Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth

Theorem 1.1

Let N ≥ 2 + 4 α α − γ + ( γ + = max { γ , 0 } ) , and assume that ( g ) , ( V 1 ) , ( V 2 ) , and ( Q ) hold. Then, there is ε * > 0 such that problem (1.1) has at least k positive solutions u ε i , i = 1 , 2 , … , k , for 0 < ε < ε * , and each u ε i possesses a maximum point y ε i ∈ R N satisfying V ( ε y ε i ) → V 0 and Q ( ε y ε i ) → Q 0 as ε → 0 . Moreover, there exist constants C i , c i > 0 such that

for ε ∈ ( 0 , ε * ) and x ∈ R N .

Remark 2.1

Since we shall find the positive solutions to problem (1.1), we can rewrite the functional J ε in (2.1) as follows:

where v + ≔ max { v , 0 } . If v ∈ E is a nontrivial critical point of J ε and let v − = min { v , 0 } , then we can obtain

which implies that v − = 0 . From the theory of elliptic regularity, we see that v ∈ C 2 ( R N ) . Then, by using strong maximum principle, we obtain v > 0 in R N .

Lemma 2.2

The functions g , G , and G − 1 have the following properties:

1. G ( ⋅ ) and G − 1 ( ⋅ ) are strictly increasing and odd;

2. G ( s ) ≤ g ( s ) s ≤ α G ( s ) for all s ≥ 0 ; α G ( s ) ≤ g ( s ) s ≤ G ( s ) for all s ≤ 0 ;

3. g ( G − 1 ( s ) ) ≥ g ( 0 ) = 1 for all s ∈ R ;

4. G − 1 ( s ) s is decreasing on ( 0 , + ∞ ) and increasing on ( − ∞ , 0 ) ;

5. ∣ G − 1 ( s ) ∣ ≤ 1 g ( 0 ) ∣ s ∣ = ∣ s ∣ for all s ∈ R ;

6. ∣ G − 1 ( s ) ∣ g ( G − 1 ( s ) ) ≤ 1 g 2 ( 0 ) ∣ s ∣ = ∣ s ∣ for all s ∈ R ;

7. G − 1 ( s ) s g ( G − 1 ( s ) ) ≤ ∣ G − 1 ( s ) ∣ 2 ≤ α G − 1 ( s ) s g ( G − 1 ( s ) ) for all s ∈ R ;

8. lim ∣ s ∣ → 0 G − 1 ( s ) s = 1 g ( 0 ) = 1 and

   lim ∣ s ∣ → ∞ G − 1 ( s ) s = 1 g ( ∞ ) , if g is b o u n d e d , 0 , if g is u n b o u n d e d ;

9. ∣ s ∣ α ≤ α β ∣ G ( s ) ∣ for all s ∈ R ;

10. ( G − 1 ( s ) ) q s g ( G − 1 ( s ) ) is increasing on ( 0 , + ∞ ) for q > 2 α − 1 ;

11. lim s → + ∞ G ( s ) s α = β α .

Proof

For the proof of (1)–(9), we refer to [35] and [27]. For (10), let l ( s ) = ( G − 1 ( s ) ) q s g ( G − 1 ( s ) ) , s > 0 . Using ( α − 1 ) g ( s ) ≥ g ′ ( s ) s and (2), we have

As for ( 11 ) , noting γ < α ,

Lemma 2.3

The functions k i , ε ( x , s ) and K i , ε ( x , s ) , i = 1 , 2 , have the following properties:

1. lim ∣ s ∣ → 0 k i , ε ( x , s ) s = 0 and lim ∣ s ∣ → 0 K i , ε ( x , s ) s 2 = 0 uniformly in x ∈ R N .

2. lim ∣ s ∣ → ∞ k i , ε ( x , s ) ∣ s ∣ 2 * − 1 = 0 and lim ∣ s ∣ → ∞ K i , ε ( x , s ) ∣ s ∣ 2 * = 0 uniformly in x ∈ R N .

Proof

Similar to the proof of Lemma 2.2 in [24], we can obtain the results.□

Lemma 2.4

Given v ∈ E with v + ≠ 0 , there exists a unique t v > 0 such that t v v ∈ N ε . Moreover, J ε ( t v v ) = max t ⩾ 0 J ε ( t v ) .

Proof

Using Lemma 2.3, for any δ > 0 , there exists C δ > 0 such that

Therefore, for t > 0 ,

When δ is small enough, we can obtain h ( t ) > 0 for small t . On the other hand, since v + ≠ 0 , there exists γ > 0 such that m ( Ω ) > 0 , where Ω ≔ { x ∈ R N : v + ( x ) ≥ γ } . Using (5) and ( 11 ) of Lemma 2.2, for t large enough, one has

It follows that h ( t ) → − ∞ as t → + ∞ .

Therefore, there exists a t v > 0 such that h ′ ( t v ) = 0 , that is, t v v ∈ N ε . Furthermore, h ( t v ) = max t ⩾ 0 h ( t ) . From h ′ ( t ) = 0 , we have

In terms of (1), (4), and (10) of Lemma 2.2, we know that

are increasing with respect to s ∈ ( 0 , + ∞ ) . Then, the uniqueness of t v can be obtained.□

Lemma 2.5

There is a positive constant θ independent of ε , such that ‖ v ‖ ε ≥ θ , v ∈ N ε .