Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians

Definition 1.1

For the given λ > 0 , a nontrivial solution w ∈ H s 2 ( R N ) of the problem

is called action ground state if it attains the infimum of the action functional

among all the nontrivial solutions.

Theorem 1.1

Suppose N ≥ 2 , f satisfies ( f 1 ) – ( f 3 ) , then the following results hold.

1. There exists c 0 ∈ [ 0 , ∞ ) such that

   E c = 0 i f 0 < c ≤ c 0 , E c < 0 i f c > c 0 .

2. E c ≔ inf u ∈ S c I ( u ) > − ∞ , c ↦ E c is nonincreasing and continuous.

3. When c > c 0 , the global infimum E c is achieved.

4. When 0 < c < c 0 , there no exists global minimizer, namely, E c is not achieved.

Remark 1.1

According to Theorem 1.1(iii), we obtain that any minimizing sequence of (1.6) is convergent in H s 2 ( R N ) by up to a subsequence and up to translation in R N . Meanwhile we obtain a sharp description of the existence and nonexistence of the minimizer based on Theorem 1.1. When 0 < c < c 0 , the global infimum E c is not achieved. But it does not mean that there are no positive critical points. If the v ∈ S c is energy ground state of the E c , then associated Lagrange multiplier λ = λ ( v ) is positive. In fact, according to Pohožaev identity corresponding to (1.1),

and I ( v ) = E c ≤ 0 , we obtain

and thus, λ > 0 .

Theorem 1.2

Assume that N ≥ 2 , f satisfies ( f 1 ) – ( f 3 ) , the following results hold.

1. c 0 = 0 if liminf t → 0 F ( t ) ∣ t ∣ 2 + 4 s 2 N = ∞ .

2. c 0 > 0 if limsup t → 0 F ( t ) ∣ t ∣ 2 + 4 s 2 N < ∞ .

Remark 1.2

In [11], Jeanjean and Lu obtained that there exist global minimizers by using concentration-compactness principle as presented in [15,16], and to obtain sharp threshold mass c 0 , the authors showed that E c is monotone and continuous. However, in our article, as I ( u ) is not bounded above by using the method of [11], this is challenging to obtain E c < 0 when c is large enough. Compared to [11], we make use of the cut-off function to obtain E c < 0 when c is large enough and demonstrate E c ≤ 0 by using different scaling functions which is uncomplicated.

Theorem 1.3

Assume that N ≥ 2 , f satisfies ( f 1 ) – ( f 3 ) , and c > 0 . If { u n } is a minimizing sequence with respect to E c , one of the following states holds.

1. lim n → ∞ sup z ∈ R N ∫ B ( z , 1 ) ∣ u n ∣ 2 d x = 0 .

2. Taking a subsequence if necessary, there exist u ∈ S c and a family { y n } ⊂ R N satisfying u n ( ⋅ − y n ) → u in H s 2 ( R N ) . In particular, u is energy ground state.

Theorem 1.4

Assume that N ≥ 2 , f satisfies ( f 1 ) – ( f 3 ) , then the following results hold.

1. If limsup t → 0 F ( t ) t 2 + 4 s 2 N ≤ 0 , then the global infimum E c 0 is achieved.

2. If there exists positive constants C and t 1 such that F ( t ) = C ∣ t ∣ 2 + 4 s 2 N if ∣ t ∣ ≤ t 1 , then E c 0 is achieved.

Remark 1.3

First, to prove the convergence of minimizing sequence, the splitting property lim n → ∞ I ( u n + v ) = lim n → ∞ I ( u n ) + I ( v ) plays an important role. Different from [22], due to the existence of fractional Laplacian, it is challenging to obtain splitting result by using the method of [22]. Fortunately, we adopt the idea given by Luo and Hajaiej in [17] to obtain the aforementioned splitting result.

Theorem 1.5

Assume that N ≥ 2 , f satisfies ( f 1 ) – ( f 3 ) , λ is the Lagrange multiplier corresponding to an arbitrary minimizer v ∈ S c of (1.6), then the following results hold.

1. Any minimizer v ∈ S c of (1.6) is action ground state. In particular,

   J λ ( w ) = E c + 1 2 λ c .

2. Any action ground state w ∈ H s 2 ( R N ) is minimizer of (1.6), and the following results hold

   ∣ w ∣ 2 2 = c , I ( w ) = E c ,

   where λ ∈ { λ ( v ) ∣ v ∈ S c is a m i n i m i z e r o f ( 1.6 ) } .

Remark 1.4

Different from [9] where Dovetta et al. showed that energy ground state is action ground state, which depending on the function t ↦ f ( t ) ⁄ t is nondecreasing on ( 0 , ∞ ) , we here make no monotonic hypothesis.

Lemma 2.1

[1] Assume that { u n } is bounded in L p ( R N ) , u ∈ L p ( R N ) and u n → u a.e. R N , then

Lemma 2.2

[17] Assume that { u n } ⊂ H s ( R N ) and u ∈ H s ( R N ) , being such that

then

Lemma 2.3

Suppose that N ≥ 2 , λ > 0 , and f satisfies ( f 1 ) and ( f 2 ) , and w ∈ H s 2 ( R N ) is nontrivial critical point of J λ . Then, for arbitrary δ > 0 and M > 0 , there exist a constant T = T ( w , δ , M ) > 0 and a continuous path ζ : [ 0 , T ] → H s 2 ( R N ) such that

1. ζ ( 0 ) = 0 , J λ ( ζ ( T ) ) < − 1 , max t ∈ [ 0 , T ] J λ ( ζ ( t ) ) = J λ ( w ) ;

2. ζ ( θ ) = w for some θ ∈ ( 0 , T ) , and

   J λ ( ζ ( t ) ) < J λ ( w )

   for any t ∈ [ 0 , T ] satisfying ‖ ζ ( t ) − w ‖ H s 2 ( R N ) ≥ δ ;

3. c ( t ) ≔ ∣ ζ ( t ) ∣ 2 2 is continuous and strictly increasing with c ( T ) > M .

Proof

Case 1. N ≥ 3 . Define

Note that c ( t ) = t N ∣ w ∣ 2 2 , by Pohožaev identity (1.9), we obtain

and

It is clear that the function J λ ( ζ ( t ) ) has a unique maximum at t = 1 and d d t J λ ( ζ ( t ) ) → − ∞ as t → ∞ . Therefore, for any M > 0 , there exists a sufficiently large constant T = T ( w , δ , M ) > 0 such that the continuous path ζ : [ 0 , T ] → H s 2 ( R N ) satisfies (i)–(iii) of Lemma 2.3.

Case 2. N = 2 . For the given ω ∈ H s 2 ( R 2 ) , we define g : ( θ , s ) ∈ ( 0 , + ∞ ) × ( 0 , + ∞ ) → R as follows:

then

It follows from Nehari and Pohožaev identity (1.9) that

thus

and

Define H : H s 2 ( R 2 ) → R as follows:

it is clear that

and there exist two positive constant 0 < θ 1 < 1 < θ 2 such that for any θ ∈ [ θ 1 , θ 2 ] ,

and

By the definition of H ,