Home Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
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Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs

  • Daniele Cassani EMAIL logo and Lele Du
Published/Copyright: October 3, 2023
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings

W 0 s , p ( Ω ) ↪ L q ( Ω ) ,

where N ≥ 1 , 0 < s < 1 , p = 1 , 2 , 1 ≤ q < p s ∗ = N p N − s p , and Ω ⊂ R N is a bounded smooth domain or the whole space R N . Our results cover the borderline case p = 1 , the Hilbert case p = 2 , N > 2 s , and the so-called Sobolev limiting case N = 1 , s = 1 2 , and p = 2 , where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.

Keywords: fractional Sobolev spaces; best constants; fractional Laplacian; nonlocal PDEs; asymptotic analysis; variational methods
MSC 2010: 35B25; 35B33; 35J61

1 Introduction

In the study of partial differential equations via a variational approach, the first step is to find a proper function space setting in which the energy functional is well defined and smooth enough to set up equivalence between (weak) solutions to Euler-Lagrange equations and critical points of related functionals: this often yields to consider Sobolev’s spaces. Once the underlying functional setting is available, on the one hand, nonlinearities that can be handled in terms of growth at infinity and near zero, are classified by embedding properties of the function space into other spaces, typically Lebesgue spaces, interpolation spaces between Lebesgue spaces, such as Lorentz spaces, and more general rearrangement invariant spaces [57]. On the other hand, optimal constants involved in integral inequalities responsible for the function space embeddings turn out to be a kind of “DNA” building blocks that can be used to describe qualitative as well as quantitative compactness features. Moreover, borderline cases in Sobolev embeddings have deep connections with geometric measure theory [56] and conformal geometry [11] and have been intensively studied for this reason by several authors during the last 50 years.

Nevertheless, there is an aspect that has gained interest in recent years, which concerns non-borderline cases where the explicit knowledge of optimal Sobolev’s constants is out of reach but still connected to compactness properties of PDEs, as we are going to develop here.

The classical Sobolev constant S 1 , p 1 ∗ , which was obtained by Aubin [11] and Talenti [56] and explicitly given by

S 1 , p 1 ∗ = π p 2 N N − p p − 1 p − 1 Γ N p Γ 1 + N − N p Γ N 2 + 1 Γ ( N ) p N ,

appears as the optimal constant of the critical Sobolev embedding

D 1 , p ( R N ) ↪ L p 1 ∗ ( R N ) , u ∈ D 1 , p ( R N ) \ { 0 } ,

where N ≥ 2 , 1 < p < N , and for the critical Sobolev exponent p 1 ∗ = N p N − p , in the sense that S 1 , p 1 ∗ is the best possible constant of the inequality

(1.1) C ∥ u ∥ L p 1 ∗ ( R N ) p ≤ ∥ ∇ u ∥ L p ( R N ) p , u ∈ D 1 , p ( R N ) \ { 0 } .

The fact that S 1 , p 1 ∗ is explicitly known is due to the invariance property by the group action of dilations and scaling of equation (1.1). As a consequence, the Sobolev constant S 1 , p 1 ∗ retains important information in studying the lack of compactness in nonlinear problems. In particular, the energy levels at which energy functionals fail to satisfy the Palais-Smale condition are quantized in terms of multiples of this constant [55]. Furthermore, this is related to the role of threshold of the Sobolev critical exponent for the existence and nonexistence of solutions to nonlinear PDEs. In fact, the infinitesimal generator of the group invariance yields the so-called Pohozaev-type identities, and in turn, nonexistence results in fairly smooth domains. This mathematical evidences reflect geometric and physical phenomena (see [13,16,18,22,42,49,50,55] and references therein).

When p = 1 , it is well known from geometric measure theory that the Sobolev constant S 1 , p 1 ∗ is equal to the isoperimetric constant

S 1 , 1 1 ∗ = N ω N 1 N ,

where ω N is the volume of unit ball, namely

ω N = π N 2 Γ N 2 + 1 ,

see [30,32,45,46].

For the subcritical Sobolev embedding,

W 0 1 , p ( Ω ) ↪ L q ( Ω ) ,

where N ≥ 2 , 1 ≤ p ≤ N , 1 ≤ q < p 1 ∗ , and Ω ⊂ R N is a bounded smooth domain or the whole space R N , there still exist optimal constants S 1 , q in the following inequalities:

C ∥ u ∥ L q ( Ω ) p ≤ ∥ ∇ u ∥ L p ( Ω ) p , u ∈ W 0 1 , p ( Ω ) \ { 0 } ; C ∥ u ∥ L q ( R N ) p ≤ ∥ ∇ u ∥ L p ( R N ) p + ∥ u ∥ L p ( R N ) p , u ∈ W 1 , p ( R N ) \ { 0 } .

The attainability of the Sobolev constants S 1 , q is well known in the literature, but there is no hope in general to obtain their explicit value. It is a general fact that there are no explicit solutions to general nonlinear equations. However, recent applications assume some sharp growth conditions, which involve the explicit knowledge of the Sobolev constant S 1 , q , see [2,6,7,20,24,26,41,48] and also [31,36] for more applications in different contexts. It turns out that some recent approaches in nonlinear PDEs make systematic use of growth conditions which involve the best constants S 1 , q . This motivates the search for fine bounds for S 1 , q as first demonstrated by Cassani et al. [21] for the Hilbert case p = 2 and then extended by Du [28] up to the general case 1 ≤ p ≤ N .

Here, we are concerned with the fractional Sobolev embeddings

W 0 s , p ( Ω ) ↪ L q ( Ω ) ,

where N ≥ 1 , 0 < s < 1 ≤ p ≤ N s , and q satisfies

1 ≤ q ≤ p s ∗ , N > s p , Ω is bounded ; 1 ≤ q < + ∞ , N = s p , Ω is bounded ; p ≤ q ≤ p s ∗ , N > s p , Ω = R N ; p ≤ q < + ∞ , N = s p , Ω = R N

with the fractional Sobolev critical exponent

p s ∗ = N p N − s p .

For the fractional critical Sobolev embedding

D s , p ( Ω ) ↪ L p s ∗ ( Ω ) ,

there exists an optimal constant S s , p s ∗ ( Ω ) such that

S s , p s ∗ ( Ω ) ∥ u ∥ L p s ∗ ( Ω ) p ≤ [ u ] W s , p ( R N ) p , u ∈ D s , p ( Ω ) \ { 0 } ,

where [ u ] W s , p ( R N ) is the standard Gagliardo semi-norm, namely

S s , p s ∗ ( Ω ) = inf u ∈ D s , p ( Ω ) \ { 0 } [ u ] W s , p ( Ω ) p ∥ u ∥ L p s ∗ ( Ω ) p .

The invariance by scaling of the quotient S s , p s ∗ ( Ω ) implies that S s , p s ∗ ( Ω ) is independent of Ω and thus

S s , p s ∗ ( Ω ) = S s , p s ∗ ( R N ) ≕ S s , p s ∗ .

In the borderline case p = 1 , the fractional isoperimetric constant S s , 1 s ∗ was given by Brasco et al. [15], namely

S s , 1 s ∗ = ω N s − N N [ χ B 1 ] W s , 1 ( R N ) ,

where [ χ B 1 ] W s , 1 ( R N ) is the nonlocal s-perimeter of the unit ball B 1 . More precisely, the explicit value of S s , 1 s ∗ can be computed by the results of Frank and Seiringer [35], namely

S s , 1 s ∗ = ω N s N N N − s A ( N , s ) ,

where A ( N , s ) is the sharp constant of the fractional Hardy-Sobolev inequality, i.e.,

A ( N , s ) = 2 ∫ 0 1 r s − 1 ( 1 − r N − s ) A N , s ( r ) d r

and

A N , s ( r ) = ( N − 1 ) ω N − 1 ∫ − 1 1 ( 1 − t 2 ) N − 3 2 ( 1 − 2 r t + r 2 ) N + s 2 d t , N ≥ 2 , 1 ( 1 − r ) 1 + s + 1 ( 1 + r ) 1 + s , N = 1 .

The fractional isoperimetric constant S s , 1 s ∗ is achieved by a scalar multiple of the characteristic function of a ball in R N . In the Hilbert case p = 2 , Lieb [39] computed the Sobolev constant

S s , 2 s ∗ = 2 π N 2 + s s ( 1 − s ) Γ ( 2 − s ) Γ N 2 − s Γ N 2 Γ ( N ) 2 s N .

In analogy to S 1 , 2 1 ∗ , the Sobolev constant S s , 2 s ∗ appears as a key ingredient in studying the lack of compactness in fractional problems as developed by Servadei and Valdinoci [53]. The extremal functions of S s , 2 s ∗ in D s , 2 ( R N ) were obtained by Lieb [39] and up to translation and dilation, given by

U s , 2 s ∗ ( x ) = 1 1 + ∣ x ∣ 2 N − 2 s 2 , x ∈ R N ,

whereas S s , 2 s ∗ has no positive minimizer on any star-shaped domain Ω ≠ R N due to the validity of a fractional Pohozaev-type identity obtained by Ros Oton and Serra [52].

However, nothing is known for S s , p s ∗ when p ∈ ( 1 , 2 ) ∪ ( 2 , + ∞ ) . Indeed, when p ≠ 2 , the Sobolev space W s , p ( R N ) and the Bessel potential space H s , p ( R N ) are no longer equivalent, making it difficult to compute S s , p s ∗ by exploiting the sharp Hardy-Littlewood-Sobolev inequality. A lower bound for S s , p s ∗ was given by Maz’ya and Shaposhnikova [47] as follows:

S s , p s ∗ ≥ ω N N ( N − s p ) p − 1 2 ( N + 1 ) ( N + 2 ) s ( 1 − s ) p p + 2 ( N + 2 p ) 3 p .

For the fractional subcritical Sobolev embedding

W 0 s , p ( Ω ) ↪ L q ( Ω ) ,

there exist optimal constants S s , q in the following inequalities:

C ∥ u ∥ L q ( Ω ) p ≤ [ u ] W s , p ( R N ) p , u ∈ W 0 s , p ( Ω ) \ { 0 } ; C ∥ u ∥ L q ( R N ) p ≤ [ u ] W s , p ( R N ) p + ∥ u ∥ L p ( R N ) p , u ∈ W s , p ( R N ) \ { 0 } ,

namely

S s , q ( Ω ) = inf u ∈ W 0 s , p ( Ω ) \ { 0 } [ u ] W s , p ( R N ) p ∥ u ∥ L q ( Ω ) p ; S s , q ( R N ) = inf u ∈ W s , p ( R N ) \ { 0 } [ u ] W s , p ( R N ) p + ∥ u ∥ L p ( R N ) p ∥ u ∥ L q ( R N ) p .

The action of the dilation group u = u ( λ x ) for the quotient S s , q ( Ω ) yields

(1.2) S s , q ( Ω ) = λ N p 1 p s ∗ − 1 q S s , q ( Ω λ ) , N > p s ; λ − N p q S s , q ( Ω λ ) , N = p s ,

which means that S 1 , q ( Ω ) strictly depends on the domain Ω when 1 ≤ q < p s ∗ .

In particular, when p = 2 , in order to be consistent with the definition of the Sobolev constant S 1 , 2 1 ∗ , one replaces the Gagliardo semi-norm [ u ] W s , 2 ( R N ) with an equivalent L 2 -norm of the fractional Laplace operator, hence one can define the optimal constant of the following fractional Sobolev inequality:

S s ( Ω ) ∥ u ∥ L 2 s ∗ ( Ω ) 2 ≤ ( − Δ ) s 2 u L 2 ( R N ) 2 , u ∈ D s , 2 ( Ω ) \ { 0 } ,

namely

S s ( Ω ) = inf u ∈ D s , 2 ( Ω ) \ { 0 } ( − Δ ) s 2 u L 2 ( R N ) 2 ∥ u ∥ L 2 s ∗ ( Ω ) 2 .

After applying the identity

(1.3) [ u ] W s , 2 ( R N ) 2 = 2 B ( N , s ) ( − Δ ) s 2 u L 2 ( R N ) 2 ,

where

B ( N , s ) = 2 2 s s π N 2 Γ N 2 + s Γ ( 1 − s ) ,

the Sobolev constant S s ≔ S s ( Ω ) = S s ( R N ) is given as follows:

S s = 2 2 s π s Γ N 2 + s Γ N 2 − s Γ N 2 Γ ( N ) 2 s N .

Let us mention that one can also apply the dual property of Hardy-Littlewood-Sobolev inequality like Cotsiolis and Tavoularis [25] to obtain the same value. Note that S s → S 1 , 2 1 ∗ as s → 1 − , hence S s can be regarded as a generalization of the Sobolev constant S 1 , 2 1 ∗ .

Moreover, for the fractional subcritical Sobolev embedding

H 0 s ( Ω ) ↪ L q ( Ω ) ,

we also replace the definition of the optimal constant S s , q ( Ω ) in the following inequality:

C ∥ u ∥ L q ( Ω ) 2 ≤ ( − Δ ) s 2 u L 2 ( R N ) 2 , u ∈ H 0 s ( Ω ) \ { 0 } ; C ∥ u ∥ L q ( R N ) 2 ≤ ( − Δ ) s 2 u L 2 ( R N ) 2 + ∥ u ∥ L 2 ( R N ) 2 , u ∈ H s ( R N ) \ { 0 } ,

namely

S s , q ( Ω ) = inf u ∈ H 0 s ( Ω ) \ { 0 } ( − Δ ) s 2 u L 2 ( R N ) 2 ∥ u ∥ L q ( Ω ) 2 ; S s , q ( R N ) = inf u ∈ H s ( R N ) \ { 0 } ( − Δ ) s 2 u L 2 ( R N ) 2 + ∥ u ∥ L 2 ( R N ) 2 ∥ u ∥ L q ( R N ) 2 .

The Sobolev constant S s , q ( Ω ) is always achieved by means of the compact embedding

H 0 s ( Ω ) ↪ L q ( Ω ) , 1 ≤ q < 2 s ∗ ,

and S s , q ( R N ) is achieved when 2 < q < 2 s ∗ by the existence results of Frank and Lenzmann [34] for N = 1 and Dipierro et al. [27] for N ≥ 2 , whereas S s , 2 ( R N ) and S s , 2 s ∗ ( R N ) are never obtained due to the fractional Pohozaev-type identity established by Chang and Wang [23] in the whole R N .

The study of quantitative aspects of fractional Sobolev constants is interesting from the theoretical point of view. In fact, the Sobolev constant S s , p s ∗ , as in the integer case s = 1 , plays an important role in compactness issues, and the fractional critical Sobolev exponent p s ∗ yields the sharp threshold for the existence and nonexistence of solutions to nonlocal PDEs.

Likewise, for classical problems [2,6,7,20,24,26,41,48], so far there are plenty of applications [3–5,10,58] that assume sharp growth conditions involving explicit knowledge of the Sobolev constants S s , q , which turn out to be crucial to determine the existence and nonexistence of solutions to partial differential equations. Those approaches essentially extend the perturbation technique of Brézis and Nirenberg [18], in which a prescribed asymptotic behavior near 0 is assumed. Hence, looking for possibly sharp bounds of Sobolev’s constants makes such sufficient conditions effective, both from the theoretical point of view and that of applications.

1.1 Main results

1.1.1 Bounds for S s , q ( Ω ) and S s , q ( R N )

Let N ≥ 1 , 0 < s < 1 ≤ p ≤ N s , and Ω be a bounded smooth domain Ω ⊂ R N . We denote the largest radius of Ω by

R Ω = sup { R : B R ( x ) ⊂ Ω , x ∈ Ω }

and B ( x , y ) is the Beta function. Let us begin with the case p = 1 .

Theorem 1.1

Let p = 1 and 1 ≤ q < 1 s ∗ . The following conditions hold:

  1. if 1 ≤ q < 1 s ∗ , then

    S s , 1 s ∗ ∣ Ω ∣ 1 1 s ∗ − 1 q ≤ S s , q ( Ω ) ≤ S s , 1 s ∗ ∣ B R Ω ∣ 1 1 s ∗ − 1 q ;

  2. S s , 1 ( R N ) = 1 ;

  3. if 1 < q < 1 s ∗ , then

    N s 1 q − 1 1 s ∗ N s 1 1 s ∗ − 1 q N s 1 − 1 q N s 1 q − 1 S s , 1 s ∗ N s 1 − 1 q ≤ S s , q ( R N ) ≤ s N 1 q − 1 1 s ∗ N s 1 1 s ∗ − 1 q 1 − 1 q N s 1 q − 1 S s , 1 s ∗ N s 1 − 1 q .

Next, we consider the Hilbert case 2 = p < N s .

Theorem 1.2

Let 2 = p < N s . The following conditions hold:

  1. if 1 ≤ q < 2 s ∗ , then

    S s ∣ Ω ∣ 2 1 2 s ∗ − 1 q ≤ S s , q ( Ω ) ≤ 2 2 s + 2 q ( ω N N ) 1 − 2 q N + 2 s Γ 2 ( s + 1 ) ⋅ B N 2 , q s + 1 − 2 q R Ω 2 N 1 2 s ∗ − 1 q ;

  2. S s , 2 ( R N ) = 1 ;

  3. if 2 < q < 2 s ∗ , then

    N s 1 q − 1 2 s ∗ N s 1 2 s ∗ − 1 q N s 1 2 − 1 q N s 1 q − 1 2 S s N s 1 2 − 1 q ≤ S s , q ( R N ) ≤ ω N 1 − 2 q s 2 2 s + 1 − 2 s N Γ 2 ( s + 1 ) ( N + 2 s ) 1 2 − 1 q N s 1 2 − 1 q ⋅ N B N 2 , q s + 1 − 2 q B N 2 , 2 s + 1 1 q − 1 2 s ∗ N s 1 q − 1 2 s ∗ .

In the limiting case N = 2 s = 1 , a lower bound for S 1 2 , q ( R ) was given by Lieb and Loss [40], namely

S 1 2 , q ( R ) ≥ ( q − 1 ) 1 − 1 q q ( q − 2 ) 2 π 2 q − 1 .

When N = 2 s = 2 , the asymptotic behavior of S 1 , q ( Ω ) and S 1 , q ( R 2 ) was obtained by Cassani et al. [21] and Ren and Wei [51], namely

(1.4) lim q → + ∞ q S 1 , q ( Ω ) = lim q → + ∞ q S 1 , q ( R 2 ) = 8 π e .

Finally, we establish bounds for S 1 2 , q ( Ω ) and S 1 2 , q ( R N ) .

Theorem 1.3

Let s = 1 2 , p = 2 , and N = 1 . The following conditions hold:

  1. if q ≥ 1 , then

    S 1 2 , q ( Ω ) ≤ 2 1 − 2 q π e q R Ω − 2 q ;

  2. S 1 2 , 2 ( R ) = 1 ;

  3. if q > 2 , then

    S 1 2 , q ( R ) ≤ 2 1 − 4 q π 1 − 2 q q ( q − 2 ) 4 q − 2 e q − 2 q ;

  4. the asymptotic behavior of S 1 2 , q ( Ω ) and S 1 2 , q ( R N ) is given as follows:

    lim q → + ∞ q S 1 2 , q ( Ω ) = lim q → + ∞ q S 1 2 , q ( R N ) = 2 π e .

Remark 1.4

  1. In Theorem 1.1, if Ω is a ball, then S s , q ( Ω ) is achieved by a scalar multiple of the characteristic function of a ball in Ω and

    S s , q ( Ω ) = S s , 1 s ∗ ∣ Ω ∣ 1 1 s ∗ − 1 q .

  2. In Theorem 1.1, when q → 1 s ∗ , we obtain S s , q ( Ω ) → S 1 , 1 s ∗ and S s , q ( R N ) → S 1 , 1 s ∗ .

  3. When q → 2 s ∗ , the lower bound for S s , q ( Ω ) and S s , q ( R N ) in Theorem 1.2 goes to S s .

1.1.2 Applications to nonlocal PDEs

Let us look for the standing wave solutions v ( t , x ) = e i ω t u ( x ) of the following nonlocal nonlinear Schrödinger equation:

i v t = ( − Δ ) s v + ( V + ω ) v − Q ∣ v ∣ q − 2 v ,

where 0 < s ≤ 1 , N ≥ 2 s , ω ∈ R , Q , V ∈ C ( R N ) , and q satisfies

2 < q ≤ 2 s ∗ , N > 2 s ; 2 < q < + ∞ , N = 2 s ,

which yields to the following equation:

(1.5) ( − Δ ) s u + V u = Q ∣ u ∣ q − 2 u , u ∈ H s ( R N ) .

We refer to the study by Laskin [38] for the physical background. When s = 1 , if V = 1 , and Q satisfies

  1. Q ≢ 1 ;

  2. Q ≥ 1 ;

  3. lim ∣ x ∣ → + ∞ Q ( x ) = 1 ,

then Ding and Ni [26] proved that equation (1.5) has a positive solution. If Q = 1 and V satisfies

  1. V ≢ 1 ;

  2. 0 < V ≤ 1 ;

  3. lim ∣ x ∣ → + ∞ V ( x ) = 1 ,

then Lions [42] proved that equation (1.5) has a positive solution, which is actually a ground state solution. In the results of Ding and Ni [26] and Lions [42], the Sobolev constant S 1 , q ( R N ) plays a key role in proving compactness by establishing the existence of a nontrivial limit of a Palais-Smale (PS) sequence, which is a solution to the equation. More precisely, let us consider the energy functional

Q s , q ( u ) = 1 2 ∫ Ω ( − Δ ) s 2 u 2 d x − 1 q ∫ Ω ∣ u ∣ q d x

and the energy level

β s , q = 1 2 − 1 q S s , q q q − 2 ( R N ) .

As β 1 , q yields the first non-compactness level of the energy functional Q 1 , q , the value β s , q is the first level of Q s , q , where the lack of compactness occurs. We have the following theorem.

Theorem 1.5

Let 0 < s < 1 , N ≥ 2 s , and

2 < q ≤ 2 s ∗ , N > 2 s ; 2 < q < + ∞ , N = 2 s = 1 .

  1. If V = 1 and Q satisfies ( Q 1 ), ( Q 2 ), and ( Q 3 ), then equation (1.5) has a positive ground state solution with

    ∥ u ∥ H s ( R N ) 2 < S s , q q q − 2 ( R N ) ;

  2. If Q = 1 and V satisfies ( V 1 ), ( V 2 ), and ( V 3 ), then equation (1.5) has a positive ground state solution with

    ∥ u ∥ L q ( R N ) < S s , q 1 q − 2 ( R N ) .

Next, we consider the following nonlinear and nonlocal scalar field equation:

(1.6) ( − Δ ) s u + u = f ( u ) , u ∈ H s ( R N ) ,

where 0 < s ≤ 1 , N = 2 s , and f ∈ C ( R ) . Equation (1.6) is a special case of the following equation:

(1.7) ( − Δ ) s u = g ( u ) , u ∈ H s ( R N ) ,

where g ( t ) = f ( t ) − t . When s = 1 , Berestycki and Lions [13] in N ≥ 3 and Berestycki et al. [12] in N = 2 studied the constraint minimization problem related to equation (1.7) with g satisfying subcritical growth conditions and proved the existence of a minimizer, which turns out to be a positive ground state solution. After that, Alves et al. [7] established the existence of a positive ground state solution of equation (1.6) in N ≥ 2 under the assumptions that f has critical growth conditions and

f ( t ) ≥ λ t q − 1 , t ≥ 0 ,

where

λ ≥ N N 2 ( q − 2 ) 2 q S 1 , 2 1 ∗ N 2 ( N − 2 ) N − 2 2 q − 2 2 S 1 , q q 2 ( R N ) , N ≥ 3 , q − 2 q q − 2 2 S 1 , q q 2 ( R 2 ) , N = 2 .

This kind of growth conditions closely rely on the lower bound for S 1 , q ( R N ) . When 0 < s < 1 , the existence of positive ground state solutions of equation (1.7) was established by Chang and Wang [23] in N ≥ 2 and Alves et al. [3] extended the results by Alves et al. [7] to the fractional problem equation (1.6), where

λ ≥ N N 2 s ( q − 2 ) 2 s q S s N 2 s ( N − 2 s ) N − 2 s 2 s q − 2 2 S s , q q 2 ( R N ) , 0 < s < 1 , N ≥ 2 , q − 2 q q − 2 2 S s , q q 2 ( R ) , s = 1 2 , N = 1 .

We have the following existence theorem for equation (1.6).

Theorem 1.6

Let 0 < s < 1 , N = 1 , and f satisfies that for all t ∈ R , there hold

  1. lim t → 0 + f ( t ) t = 0 ;

  2. ∣ f ( t ) ∣ ≤ C e π t 2 ;

  3. f ( t ) ≥ q 2 S 1 2 , q q 2 ( R ) ∣ t ∣ q − 2 t .

Then, equation (1.6) has a positive ground state solution.

Let us stress the fact that growth condition on f as in Theorem 1.6-( f 3 ) was just theoretical up to the bound provided in Theorem 1.3 and, for instance, could not be implemented in numerical applications.

Finally, we study the following class of system of strongly coupled nonlocal and nonlinear Schrödinger equations:

(1.8) ( − Δ ) s u + u = ∣ u ∣ p − 2 u + λ v , ( − Δ ) s v + v = ∣ v ∣ q − 2 v + λ u , u , v ∈ H s ( R N ) ,

where 0 < s ≤ 1 , N > 2 s , 2 < p , q ≤ 2 s * and 0 < λ < 1 . When s = 1 , if N = 3 and λ is small, Ambrosetti et al. [9] proved that there exists multi-bump solitons in (1.8), provided 2 < p = q < 2 1 * and 0 < λ < 1 and then in Ambrosetti et al. [8] proved that equation (1.8) has a positive ground state solution. If N ≥ 3 , 2 < p , q < 2 1 * and 0 < λ < 1 , Brézis and Lieb [17] proved that equation (1.8) has a positive ground state solution.

Chen and Zou [24] established the following theorem.

Theorem 1.7

Let s = 1 , N ≥ 3 , and 0 < λ < 1 .

  1. If 2 < p < 2 1 * and q = 2 1 * , let

    α 1 = S 1 , 2 1 ∗ N 2 N 1 2 − 1 q S 1 , q q q − 2 ( R N ) 1 q q − 2 − N 2 ,

    then there exists λ 1 ∈ [ 1 − α 1 , 1 ) such that

    1. if λ < λ 1 , then equation (1.8) has no ground state solution;

    2. if λ > λ 1 , then equation (1.8) has a positive radial decreasing ground state solution.

  2. If p = q = 2 1 * , then equation (1.8) has no nontrivial solution.

The sharp classification of existence and nonexistence of solutions to equation (1.8) relies on the value of α 1 and λ 1 , which can be estimated as in [21,28]. We next extend the results of Theorem 1.7 to the fractional case where α s and λ s have explicit bounds by Theorem 1.2.

Theorem 1.8

Let 0 < s < 1 , N > 2 s , and 0 < λ < 1 .

  1. If 2 < p , q < 2 s * , then equation (1.8) has a positive ground state solution.

  2. If 2 < p < 2 s * and q = 2 s * , let

    α s = S s N 2 s N s 1 2 − 1 q S s , q q q − 2 ( R N ) 1 q q − 2 − N 2 s ,

    then there exists λ s ∈ [ 1 − α s , 1 ) such that

    1. if λ < λ s , then equation (1.8) has no ground state solution;

    2. if λ > λ s , then equation (1.8) has a positive radial decreasing ground state solution.

  3. If p = q = 2 s * , then equation (1.8) has no nontrivial solution.

2 Preliminaries

2.1 Fractional Sobolev spaces and the fractional Laplacian

Let 0 < s < 1 ≤ p < + ∞ , and the so-called Gagliardo semi-norm is given as follows:

[ u ] W s , p ( R N ) = ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s p d x d y 1 p .

The fractional Sobolev space W s , p ( R N ) is defined as the completion of C 0 ∞ ( R N ) with respect to the norm

∥ u ∥ W s , p ( R N ) = ( [ u ] W s , p ( R N ) p + ∥ u ∥ L p ( R N ) p ) 1 p ,

then W 0 s , p ( Ω ) is defined as follows:

W 0 s , p ( Ω ) ≔ { u ∈ W s , p ( R N ) , u ≡ 0 in R N \ Ω } ,

and D s , p ( Ω ) is the completion of C 0 ∞ ( Ω ) with respect to [ u ] W s , p ( R N ) .

In the Hilbert case p = 2 , let ℱ be the standard Fourier transform

ℱ u ( ξ ) = ∫ R N u ( x ) e − 2 π i ξ ⋅ x d x ,

then the fractional Laplace operator ( − Δ ) s is defined as follows

(2.1) ( − Δ ) s u = ℱ − 1 ( ∣ 2 π ξ ∣ 2 s ℱ u ) .

The Hilbert space W s , 2 ( R N ) coincides with the Bessel potential space H s ( R N ) defined via Fourier transform

H s ( R N ) ≔ u ∈ L 2 ( R N ) , ℱ − 1 ( 1 + ∣ 2 π ξ ∣ 2 s ) 1 2 ℱ u ∈ L 2 ( R N )

with the inner product

⟨ u , v ⟩ H s ( R N ) = ∫ R N ( − Δ ) s 2 u ( − Δ ) s 2 v + u v d x

and endowed with the norm

∥ u ∥ H s ( R N ) = ⟨ u , u ⟩ H s ( R N ) .

Similarly, H 0 s ( Ω ) is defined as follows:

H 0 s ( Ω ) ≔ { u ∈ H s ( R N ) , u ≡ 0 in R N \ Ω } .

Let us mention that for all u ∈ H 0 1 ( Ω ) , on the one hand, by the formula of Bourgain et al. [14], we have

lim s → 1 − ( 1 − s ) [ u ] W s , 2 ( R N ) 2 = π N 2 2 Γ N 2 + 1 ∥ ∇ u ∥ L 2 ( Ω ) 2 .

On the other hand, by the formula of Maz’ya and Shaposhnikova [47], we have

lim s → 0 + s [ u ] W s , 2 ( R N ) 2 = N π N 2 Γ N 2 + 1 ∥ u ∥ L 2 ( Ω ) 2 ,

and hence, by the identity (1.3), one has

lim s → 1 − ( − Δ ) s 2 u L 2 ( R N ) = ∥ ∇ u ∥ L 2 ( Ω ) , lim s → 0 + ( − Δ ) s 2 u L 2 ( R N ) = ∥ u ∥ L 2 ( Ω ) ,

and thus, the Fourier characterization of H 0 s ( Ω ) recovers both the norm of W 0 1 , 2 ( Ω ) and L 2 ( Ω ) .

2.2 Localizing issues

Let us briefly discuss an alternative definition of S s , 2 s ∗ , which involves the localized Gagliardo semi-norm

[ u ] W s , 2 ( Ω ) = ∫ Ω ∫ Ω ∣ u ( x ) − u ( y ) ∣ 2 ∣ x − y ∣ N + 2 s d x d y 1 2 .

Let 0 < s < 1 ≤ p < + ∞ , and the fractional Sobolev space W ˜ s , 2 ( Ω ) is defined as the completion of C 0 ∞ ( R N ) with respect to the norm

∥ u ∥ W ˜ s , 2 ( Ω ) = ( [ u ] W s , 2 ( Ω ) 2 + ∥ u ∥ L 2 ( Ω ) 2 ) 1 2 ,

then W ˜ 0 s , 2 ( Ω ) is defined as follows:

W ˜ 0 s , 2 ( Ω ) ≔ { u ∈ W ˜ s , 2 ( Ω ) , u ≡ 0 i n R N \ Ω } ,

and the space D ˜ s , 2 ( Ω ) is the completion of C 0 ∞ ( R N ) with respect to the norm [ u ] W s , 2 ( Ω ) . Obviously, we have

W ˜ s , 2 ( R N ) = W s , 2 ( R N ) .

Moreover, Brasco et al. [15] showed that W ˜ 0 s , 2 ( Ω ) and W 0 s , 2 ( Ω ) do coincide if s ≠ 1 2 .

Actually, for N > 2 s , let us consider the possible embedding

(2.2) D ˜ s , 2 ( Ω ) ↪ L 2 s ∗ ( Ω )

and denote S ˜ s , 2 s ∗ ( Ω ) as the optimal constant such that

S ˜ s , 2 s ∗ ( Ω ) ∥ u ∥ L 2 s ∗ ( Ω ) 2 ≤ [ u ] W s , 2 ( Ω ) 2 , u ∈ D ˜ s , 2 ( Ω ) \ { 0 } ,

namely

S ˜ s , 2 s ∗ ( Ω ) = inf u ∈ D ˜ s , 2 ( Ω ) \ { 0 } [ u ] W s , 2 ( Ω ) 2 ∥ u ∥ L 2 s ∗ ( Ω ) 2 .

Nevertheless, in contrast to S s , 2 s ∗ ( Ω ) , there is no scale invariance for S ˜ s , 2 s ∗ ( Ω ) , which means that S ˜ s , 2 s ∗ ( Ω ) strictly depends on Ω . Moreover, when 0 < s < 1 2 , Frank et al. [33] proved that S ˜ s , 2 s ∗ ( Ω ) = 0 , which implies that equation (2.2) fails. When N ≥ 2 and 1 2 < s < 1 , the constant S ˜ s , 2 s ∗ ( Ω ) can be achieved provided some additional conditions are assumed as done in the study by Frank et al. [33]. Furthermore, Dyda and Frank [29] showed that there exists a uniform constant

S ˜ s = inf Ω ⊂ R N S ˜ s , 2 s ∗ ( Ω )

such that for any Ω ≠ R N , there holds

S ˜ s ∥ u ∥ L 2 s ∗ ( Ω ) 2 ≤ [ u ] W s , 2 ( Ω ) 2 , u ∈ D ˜ s , 2 ( Ω ) \ { 0 } .

One possible explanation for such phenomena goes back to the results obtained by Brézis and Nirenberg [18]. Indeed, let us rewrite the norm

[ u ] W s , 2 ( Ω ) = [ u ] W s , 2 ( R N ) − 2 ∫ R N − Ω ∫ Ω ∣ u ( x ) ∣ 2 ∣ x − y ∣ N + 2 s d x d y ,

so that we see how the negative part in the right-hand side lowers the value S s , 2 s ∗ and as a consequence S ˜ s , 2 s ∗ ( Ω ) retrieves a minimizer.

3 Bounds for best constants of fractional subcritical Sobolev embeddings

In this section, we establish fine bounds for S s , q ( Ω ) and S s , q ( R N ) in the borderline case p = 1 , the Hilbert case 2 = p < N s , and the limiting case s = 1 2 , p = 2 , and N = 1 . Moreover, we also establish sharp asymptotics for the limiting case s = 1 2 , p = 2 , and N = 1 .

3.1 The borderline case p = 1 : proof of Theorem 1.1

We first prove Theorem 1.1-(1). By Hölder’s inequality, we have

S s , q ( Ω ) ≥ ∣ Ω ∣ 1 1 s ∗ − 1 q inf u ∈ W 0 s , 1 ( Ω ) \ { 0 } [ u ] W s , 1 ( R N ) ∥ u ∥ L 1 s ∗ ( Ω ) = S s , 1 s ∗ ∣ Ω ∣ 1 1 s ∗ − 1 q .

Let us take the characteristic function χ B 1 to obtain

S s , q ( B 1 ( 0 ) ) ≤ ω N 1 1 s ∗ − 1 q [ χ B 1 ] W s , 1 ( R N ) ∥ χ B 1 ∥ L 1 s ∗ ( B 1 ( 0 ) ) = ω N 1 1 s ∗ − 1 q S s , 1 s ∗ .

Next, translate the center of B R Ω into the origin and we apply the dilation group action (1.2) to obtain

S s , q ( Ω ) ≤ ω N 1 1 s ∗ − 1 q S s , 1 s ∗ R Ω N 1 1 s ∗ − 1 q = S s , 1 s ∗ ∣ B R Ω ∣ 1 1 s ∗ − 1 q .

Next, we prove Theorem 1.1-(3). For any u ∈ W s , 1 ( R N ) \ { 0 } , by interpolation inequality, we have

∥ u ∥ L q ( R N ) = ∥ u ∥ L 1 ( R N ) λ 1 ∥ u ∥ L 1 s ∗ ( R N ) 1 − λ 1 ,

where

λ 1 = N s − 1 1 s ∗ q − 1 .

By Young’s inequality, we obtain

∥ u ∥ L 1 ( R N ) λ 1 ∥ u ∥ L 1 s ∗ ( R N ) 1 − λ 1 = ( ε 1 ∥ u ∥ L 1 ( R N ) λ 1 ) 1 ε 1 ∥ u ∥ L 1 s ∗ ( R N ) 1 − λ 1 ≤ λ 1 ε 1 1 λ 1 ∥ u ∥ L 1 ( R N ) + ( 1 − λ 1 ) ε 1 1 λ 1 − 1 ∥ u ∥ L 1 s ∗ ( R N ) .

Let us choose

λ 1 ε 1 1 λ 1 = ρ 1 ; ( 1 − λ 1 ) ε 1 1 λ 1 − 1 = ρ 1 S s , 1 s ∗ ,

hence

∥ u ∥ L q ( R N ) ≤ ρ 1 ( ∥ u ∥ L 1 ( R N ) + S s , 1 s ∗ ∥ u ∥ L 1 s ∗ ( R N ) ) ,

where

ε 1 = λ 1 S s , 1 s ∗ 1 − λ 1 λ 1 ( λ 1 − 1 ) ; ρ 1 = λ 1 λ 1 S s , 1 s ∗ 1 − λ 1 λ 1 − 1 .

Therefore, we have

∥ u ∥ W s , 1 ( R N ) ≥ 1 ρ 1 ∥ u ∥ L q ( R N ) ,

and we conclude

S s , q ( R N ) ≥ N s − 1 1 s ∗ q − 1 N s − 1 1 − 1 1 ∗ q N s 1 − 1 q N s 1 q − 1 S s , 1 s ∗ N s 1 − 1 q .

Let us choose the characteristic function χ B k to have

S s , q ( R N ) ≤ ( ω N k N ) 1 1 s ∗ − 1 q [ χ B k ] W s , 1 ( R N ) ∥ χ B k ∥ L 1 s ∗ ( R N ) + ∥ χ B k ∥ L 1 ( R N ) ∥ χ B k ∥ L q ( R N ) = ω N 1 1 s ∗ − 1 q S s , 1 s ∗ k N 1 1 s ∗ − 1 q + ω N 1 − 1 q k N 1 − 1 q ≕ g 1 ( k ) .

We conclude the following:

S 1 , q ( R N ) ≤ inf k > 0 g 1 ( k ) = g 1 S s , 1 s ∗ 1 q − 1 1 s ∗ ω N s N 1 − 1 q 1 s = s N 1 q − 1 1 s ∗ N s 1 1 s ∗ − 1 q 1 − 1 q N s 1 q − 1 S s , 1 s ∗ N s 1 − 1 q .

Finally, when q = 1 , there holds

S s , 1 ( R N ) ≤ lim k → + ∞ g 1 ( k ) = 1 ,

therefore S s , 1 ( R N ) = 1 , which yields Theorem 1.1-(2).

3.2 The Hilbert case 2 = p < N s : proof of Theorem 1.2

We start by proving Theorem 1.2-(1). By Hölder’s inequality, we have

S s , q ( Ω ) ≥ ∣ Ω ∣ 2 1 2 s ∗ − 1 q inf u ∈ H 0 s ( Ω ) \ { 0 } ( − Δ ) s 2 u L 2 ( R N ) 2 ∥ u ∥ L 2 s ∗ ( Ω ) 2 = S s ∣ Ω ∣ 2 1 2 s ∗ − 1 q .

Next, we establish an upper bound for S 1 , q ( Ω ) using the classical Fourier transform of radial functions by Stein and Weiss [54] involving the standard Bessel function

J v ( t ) = t 2 v Γ v + 1 2 Γ 1 2 ∫ − 1 1 ( 1 − s 2 ) v − 1 2 e i t s d s , v > − 1 2 ; 2 π cos t t , v = − 1 2 .

Consider the radial decreasing function

f 1 ( x ) = ( k 2 − ∣ x ∣ 2 ) s , 0 ≤ ∣ x ∣ ≤ k , 0 , k ≤ ∣ x ∣ ≤ 1 ,

we have

ℱ f 1 ( ξ ) = 2 π ∣ ξ ∣ − N 2 + 1 ∫ 0 + ∞ f 1 ( r ) r N 2 J N 2 − 1 ( 2 π ∣ ξ ∣ r ) d r = π − s ∣ ξ ∣ − N 2 − s k N 2 + s Γ ( s + 1 ) J N 2 + s ( 2 π k ∣ ξ ∣ ) ,

where we apply the following identity:

∫ 0 1 ( 1 − r 2 ) s r N 2 J N 2 − 1 ( 2 π ∣ ξ ∣ k r ) d r = 2 − 1 ( π ∣ ξ ∣ k ) − s − 1 Γ ( s + 1 ) J N 2 + s ( 2 π ∣ ξ ∣ k ) .

Hence, (2.1) implies

( − Δ ) s 2 f 1 L 2 ( R N ) 2 = 2 2 s ω N N N + 2 s [ Γ ( s + 1 ) ] 2 k N + 2 s .

Moreover, from

∥ f 1 ∥ L q ( B 1 ( 0 ) ) q = ω N N 2 B N 2 , q s + 1 k N + 2 q s ,

we conclude that

S s , q ( B 1 ( 0 ) ) ≤ 2 2 s + 2 q ( ω N N ) 1 − 2 q N + 2 s [ Γ ( s + 1 ) ] 2 B N 2 , q s + 1 − 2 q k 2 N 1 2 s ∗ − 1 q .

We reach the desired result letting k → 1 . By using equation (1.2) again, we obtain

S s , q ( Ω ) ≤ 2 2 s + 2 q ( ω N N ) 1 − 2 q N + 2 s [ Γ ( s + 1 ) ] 2 B N 2 , q s + 1 − 2 q R Ω 2 N 1 2 s ∗ − 1 q .

Next, we prove Theorem 1.2-(3). For any u ∈ H s ( R N ) \ { 0 } , by interpolation inequality, we have

∥ u ∥ L q ( R N ) 2 = ∥ u ∥ L 2 ( R N ) 2 λ 2 ∥ u ∥ L 2 s ∗ ( R N ) 2 ( 1 − λ 2 ) ,

where

λ 2 = N s 1 q − 1 2 s ∗ .

By Young’s inequality, we obtain

∥ u ∥ L 2 ( R N ) 2 λ 2 ∥ u ∥ L 2 s ∗ ( R N ) 2 ( 1 − λ 2 ) = ( ε 2 ∥ u ∥ L 2 ( R N ) 2 λ 2 ) 1 ε 2 ∥ u ∥ L 2 s ∗ ( R N ) 2 ( 1 − λ 2 ) ≤ λ 2 ε 2 1 λ 2 ∥ u ∥ L 2 ( R N ) 2 + ( 1 − λ 2 ) ε 2 1 λ 2 − 1 ∥ u ∥ L 2 s ∗ ( R N ) 2 .

Let us set

λ 2 ε 2 1 λ 2 = ρ 2 ; ( 1 − λ 2 ) ε 2 1 λ 2 − 1 = ρ 2 S s ,

hence

∥ u ∥ L q ( R N ) 2 ≤ ρ 2 ( ∥ u ∥ L 2 ( R N ) 2 + S s ∥ u ∥ L 2 s ∗ ( R N ) 2 ) ,

where

ε 2 = λ 2 S s 1 − λ 2 λ 2 ( λ 2 − 1 ) ; ρ 2 = λ 2 λ 2 S s 1 − λ 2 λ 2 − 1 .

Therefore, one has

∥ u ∥ H s ( R N ) 2 ≥ 1 ρ 2 ∥ u ∥ L q ( R N ) 2 ,

from which we deduce

S s , q ( R N ) ≥ N s 1 q − 1 2 s ∗ N s 1 2 s ∗ − 1 q N s 1 2 − 1 q N s 1 q − 1 2 S s N s 1 2 − 1 q .

Let us now consider the following test function:

f 2 ( x ) = ( k 2 − ∣ x ∣ 2 ) s , 0 ≤ ∣ x ∣ < k , 0 , ∣ x ∣ ≥ k ,

and direct calculations as for f 1 give

S s , q ( R N ) ≤ ( ω N N ) 1 − 2 q 2 2 s + 2 q N + 2 s Γ 2 ( s + 1 ) B N 2 , q s + 1 − 2 q k 2 N 1 2 s ∗ − 1 q + 2 2 q − 1 B N 2 , 2 s + 1 B N 2 , q s + 1 − 2 q k N 1 − 2 q ≕ g 2 ( k ) .

We conclude that

S s , q ( R N ) ≤ inf k > 0 g 2 ( k ) = g 2 2 2 s + 1 Γ 2 ( s + 1 ) 1 q − 1 2 s ∗ ( N + 2 s ) B N 2 , 2 s + 1 1 2 − 1 q 1 2 s = 2 2 N + N s − 2 1 2 − 1 q ω N 1 − 2 q s N B N 2 , q s + 1 − 2 q ⋅ Γ 2 ( s + 1 ) ( N + 2 s ) 1 2 − 1 q N s 1 2 − 1 q B N 2 , 2 s + 1 1 q − 1 2 s ∗ N s 1 q − 1 2 s ∗ .

Finally, when q = 2 , we have

S s , 2 ( R N ) ≤ lim k → + ∞ g 2 ( k ) = 1 ,

therefore S s , 2 ( R N ) = 1 , which yields Theorem 1.2-(2).

3.3 The limiting case s = 1 2 , p = 2 and N = 1 : proof of Theorem 1.3

Let us prove Theorem 1.3-(1). Consider the following so-called Moser-type function defined in the interval [ − 1 , 1 ] :

f 3 ( x ) = ln K − ln k , 0 ≤ ∣ x ∣ ≤ k ; ln K − ln ∣ x ∣ , k ≤ ∣ x ∣ ≤ K ; 0 , K ≤ ∣ x ∣ ≤ 1 .

A direct computation gives

ℱ f 3 ( ξ ) = 1 π ξ ∫ k K sin ( 2 π ξ x ) x d x .

Hence, by equation (2.1) and Fubini’s theorem, we obtain

( − Δ ) 1 4 f 3 L 2 ( R ) 2 = 4 π ∫ k K ∫ k K 1 x y ∫ 0 + ∞ sin ( 2 π x ξ ) sin ( 2 π y ξ ) ξ d ξ d x d y = 2 π ∫ k K ∫ k K 1 x y ln x + y x − y d x d y = 2 π ∫ k K 1 y ∫ k y K y 1 t ln t + 1 t − 1 d t d y .

Note that

∫ k y K y 1 t ln t + 1 t − 1 d t ≤ ∫ 0 + ∞ 1 t ln t + 1 t − 1 d t = π 2 2 ,

and thus

( − Δ ) 1 4 f 3 L 2 ( R ) 2 ≤ π ( ln K − ln k ) .

Moreover, we have

∥ f 3 ∥ L q ( [ − 1 , 1 ] ) q ≥ 2 k ( ln K − ln k ) q ,

and hence, there holds

S 1 2 , q ( [ − 1 , 1 ] ) ≤ 2 − 2 q π k − 2 q ( ln K − ln k ) − 1 ≕ g 3 ( k , K ) .

We know that

S 1 2 , q ( B 1 ( 0 ) ) ≤ inf 0 < k < K ≤ 1 g 3 ( k , K ) = g 3 e − q 2 , 1 = 2 1 − 2 q π e q .

By equation (1.2), we have

S 1 2 , q ( Ω ) ≤ 2 1 − 2 q π e q R Ω − 2 q .

Next, we prove Theorem 1.3-(3). Let us consider

f 4 ( x ) = ln K − ln k , 0 ≤ ∣ x ∣ ≤ k ; ln K − ln ∣ x ∣ , k ≤ ∣ x ∣ ≤ K ; 0 , ∣ x ∣ ≥ K .

In a similar computation as for f 3 , we obtain

S 1 2 , q ( R ) ≤ 2 − 2 q π k − 2 q ( ln K − ln k ) − 1 + 2 k − 2 q K ≕ g 4 ( k , K ) .

We conclude that

S 1 2 , q ( R ) ≤ inf 0 < k < K g 4 ( k , K ) = g 4 2 π ( q − 2 ) 2 e q − 2 2 , 2 π ( q − 2 ) 2 = 2 1 − 4 q π 1 − 2 q q ( q − 2 ) 4 q − 2 e q − 2 q .

When q = 2 , there holds

S 1 2 , 2 ( R ) ≤ lim K → + ∞ g 4 K e − 1 K , K = 1 ,

therefore S 1 2 , 2 ( R ) = 1 , which yields Theorem 1.3-(2).

Finally, we prove Theorem 1.3-(4). The bounds we established in Theorem 1.3 do not obviously show the asymptotic behavior of S s , q ( Ω ) and S s , q ( R N ) . Actually, Ren and Wei [51] used the Trudinger-Moser inequality

∫ Ω exp 4 π u ∥ ∇ u ∥ L 2 ( Ω ) 2 d x ≤ C ∣ Ω ∣

to obtain equation (1.4). Motivated by Ren and Wei [51], we recall the fractional Trudinger-Moser inequality on the bounded domain Ω established by Martinazzi [44].

Proposition 3.1

Let N = 1 , for any u ∈ H 0 1 2 ( Ω ) and 0 < γ ≤ π , there exists a positive constant C 1 such that

sup ( − Δ ) 1 4 u L 2 ( R ) ≤ 1 ∫ Ω e γ u 2 d x ≤ C 1 ∣ Ω ∣ .

Another version of the fractional Trudinger-Moser inequality in the whole space R N was obtained by Iula et al. [37].

Proposition 3.2

For any u ∈ H 1 2 ( R ) and 0 < γ ≤ π , there exists a positive constant C 2 such that

sup ∥ u ∥ H 1 2 ( R ) ≤ 1 ∫ R e γ u 2 − 1 d x ≤ C 2 .

On the one hand, for any u ∈ H 0 1 2 ( Ω ) , we have

∥ u ∥ L q ( Ω ) q = π − q 2 ( − Δ ) 1 4 u L 2 ( R ) q ∫ Ω π u ( − Δ ) 1 4 u L 2 ( R ) 2 q 2 d x ≤ C 1 π − q 2 Γ q 2 + 1 ∣ Ω ∣ ( − Δ ) 1 4 u L 2 ( R ) q ,

where we apply Proposition 3.1 and the inequality

x t ≤ Γ ( t + 1 ) e x , x , t ≥ 0 .

Hence, we obtain

S 1 2 , q ( Ω ) ≥ C 1 − 2 q π Γ q 2 + 1 − 2 q ∣ Ω ∣ − 2 q .

By Stirling’s formula

Γ ( t + 1 ) ∼ 2 π t t + 1 2 e − t , t → + ∞

and Theorem 1.3-(1), we obtain the first asymptotic behavior

lim q → + ∞ q S 1 2 , q ( Ω ) = 2 π e .

On the other hand, let us consider the symmetric decreasing rearrangement u # . By the following fractional Pólya-Szegö inequality proved by Almgren and Lieb [1]:

(3.1) [ u # ] W 1 2 , 2 ( R N ) ≤ [ u ] W 1 2 , 2 ( R N ) ,

we can replace u by u # . Let us split the norm ∥ u # ∥ L q ( R ) q into two parts

(3.2) ∥ u # ∥ L q ( R ) q = ∥ u # ∥ L q ( ∣ x ∣ ≤ 1 ) q + ∥ u # ∥ L q ( ∣ x ∣ ≥ 1 ) q .

By Proposition 3.2, for any u ∈ H 1 2 ( R ) , we have

(3.3) ∥ u # ∥ L q ( ∣ x ∣ ≤ 1 ) q = π − q 2 ∥ u # ∥ H 1 2 ( R ) q ∫ ∣ x ∣ ≤ 1 π u # ∥ u # ∥ H 1 2 ( R ) 2 q 2 d x ≤ ( C 2 + 2 ) π − q 2 Γ q 2 + 1 ∥ u # ∥ H 1 2 ( R ) q .

Since

u # ( r ) ≤ ∥ u # ∥ L 2 ( R ) 2 r ,

one has

(3.4) ∥ u # ∥ L q ( ∣ x ∣ ≥ 1 ) q ≤ 2 2 − q 2 q − 2 ∥ u # ∥ H 1 2 ( R ) q .

Let us combine equations (3.1), (3.2), (3.3), and (3.4) to obtain

S 1 2 , q ( R ) ≥ ( C 2 + 2 ) π − q 2 Γ q 2 + 1 + 2 2 − q 2 q − 2 − 2 q .

By Stirling’s formula and Theorem 1.3-(3), we obtain the second asymptotic behavior

lim q → + ∞ q S 1 2 , q ( R ) = 2 π e .

4 Applications to nonlocal PDEs

4.1 A nonlocal nonlinear Schrödinger equation: proof of Theorem 1.5

Let us prove Theorem 1.5-(1). Define the functional E ∈ C 1 ( H s ( R N ) , R ) by

E ( u ) = 1 2 ∫ R N ( − Δ ) s 2 u 2 + ∣ u ∣ 2 d x − 1 q ∫ R N Q ∣ u + ∣ q d x ,

for which

⟨ E ′ ( u ) , v ⟩ = ⟨ u , v ⟩ H s ( R N ) − ∫ R N Q ∣ u + ∣ q − 1 v d x , v ∈ H s ( R N ) .

We claim that E satisfies the Palais-Smale condition at level c (in the sequel ( P S ) c condition) for any

(4.1) c < c ∗ = 1 2 − 1 q S s , q q q − 2 ( R N ) .

Indeed, for any sequence { u n } ⊂ H s ( R N ) such that

E ( u n ) → c ; E ′ ( u n ) → 0 ,

we have, by standard computations, that u n stays bounded in H s ( R N ) . Passing if necessary to the subsequence, we assume

u n ⇀ u , in H s ( R N ) ; u n → u , in L loc q ( R N ) ; u n → u , a.e. on R N .

Obviously u satisfies

(4.2) ( − Δ ) s u + u = Q ∣ u + ∣ q − 2 u + .

We obtain E ( u ) ≥ 0 . Let v n = u n − u . On the one hand, by Brezis-Lieb Lemma, there holds

(4.3) E ( v n ) = E ( u n ) − E ( u ) + o ( 1 ) ≤ c + o ( 1 ) .

On the other hand, by ( Q 3 ) and

⟨ E ( u ′ n ) , u n ⟩ → 0 ; ⟨ E ′ ( u ) , u ⟩ = 0 ,

we obtain

∥ v n ∥ H s ( R N ) 2 ≤ ∥ v n ∥ L q ( R N ) q + o ( 1 ) .

As n → ∞ , suppose that ∥ v n ∥ H s ( R N ) ≠ o ( 1 ) in H s ( R N ) , so that

∥ v n ∥ H s ( R N ) ≥ S s , q q 2 ( q − 2 ) ( R N ) + o ( 1 ) ,

and thus

(4.4) E ( v n ) = 1 2 − 1 q ∥ v n ∥ H s ( R N ) 2 + o ( 1 ) ≥ c ∗ + o ( 1 ) .

The inequalities (4.3) and (4.4) contradicts (4.1), therefore v n → 0 in H s ( R N ) .

For any c < c ∗ , let u s , q be a positive minimizer of S s , q ( R N ) . Since E satisfies the mountain pass geometry, by the Ekeland variational principle, there exists a sequence { u n } ⊂ H s ( R N ) such that

E ( u n ) → c 0 ; E ′ ( u n ) → 0 ,

where

c 0 = inf γ ∈ Γ max t ∈ [ 0 , 1 ] E ( γ ( t ) )

and

Γ ≔ { γ ∈ C ( [ 0 , 1 ] , H s ( R N ) ) : γ ( 0 ) = 0 , γ ( 1 ) = e } .

Note that by ( Q 1 ) and ( Q 2 ), we have

c ≤ max t ≥ 0 E ( t u s , q ) < c ∗ .

By the mountain pass theorem, there exists a nontrivial solution u of equation (4.2). Let us multiply equation (4.2) by u − and integrate by parts, we have u − = 0 , which means u is a nonnegative solution of equation (1.5), and thus positive by the maximum principle for the fractional Laplacian established by Cabré and Sire [19]. By E ( u ) < c ∗ , we conclude that

∥ u ∥ H s ( R N ) 2 < S s , q q q − 2 ( R N ) .

Next, we prove Theorem 1.5-(2). Let us define the following energy functional I ∈ C 1 ( H s ( R N ) , R ) by

I ( u ) = 1 2 ∫ R N ( − Δ ) s 2 u 2 + V ∣ u ∣ 2 d x

with

⟨ I ′ ( u ) , v ⟩ = ∫ R N ( − Δ ) s 2 u ( − Δ ) s 2 v + V u v d x .

We also define the unit sphere manifold

ℳ = { u ∈ H s ( R N ) , ∥ u ∥ L q ( R N ) = 1 }

and

I 0 = inf u ∈ ℳ I ( u ) .

By ( V 1 ) and ( V 2 ), we have

I 0 < 1 2 S s , q ( R N ) .

Let { u n } ⊂ ℳ be a minimizing sequence of I 0 , it is standard to prove that { u n } is bounded in H s ( R N ) . Passing if necessary to the subsequence, we may assume

u n ⇀ u , in H s ( R N ) ; u n → u , in L loc 2 ( R N ) ; u n → u , a.e. on R N .

Let v n = u n − u , by Brezis-Lieb Lemma, we obtain

(4.5) 1 = ∥ u n ∥ L q ( R N ) q = ∥ u ∥ L q ( R N ) q + ∥ v n ∥ L q ( R N ) q + o ( 1 ) ;

moreover, we obtain

(4.6) I ( v n ) = I ( u n ) − I ( u ) + o ( 1 ) .

Note that

(4.7) I ( u ) ≥ I 0 ∥ u ∥ L q ( R N ) 2 ,

and by ( V 3 ), we have

(4.8) I ( v n ) ≥ 1 2 S s , q ( R N ) ∥ v n ∥ L q ( R N ) 2 + o ( 1 ) .

By combining equations (4.5), (4.6), (4.7), and (4.8), we conclude that

I 0 = I ( u n ) + o ( 1 ) ≥ I 0 ∥ u ∥ L q ( R N ) 2 + S s , q 2 ( 1 − ∥ u ∥ L q ( R N ) q ) 2 q + o ( 1 ) .

This means that ∥ u ∥ L q ( R N ) = 1 and then u ∈ ℳ . By the lower semicontinuity of I , we have that u is a minimizer for I in ℳ . Finally, assume u is nonnegative, and by the Lagrange multiplier rule, we have that

u 0 = ( 2 I 0 ) 1 q − 2 u

is a positive solution of equation (1.5) by the maximum principle again, which satisfies

∥ u 0 ∥ L q ( R N ) < S s , q 1 q − 2 ( R N ) .

Finally, the positive solution obtained by equation (1.5) is also a ground state solution, see Berestycki and Lions [13].

4.2 A nonlocal scalar field equation: proof of Theorem 1.6

Define the functional J ∈ C 1 H 1 2 ( R ) , R as follows:

J ( u ) = 1 2 ∫ R ( − Δ ) 1 4 u 2 d x

and the manifold

N = u ∈ H 1 2 ( R ) \ { 0 } , 1 2 ∫ R ∣ u ∣ 2 d x = ∫ R F ( u ) d x ,

where F ( t ) = ∫ 0 t f ( s ) d s . Note that N is a C 1 manifold and nonempty by ( f 1 ). Let us denote

J 0 = inf u ∈ N J ( u )

and

U 1 2 , q = u 1 2 , q u 1 2 , q H 1 2 ( R ) ∈ H 1 2 ( R ) ,

where u 1 2 , q is a minimizer for S 1 2 , q ( R ) . By ( f 3 ), we have

∫ R F U 1 2 , q d x ≥ 1 2 > 1 2 U 1 2 , q L 2 ( R ) 2 ,

and by ( f 2 ), for any sufficiently small t 1 > 0 , we also have

∫ R F t 1 U 1 2 , q d x < 1 2 t 1 U 1 2 , q L 2 ( R ) 2 .

Hence, the continuous function

h 1 ( t ) = 1 2 t U 1 2 , q L 2 ( R ) 2 − ∫ R F t U 1 2 , q d x

satisfies

h 1 ( 0 ) = 0 ; h 1 ( t 1 ) > 0 ; h 1 ( 1 ) < 0 ,

which means that there exists t 1 ¯ ∈ ( 0 , 1 ) such that h 1 ( t 1 ¯ ) = 0 and then t 1 ¯ U 1 2 , q ∈ N . Therefore, we have

J 0 ≤ J t 1 ¯ U 1 2 , q < 1 2 .

Let { u n } ⊂ N be a minimizing sequence for J 0 and consider the radially symmetric sequence

U n = u n # ( λ ∥ u n # ∥ L 2 ( R ) 2 x ) ,

where

λ ≥ 1 1 − 2 J 0 ,

to obtain { U n } ⊂ N and

lim n → + ∞ J ( U n ) ≤ J 0 .

Therefore, { U n } is also a minimizing sequence for J 0 . Passing if necessary to a subsequence, we have

sup n ∥ U n ∥ H 1 2 ( R ) 2 ≤ sup n 2 J 0 + 1 λ + o ( 1 ) ≤ 1 .

and we may assume

U n ⇀ U , in H rad 1 2 ( R ) ; U n → U , a.e. on R .

Let

Q ( t ) = e π t 2 − 1 ,

and note that by ( f 1 ), ( f 2 ) and L’Hôpital’s rule, we have

(4.9) lim t → 0 F ( t ) Q ( t ) = lim ∣ t ∣ → + ∞ F ( t ) Q ( t ) = 0 ,

and by Proposition 3.2, we know

(4.10) sup n ∫ R ∣ Q ( U n ) ∣ d x < + ∞ .

Therefore, equations (4.9) and (4.10) together with

F ( U n ) → F ( U ) a.e. on R

satisfy the conditions of the compactness lemma of Strauss established by Berestycki and Lions [13], which gives

∫ R F ( U n ) d x → ∫ R F ( U ) d x ,

and in turn, we have

1 2 ∥ U ∥ L 2 ( R ) ≤ 1 2 liminf n → + ∞ ∥ U n ∥ L 2 ( R ) = ∫ R F ( U ) d x .

If the strict inequality holds

1 2 ∥ U ∥ L 2 ( R ) < ∫ R F ( U ) d x ,

by ( f 1 ) for sufficiently small t 2 > 0 , we obtain

∫ R F ( t 2 U ) d x < 1 2 ∥ t 2 U ∥ L 2 ( R ) 2 ,

so that the continuous function

h 2 ( t ) = 1 2 ∥ t U ∥ L 2 ( R ) 2 − ∫ R F ( t U ) d x

satisfies

h 2 ( 0 ) = 0 ; h 2 ( t 2 ) > 0 ; h 2 ( 1 ) < 0 ,

hence there exists t 2 ¯ ∈ ( 0 , 1 ) such that h 2 ( t 2 ¯ ) = 0 and then t 2 ¯ U ∈ N , which leads to

J 0 ≤ J ( t 2 U ) < liminf n → + ∞ J ( U n ) = J 0 ,

and then a contradiction. Therefore, we deduce that

1 2 ∥ U ∥ L 2 ( R ) = ∫ R F ( U ) d x ,

i.e. U ∈ N . By lower semicontinuity of J , U is a minimizer for J in N . Following the standard procedures of Berestycki and Lions [13], we also have that U is a positive ground state solution of equation (1.6).

4.3 A nonlocal Schrödinger system: proof of Theorem 1.8

Let us recall from the study by Lu and Peng [43] the following system:

(4.11) ( − Δ ) s u + u = f ( u ) + λ v , ( − Δ ) s v + v = g ( v ) + λ u , u , v ∈ H s ( R N ) ,

where 0 < s < 1 , f , g ∈ C 1 ( R ) , and 0 < λ < 1 , for which they proved the following proposition.

Proposition 4.1

Let f and g satisfy

  • lim t → 0 + f ( t ) t = lim t → 0 + g ( t ) t = 0 ;

  • there exist p , q ∈ ( 2 , 2 s ∗ ) such that

    lim ∣ t ∣ → + ∞ f ( t ) ∣ t ∣ p − 1 = lim ∣ t ∣ → + ∞ g ( t ) ∣ t ∣ q − 1 = 0 ;

  • there exist ζ 1 , ζ 2 > 0 such that

    ∫ 0 ζ 1 f ( t ) d t > ζ 1 2 2 , ∫ 0 ζ 2 g ( t ) d t > ζ 2 2 2 .

For any λ ∈ ( 0 , 1 ) , the system (4.11) has a ground state solution.

Let

f ( t ) = ∣ t ∣ p − 2 t ; g ( t ) = ∣ t ∣ q − 2 t ,

then the system (1.8) satisfies Proposition 4.1, and thus we have Theorem 1.8-(1). Next, we prove Theorem 1.8-(2). Zhen et al. [58] studied the following Schrödinger system:

(4.12) ( − Δ ) s u + α u = ∣ u ∣ 2 s ∗ − 2 u + λ v , ( − Δ ) s v + β v = ∣ v ∣ q − 2 v + λ u , u , v ∈ H s ( R N ) ,

and obtained the following result in the spirit of Theorem 1.7.

Proposition 4.2

Let 0 < s < 1 , N > 2 s , α , β > 0 , and 0 < λ < α β . Let

α s = S s N 2 s N s 1 2 − 1 q S s , q q q − 2 ( R N ) 1 q q − 2 − N 2 s ,

  1. if 0 < α ≤ α s , then equation (4.12) has a positive radial decreasing ground state solution.

  2. if α > α s , then there exists λ s ∈ [ ( α − α 0 ) β , α β ) such that

    • (I) if λ < λ s , then equation (4.12) has no ground state solution;

    • (II) if λ > λ s , then equation (4.12) has a positive radial decreasing ground state solution.

Let

τ 1 = N s 1 2 − 1 q , τ 2 = N s 1 q − 1 2 s ∗ ,

for which the following holds:

τ 1 + τ 2 = 1 .

By Theorem 1.2-(3), we have

S s , q ( R N ) ≥ τ 2 − τ 2 τ 1 − τ 1 S s τ 1 ,

hence

α s ≤ τ 2 τ 1 N 2 s − 1 q q − 2 − N 2 s < 1 .

As a consequence, equation (1.8) satisfies Proposition 4.2-(2), which proves Theorem 1.8-(2). Finally, we prove Theorem 1.8-(3). Let us use u and v as test functions in equation (1.8) to obtain

(4.13) ∫ R N ( − Δ ) s 2 u 2 d x + ∫ R N ∣ u ∣ 2 d x = ∫ R N ∣ u ∣ 2 s ∗ d x + λ ∫ R N u v d x , ∫ R N ( − Δ ) s 2 v 2 d x + ∫ R N ∣ v ∣ 2 d x = ∫ R N ∣ v ∣ 2 s ∗ d x + λ ∫ R N u v d x .

Moreover, by the Pohožaev identity established by Chang and Wang [23], we also have

(4.14) ∫ R N ( − Δ ) s 2 u 2 d x + ∫ R N ( − Δ ) s 2 v 2 d x + 2 s ∗ 2 ∫ R N ∣ u ∣ 2 d x + ∫ R N ∣ v ∣ 2 d x = ∫ R N ∣ u ∣ 2 s ∗ d x + ∫ R N ∣ v ∣ 2 s ∗ d x + 2 s ∗ λ ∫ R N u v d x .

By combining equations (4.13) and (4.14), we obtain

∫ R N ∣ u ∣ 2 d x + ∫ R N ∣ v ∣ 2 d x = 2 λ ∫ R N u v d x ,

which yields u = v = 0 by Cauchy-Schwarz inequality and the assumption 0 < λ < 1 .

  1. Conflict of interest: Prof. Daniele Cassani is a member of the editorial board. The authors declare that there is no conflict of interest.

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Received: 2023-04-12
Accepted: 2023-07-25
Published Online: 2023-10-03

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2023-0103
Keywords for this article
fractional Sobolev spaces; best constants; fractional Laplacian; nonlocal PDEs; asymptotic analysis; variational methods
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Articles in the same Issue

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
  5. Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
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