Weighted Fractional Sobolev Inequality in ℝ^N

1 Introduction

In this paper, we investigate the minimizing problem

which is associated with fractional Sobolev–Hardy inequalities with the weight 1|y|α:

( ∫ ℝ N | u ⁢ ( x ) | 2 s , α * | y | α ⁢ 𝑑 x ) 2 2 s , α * ≤ Λ s , N , k , α - 1 ⁢ ∫ ℝ N | ( - Δ ) s 2 ⁢ u ⁢ ( x ) | 2 ⁢ 𝑑 x ,

where x=(y,z)∈ℝk×ℝN-k, 0<α<2⁢s, N>2⁢s and 2s,α*=2⁢(N-α)N-2⁢s. We refer to the weight 1|y|α as partial weight. The space H˙s⁢(ℝN) is defined as the completion of C0∞⁢(ℝN) under the norm

∥ u ∥ H ˙ s ⁢ ( ℝ N ) = ( ∫ ℝ N | ξ | 2 ⁢ s ⁢ | u ^ ⁢ ( ξ ) | 2 ⁢ 𝑑 ξ ) 1 2 ,

where u^ denotes the Fourier transform of u. We define the operator (-Δ)s2, s∈ℝ, by the Fourier transform

( ( - Δ ) s 2 ⁢ u ⁢ ( ξ ) ) ^ = | ξ | s ⁢ u ^ ⁢ ( ξ )

for u∈C0∞⁢(ℝN). Therefore, for s>0, we have

∥ ( - Δ ) s 2 ⁢ u ∥ L 2 ⁢ ( ℝ N ) 2 = ∫ ℝ N | ξ | 2 ⁢ s ⁢ | u ^ ⁢ ( ξ ) | 2 ⁢ 𝑑 ξ .

It is well known that a minimizer u of Λs,N,k,α weakly solves the problem

( - Δ ) s ⁢ u = | u | 2 s , α * - 2 ⁢ u | y | α   in ⁢ ℝ N

up to a multiplying constant. The constant Λs,N,k,α is closely related to various best constants of Sobolev type embedding. Indeed, if s=1 and α=0, the constant Λs,N,k,α is reduced to the best Sobolev constant

S = inf u ∈ D 1 , 2 ⁢ ( ℝ N ) , u ≢ 0 ⁡ ∫ ℝ N | ∇ ⁡ u ⁢ ( x ) | 2 ⁢ 𝑑 x ( ∫ ℝ N | u ⁢ ( x ) | 2 * ⁢ 𝑑 x ) 2 2 * ,

where 2*=2⁢NN-2, N≥3 and D1,2⁢(ℝN) is the closure of C0∞⁢(ℝN) under the norm

∥ u ∥ D 1 , 2 ⁢ ( ℝ N ) = ( ∫ ℝ N | ∇ ⁡ u | 2 ⁢ 𝑑 x ) 1 2 .

It was shown in [2, 22] that S is achieved in ℝN by a family of functions

U λ ⁢ ( x ) = ( λ λ 2 + | x - x 0 | 2 ) N - 2 2

for λ>0. Such functions play an important role in studying the existence of solutions of critical elliptic problems, see [6].

If s=1, α≠0, then the constant Λs,N,k,α becomes the best Sobolev–Hardy constant

where 2α*=2⁢(N-α)N-2, N≥3, 2≤k≤N. The extremal problem (1.2) is attained by a cylindrically symmetric decreasing function, see [3, 19] for details. A minimizer of SN,k,α weakly solves the problem

up to a multiplying constant. Equation (1.3) is related to a model describing the dynamics of elliptic galaxies, see [4, 9]. Existence and symmetry of solutions were studied in [3, 19]. In the case α=1, all positive finite energy solutions of (1.3) were classified in [18]. Such solutions are given by

V ⁢ ( y , z ) = ( ( N - 2 ) ⁢ ( k - 1 ) ) N - 2 2 ( ( 1 + | y | ) 2 + | z | 2 ) N - 2 2

or its scaling and translations in the z-variable.

For N>2⁢s, the fractional Sobolev embedding H˙s⁢(ℝN)↪L2⁢NN-2⁢s⁢(ℝN) was considered in [8, 10]. The continuity of this inclusion corresponds to the inequality

with 2s*=2⁢NN-2⁢s. The best constant Λs in (1.4) was computed in [10]. Using the moving plane method for integral equations, Chen, Li and Ou classified in [8] the solutions of an integral equation related to the problem

They showed that positive regular solutions of (1.5) are precisely given by

for λ>0 and x0∈ℝN.

Recently, the second author showed in [25] that in the case k=N, the minimizing problem (1.1) is achieved by a positive, radially symmetric and strictly decreasing function.

In this paper, we investigate minimization problem (1.1). We first show that Λs,N,k,α is achieved, then we consider properties of minimizers, including symmetry and decay at infinity.

We commence with the existence of minimizers of problem (1.1). At the beginning, we justify that the problem is well defined in the sense that Λs,N,k,α>0. In the proof of this fact, we need to establish a fractional Hardy inequality with partial weights, or precisely the following result.

Inequality (1.7) is a fractional counterpart of Hardy’s inequality with partial weight. The classical Hardy’s inequality states that the inequality

holds for u∈C0∞⁢(ℝN) if 1≤p<N and for u∈C0∞⁢(ℝN\{0}) if p>N. The constant on the right-hand side of (1.9) is sharp and for p>1 this constant is not attained in the corresponding homogeneous Sobolev spaces W˙1,p⁢(ℝN) and W˙1,p⁢(ℝN\{0}), respectively, which are the completion of C0∞⁢(ℝN) and C0∞⁢(ℝN\{0}) with respect to the norm on the left-hand side of (1.9). Similarly, in [13], Frank and Seiringer proved a fractional analog of Hardy’s inequality (1.9):

with 𝒞N,s,p given by (1.8), for u∈W˙s,p⁢(ℝN) if 1≤p<Ns and for u∈W˙s,p⁢(ℝN\{0}) if p>Ns. The best constant 𝒞N,s,p was shown in [24]. Inequality (1.10) is the particular case k=N of Theorem 1.1.

Now, we consider the existence of minimizers of problem (1.1). In general, this can be done by using the concentration-compactness principle in [16, 17]. Technically, the method involves an estimate of the commutator of smooth functions and fractional Laplacian if problem (1.1) is considered. Another way to deal with the problem is given in [20, 25] by a refinement of Sobolev–Hardy inequalities in terms of Morrey space ℒp,γ (see Section 3 for its definition). Observing that Lévy concentration functions are special cases of norms of Morrey spaces, it is then possible to avoid directly using the concentration-compactness principle. For this purpose, we establish a refinement of Sobolev–Hardy inequalities with partial weight, which is of interest itself and stated as follows.

Using inequality (1.11), we show that every minimizing sequence of Λs,N,k,α is compact up to translations and dilations. Hence, Λs,N,k,α is achieved. The result is as follows.

It seems difficult to give a precise form as (1.6) for minimizers of Λs,N,k,α. In many cases, special properties of minimizers are adequate for further applications. Particularly, besides it is important itself, symmetric and decay property of minimizers are significant in studying existence of solutions for problems involving critical Sobolev–Hardy terms. Our next result concerns the symmetry of minimizers. We show that minimizers of Λs,N,k,α are cylindrically symmetric.

Theorem 1.4 is proved by the argument in [19]. It requires an equivalent partial weighted Hardy–Littlewood–Sobolev inequality, which is established in Section 5. This inequality, combined with a weighted Riesz’s rearrangement inequality, enables us to prove Theorem 1.4.

Finally, we establish the following decaying laws for the minimizers.

In proving Theorem 1.5, we consider the related extension problem. Hence, we show a Harnack inequality by Nash–Moser iteration. An application of the Harnack inequality to the Kelvin transform of the minimizer yields the result.

The paper is organized as follows. In Section 2, we establish a sharp fractional Hardy inequality with partial weight. Then, we establish the improved Sobolev–Hardy inequality in Section 3. Theorem 1.3 and Theorem 1.4 are proved in Sections 4 and 5, respectively. Finally, in Section 6, we establish the decaying law for the minimizer u of Λs,N,k,α.

2 Fractional Hardy Inequality with Partial Weight

In this section, we show the Hardy type inequality with the fractional Laplacian, that is, we will prove Theorem 1.1. The following result was shown in [13].

Suppose N≥2⁢s, k≥1, 0<s<1 and denote x=(y,z)∈ℝk×ℝN-k, x′=(y′,z′)∈ℝk×ℝN-k. We have a counterpart of Lemma 2.1 for partial Hardy weights.

Now, inequality (1.7) follows by the approach of Frank and Seiringer [13]. Their proof relies on a ground state substitution, i.e., it relies on writing u⁢(x)=ω⁢(x)⁢v⁢(x), where ω⁢(x)=|x|-(N-p⁢s)/p is a solution of the Euler–Lagrange equation corresponding to (1.10).

3 Improved Sobolev–Hardy Inequality

In this section, we prove inequality (1.11). For each s≥0, we recall that the fractional Sobolev space H˙s⁢(ℝN) is defined by

H ˙ s ⁢ ( ℝ N ) = { u : | ξ | s ⁢ u ^ ⁢ ( ξ ) ∈ L 2 ⁢ ( ℝ N ) }

via the Fourier transform

u ^ ⁢ ( ξ ) = 1 ( 2 ⁢ π ) N 2 ⁢ ∫ ℝ N e - i ⁢ x ⋅ ξ ⁢ u ⁢ ( x ) ⁢ 𝑑 x .

For s∈(0,1), it is known from [11] that there holds

where

C s , N = 2 - 2 ⁢ s ⁢ π - s ⁢ Γ ⁢ ( N - 2 ⁢ s 2 ) Γ ⁢ ( N + 2 ⁢ s 2 ) ⁢ [ Γ ⁢ ( N ) Γ ⁢ ( N 2 ) ] 2 ⁢ s N .

This provides an alternative norm

∥ u ∥ H ˙ s ⁢ ( ℝ N ) = ( ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) 1 2

of H˙s⁢(ℝN). Equation (3.1) and Theorem 1.1 imply, for N>2⁢s, the fractional partial Hardy inequality

where u∈H˙s⁢(ℝN). Let

L 2 s , α * ⁢ ( ℝ N , | y | - α ) = { u : ∫ ℝ N | u | 2 s , α * | y | α ⁢ 𝑑 x < ∞ }

with 2s,α*=2⁢(N-α)N-2⁢s. We claim that Λs,N,k,α defined in (1.1) is positive.

Now, we recall the definition of Morrey space. A measurable function u:ℝN→ℝ belongs to the Morrey space ℒp,γ⁢(ℝN) with p∈[1,∞) and γ∈(0,N], if and only if

∥ u ∥ ℒ p , γ ⁢ ( ℝ N ) p = sup R > 0 , x ∈ ℝ N ⁡ R γ - N ⁢ ∫ B R ⁢ ( x ) | u ⁢ ( x ) | p ⁢ 𝑑 y < ∞ .

By Hölder’s inequality, we may verify (as in [11]) that

L 2 s * ⁢ ( ℝ N ) ↪ ℒ p , N - 2 ⁢ s 2 ⁢ p ⁢ ( ℝ N )   for ⁢ 1 ≤ p < 2 s *

and

ℒ p , N - 2 ⁢ s 2 ⁢ p ⁢ ( ℝ N ) ↪ ℒ q , N - 2 ⁢ s 2 ⁢ q ⁢ ( ℝ N )   for ⁢ 1 < q < p < 2 s * .

Moreover, we have ℒp,γ⁢(ℝN)↪ℒ1,γp⁢(ℝN) provided that p∈(1,∞) and γ∈(0,N).

Inequality (1.11) will be proved by the weighted estimates on Riesz potential in [21]. For any u∈H˙s⁢(ℝN), let g∈L2⁢(ℝN) be a function such that g^⁢(ξ)=|ξ|s⁢u^⁢(ξ). By Plancherel’s theorem we have

∥ u ∥ H ˙ s ⁢ ( ℝ N ) = ∥ ( - Δ ) s 2 ⁢ u ∥ L 2 ⁢ ( ℝ N ) = ∥ g ∥ L 2 ⁢ ( ℝ N ) .

Thus,

u ⁢ ( x ) = ( 1 | ξ | s ) ∨ * g ⁢ ( x ) = ℐ s ⁢ g ⁢ ( x ) ,

where ℐs⁢g denotes the Riesz potential of order s, namely

ℐ s ⁢ g ⁢ ( x ) = ∫ ℝ N g ⁢ ( y ) | x - y | N - s ⁢ 𝑑 y .

The following result was established in [21].