The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity

Theorem 1.1.

Suppose (A0), (A1) and (f1)–(f4) hold. Then we have the following statements for sufficiently small ε>0.

1. Problem ( 1.1 ) admits a positive ground state solution w ε .

2. w ε has a global maximum point x ε such that, up to a subsequence,

   lim ε → 0 ⁡ x ε = x 0 , lim ε → 0 ⁡ dist ⁡ ( ε ⁢ y ε , ℋ 1 ) = 0 .

   Moreover, v ε ( x ) : = w ε ( ε x + x ε ) converges in H 1 ⁢ ( ℝ 3 ) to a positive ground state solution of

   { - ( a + b ⁢ ∫ ℝ 3 | ∇ ⁡ u | 2 ⁢ d ⁢ x ) ⁢ Δ ⁢ u + V ⁢ ( x 0 ) ⁢ u = K 2 ⁢ ( x 0 ) ⁢ ( ∫ ℝ 3 F ⁢ ( u ⁢ ( y ) ) | x - y | μ ⁢ d ⁢ y ) ⁢ f ⁢ ( u ) , u ∈ H 1 ⁢ ( ℝ 3 ) .

   In particular, if 𝒱 ∩ 𝒦 ≠ ∅ , then lim ε → 0 ⁡ dist ⁡ ( x ε , 𝒱 ∩ 𝒦 ) = 0 , and v ε (up to a subsequence) converges in H 1 ⁢ ( ℝ 3 ) to a positive ground state solution of

   { - ( a + b ⁢ ∫ ℝ 3 | ∇ ⁡ u | 2 ⁢ d ⁢ x ) ⁢ Δ ⁢ u + V min ⁢ u = K max 2 ⁢ ( ∫ ℝ 3 F ⁢ ( u ⁢ ( y ) ) | x - y | μ ⁢ d ⁢ y ) ⁢ f ⁢ ( u ) , u ∈ H 1 ⁢ ( ℝ 3 ) .

3. There exists C > 0 such that

   w ε ⁢ ( x ) ≤ C ⁢ e - τ ε ⁢ | x - η ε | .

Theorem 1.2.

The conclusions of Theorem 1.1 remain valid if we replace H1 and (A1) by H2 and (A2), respectively.

Proposition 2.1.

Let t,r>1 and 0<μ<N with 1t+1r+μN=2, f∈Lt⁢(RN) and h∈Lr⁢(RN). There exists a positive constant C⁢(t,r,N,μ) (independent of f and h) such that

Lemma 2.1.

Let {un} be a bounded sequence in H1⁢(R3), and it satisfies

where R>0. Then un→0 in Lt⁢(R3) for every 2<t<6.

Lemma 2.2.

Assume that (f1)–(f4) hold. Then

1. for each δ > 0 , there is C δ > 0 such that | f ⁢ ( t ) | ≤ δ ⁢ | t | + C δ ⁢ | t | q - 1 and | F ⁢ ( t ) | ≤ δ ⁢ | t | 2 + C δ ⁢ | t | q for all t ∈ ℝ ,

2. F ⁢ ( t ) > 0 and t ⁢ f ′ ⁢ ( t ) > f ⁢ ( t ) if t > 0 ,

3. f ⁢ ( t ) ⁢ t ≥ 2 ⁢ F ⁢ ( t ) > 0 for all t > 0 , and f ⁢ ( t ) ⁢ t - 2 ⁢ F ⁢ ( t ) is increasing for t > 0 ,

4. F ⁢ ( t ) t 2 is increasing for t > 0 and lim t → + ∞ ⁡ F ⁢ ( t ) t 2 = + ∞ .

Lemma 2.3.

The following statements hold for any u∈Hε∖{0}.

1. There exists a unique t u > 0 such that t u ⁢ u ∈ 𝒩 ε . Moreover, t ⁢ u ∈ 𝒩 ε if and only if t = t u .

2. There exist T 1 > T 2 > 0 independent of ε > 0 such that T 2 ≤ t ε ≤ T 1 .

Proof.

(i) Let u∈Hε∖{0} be fixed, and, for t>0, by Lemma 2.2, we have

which implies h⁢(t)>0 for small t>0. Moreover, by the above equation, we get

Hence h has a positive maximum, that is, there exists tu>0 such that h′⁢(tu)=0 and tu⁢u∈𝒩ε. It follows from h′⁢(t)=0 that

If there exists tu′>tu such that tu′⁢u, tu⁢u∈𝒩ε, we have

which is impossible in view of Lemma 2.2.

(ii) By tε⁢u∈𝒩ε, we have

So there exists T1>0 independent of ε such that T1≥tε.

By tε∈𝒩ε and Proposition 2.1, we have

which implies that there exists T2>0 independent of ε such that tε≥T2. ∎

Lemma 2.4.

The functional Iε satisfies the mountain pass geometry.

1. There exist α , ρ > 0 such that I ε ⁢ ( u ) ≥ α if ∥ u ∥ ε = ρ .

2. There exists an e ∈ H ε with ∥ e ∥ ε ≥ ρ such that I ε ⁢ ( e ) < 0 .

Proof.

(i) For any u∈Hε∖{0}, by Proposition 2.1 and Lemma 2.1, we have

Hence there exist α,ρ>0 such that Iε⁢(u)≥α with ∥u∥ε=ρ.

(ii) Similar to Lemma 2.3, for fixed u∈Hε∖{0}, we have Iε⁢(t⁢u)→-∞ as t→+∞. Thus there exists t*>0 large enough such that Iε⁢(e)<0, where e:=t*u and ∥e∥ε≥ρ. ∎

Lemma 2.5.

If {un} is a (PSc) sequence for Iε, then {un} is bounded in Hε.

Proof.

Assume that {un} is a (PSc) sequence for Iε, i.e.,

By Lemma 2.2, we obtain

The conclusion follows. ∎

Lemma 2.6.

Assume that (A0), (A1) and (f1)–(f4) are satisfied. Then

where Γ={γ∈([0,1],Hε):γ⁢(0)=0,Iε⁢(γ⁢(1))<0}.

Lemma 3.1.

Let α2≥α1>0 and β1>β2>0. Then γα1⁢β1≤γα2⁢β2. Moreover, γα1⁢β1<γα2⁢β2 if one of the inequalities is strict.

Proof.

Let u∈𝒩α2⁢β2 satisfy γα2⁢β2=Iα2⁢β2⁢(u)=maxt>0⁡Iα2⁢β2⁢(t⁢u). Define

Then we have