ‖ V 0 > ρ such that I V 0 ( e ) < 0 .

Proof

(i) By Sobolev’s embedding inequality, one has

where C > 0 is a constant. Note that 1 + p > 2 , we choose 0 < ρ < ( 2 C λ ) 1 1 − p such that ‖ u ‖ V 0 = ρ , then, ( i ) holds.

(ii) Let u t ≔ t a u ( t b x ) here and in the sequel, where a = α + 2 2 b > 0 . Then

Note that p > ( α + 6 ) / ( α + 2 ) , we have ( 1 + p ) a > 2 a + 2 b = 4 a − α b , and it follows that lim t → ∞ I V 0 ( u t ) = − ∞ . Hence, we can choose t ¯ > 0 such that e = u t ¯ ∈ { u ∈ H ε ( R N ) : ‖ u ‖ V 0 > ρ } and I V 0 ( e ) < 0 .□

Lemma 2.2

Let { u n } ⊂ H ε ( R N ) be a (PS) sequence for I V 0 , that is,

Then, going to a subsequence, there is u ∈ H ε ( R N ) such that u n ⇀ u in H ε ( R N ) , and u is a weak solution of (2.1).

Proof

Note that { u n } is a (PS) sequence for I V 0 , then

Now we prove the boundedness of { u n } in H ε ( R N ) . Observe that

Now we distinguish two cases. When p ⩾ 3 , we have

which means the boundedness of ‖ u n ‖ V 0 . When α + 6 α + 2 < p < 3 , it follows from Sobolev’s embedding inequality that

Choosing 0 < λ ∗ ⩽ ( 8 C ‖ u n ‖ V 0 p − 1 ) − 1 such that λ ∈ ( 0 , λ ∗ ] , we can found that ‖ u n ‖ V 0 is bounded. Thus,

Since for any ϕ ∈ C 0 ∞ ( R N ) ,

From (2.6), we obtain

for any ϕ ∈ C 0 ∞ ( R N ) . Thus, u is a weak solution of (2.1).□

Lemma 2.3

Problem (2.1) has a ground state solution in H ε ( R N ) .

Proof

From Lemmas 2.1 and 2.2, we obtain that there exists a (PS) sequence { u n } weakly converging to the weak solution of (2.1), and then it remains only to prove that the weak solution u is nontrivial. With the reduction to absurdity, we suppose that u ≡ 0 . Since { u n } is bounded, there are two cases need to be distinguished: { u n } is vanishing or nonvanishing.

Case (i): { u n } is vanishing. By Lion’s concentration compactness lemma (see [41, Lemma 1.21]), for any 2 ⩽ s < 2 ∗ , one has

Noticing that { u n } is a ( PS ) c V 0 sequence, applying Lemma 2.2, we have

Since u n → 0 in H ε ( R N ) , then I V 0 ( u n ) → 0 , that is, c V 0 = 0 . Impossible.

Case (ii): { u n } is nonvanishing. Thus, there exist δ , r > 0 and a sequence { y n } ∈ Z N such that

Let v n ≔ u n ( x − y n ) , we obtain

By using the fact that I V 0 and I V 0 ′ are invariant by Z N translations, one has that { v n } is also a (PS) sequence. Hence, v n ⇀ v in H ε ( R N ) . From (2.8) and v n → v in L loc 2 ( R N ) , one has v ≠ 0 . Then, v is a nontrivial critical point of I V 0 .□

Lemma 3.1

There exists α 1 > 0 independent of ε , such that

Proof

Since u ∈ N ε ,

it follows from Sobolev’s embedding inequality that

Then, ‖ u ‖ ε is bounded from below.□

Lemma 3.2

Assume that ( V ) holds. Then

where c V ∞ is defined as (2.2).

Proof

Applying Lemma 2.3, we know that there exists a ground state solution w ∈ H ε ( R N ) , that is,