Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

Qi Han

doi:10.1515/anona-2020-0133

Open Access Published by De Gruyter September 4, 2021

Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

Qi Han

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0133

Abstract

In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ)^s in ℝⁿ, for n ≥ 2, such as

(0.1) (−Δ)su+E(x)u+V(x)uq−1=K(x)f(u)+u2s⋆−1.

Here, s ∈ (0, 1), q∈2,2s⋆ with 2s⋆:=2nn−2s being the fractional critical Sobolev exponent, E(x), K(x), V(x) > 0 : ℝⁿ → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f (u) is a continuous function on ℝ of quasi-critical, super-q-linear growth with f (u) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝⁿ such as

(0.2) (−Δ)su+E(x)u+V(x)uq−1=λK(x)ur−1,

where 2 < r < q < ∞(both possibly (super-)critical), E(x), K(x), V(x) > 0 : ℝⁿ → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces M^s^;q,p(ℝⁿ) as well as their associated compact embedding results.

Keywords: Sobolev-Slobodeckij spaces; compact embedding; fractional Laplacian; positive solutions

MSC 2010: Primary 46E35; 35R11; Secondary 35J20; 35P05

1 Introduction

In this paper, we are interested in seeking a positive solution to

(1.1) (−Δ)su+E(x)u+V(x)uq−1=K(x)f(u)+u2∗−1

in ℝⁿ when n ≥ 2. Here, and henceforth, (−Δ)^s denotes the fractional Laplace operator that, up to a normalization constant and for 0 < s < 1, is defined to be

(1.2) (−Δ)su(x):=−12∫Rnu(x+y)+u(x−y)−2u(x)|y|n+2sdy,

2≤q<2s⋆ with 2s⋆:=2nn−2s the fractional critical Sobolev index, while E(x), K(x), V(x) > 0 : ℝⁿ → ℝ and f (u) : ℝ → ℝ are functions satisfying the conditions below.

(EKV1) K(x)∈L∞Rn,infBRV(x)≥VBR>0 for each R > 0, while for all c > 0 and some ϑ∈q,2s⋆,L({x∈ Rn:K(x)E−2s∗−92s∗−2−τ2s∗−q2s∗−2(x)V−τ(x)≥c<∞withτ∈0,2s∗−ϑ2s∗−q.

(f1) f (u) is continuous and satisfies limu→0+f(u)uq−1=limu→∞f(u)u2s∗−1=0.

(f2) f (u) ≥ A_ϑu^ϑ⁻¹ ≥ 0 when u ≥ 0, Aϑ>max2,2θ2sϑn(9−2)∫RnKv09dxSn2s2−92 > 0 is a constant, v₀ > 0 is a mountain pass solution to

(1.3) (−Δ)su+E(x)u+V(x)uq−1=K(x)uϑ−1,

and 𝒮 represents the best fractional critical Sobolev embedding constant.

(f3) There is some θ∈q,2s⋆ such that, for F(u):=∫0uf(v)dv with u, v ≥ 0,

(1.4) K(x)(uf(u)−θF(u))≥−η1K1(x)ur1−η2K2(x)ur2,

where r1∈1,2s⋆ and r₂ ∈ [1, 2), K₁(x), K₂(x) > 0 : ℝⁿ → ℝ are functions satisfying K12s⋆2s∗−r1(x)∈ L¹(ℝⁿ) and K22s∗+κ2s⋆−q2s∗−r2(x)V−κ(x)∈L1Rn with κ∈0,r2q−r2, while 0≤η1≤2⋅2s⋆r1sθ3n2s∗−r1Sn2s2s∗−r12s∗⋅2s∗−θr12s⋆6r1CK1 and 0≤η2≤min4r2sθ3n2−r2Sn2s2−r22⋅(θ−2)r226r2CK2,qr2sθ9nq−r2Sn2sq−r2q⋅(θ−q)r2qr2CK2 are some constants with CK1:= ∫RnK12s∗2s∗−r1(x)dx2s∗−r12s⋆>0 and C _K2 > 0 a constant that depends on some embedding constants (see Section 5 for details).

Note the hypothesis A_ϑ > 2 in condition (f2) guarantees that

(1.5) 0≤supt≥0t22+tqq−Aϑtθϑ=sup0≤t≤1t22+tqq−A9tθϑ≤supt≥0t2−Aϑtϑϑ.

It may be helpful to demonstrate f (u) using an example. For ε > 0 small, set

f(u):={c1(q+ε)uq+ε−1+c22s∗u2s∗−1ln(1+u)+2s∗u2s∗ln(1+u)−u2s∗(1+u)ln2(1+u)≥0ifu≥00otherwise

for appropriate constants c₁, c₂ > 0 to observe F(u)={c1uq+ε+c2u2s∗ln(1+u)ifu≥00otherwise. So, one derives

uf(u)−θF(u)=−c1(θ−q−ε)uq+ε+c22s⋆−θu2∗ln(1+u)−c2u2s∗+1(1+u)ln2(1+u)

for u ≥ 0. When u is small, one has ln(1 + u) u and thus

uf(u)−θF(u)≥−c1(θ−q−ε)uq+ε+c22s⋆−θ−1u2s∗−1≥−η2ur2

for some small η₂ > 0 if 2s⋆−1≥q+ε , since q ≥ 2 > r₂ ≥ 1; when u is large, one has

uf(u)−θF(u)≥−c1(θ−q−ε)uq+ε+c22s⋆−θ−1u2s∗−ε≥−η1ur1

for some small η₁ > 0 if 2s⋆−q≥ max{1, 2ε}, provided that q+ε< r1<2S⋆ in the case where 2s⋆−θ−1≥0 and 2s⋆−ε< r1<2s⋆ in the case where 2s∗−θ−1<0. As a consequence, our condition (1.4) is fulfilled upon the introduction of K₁(x), K₂(x) ≥ K(x) > 0.

Since the seminal work by Brézis and Nirenberg [17], there has been a lot of research on the existence of positive solutions to elliptic problems involving the critical Sobolev exponent; see also the renowned papers of Ambrosetti, García-Azorero and Peral [5], and Benci and Cerami [12]. Problem (1.1) is motivated by the recent paper of do Ó, Miyagaki and Squassina [26], but our approach applies the insightful idea from García-Azorero and Peral [27] using the profound concentration-compactness principle of Lions [35]; see Palatucci and Pisante [39], and Bonder, Saintier and Silva [14] in the nonlocal setting. The authors in [26] used the ingenious approach of Caffarelli and Silvestre [18] which transfers a fractional Laplacian problem on ℝⁿ to a locally possibly degenerate Laplacian problem on ℝⁿ × ℝ⁺ via the s-harmonic extension; the relations of K (x), V(x) in [26] depend essentially on those of Alves and Souto [2], but ours are different. Finally, our condition (1.4) utilizes the embedding results and we have

uf(u)−θF(u)≥−η1K1(x)K−1(x)ur1−η2K2(x)K−1(x)ur2;

noting that the right-hand side term is nonpositive, (1.4) is indeed weaker than the well-known condition of Ambrosetti-Rabinowitz that demands uf (u) ≥ θF(u) > 0.

Now, our first main result can be formulated as follows.

Theorem 1.1

Suppose n ≥ 2, 0 < s < 1, 2 ≤ q<2s∗,E(x), K(x), V(x) > 0 : ℝⁿ → ℝ satisfy condition (EKV1), and f (u) : ℝ → ℝ satisfies conditions (f1)-(f3). Then, equation (1.1) has a positive solution.

Below, we proceed to discuss the existence of multiple positive solutions to

(1.6) (−Δ)su+E(x)u+V(x)uq−1=λK(x)ur−1,

where 2 < r < q < ∞ both possibly being (super-)critical, E(x), K(x), V(x) > 0 : ℝⁿ → ℝ are functions satisfying the subsequent conditions, and λ > 0 is a parameter.

(EKV2-1) 2< r<min{2s∗,q},K(x)∈Lloc∞,infBRV(x)≥VBR>0 for each R > 0, while for some β ∈ [0,∞) and κ∈β(r−2)q−r+r−2q−r⋅n2s,β(r−2)q−r+rq−r,Kψ2s⋆(β,κ)(x)E−β(x)V−κ(x)∈L1Rn with ψ2s⋆∗(β,κ):=2s∗+β2s∗−2+κ2s∗−q2s∗−r. (EKV2-2) 2s⋆≤r< q,K(x)∈Lloc∞Rn,infBRV(x)≥VBR>0 for each R > 0, while for some β ∈ [0,∞) and κ∈β(r−2)q−r+rq−r,β(r−2)q−r+r−2q−r⋅n2s,Kψ2s∗(β,κ)(x)E−β(x)V−k(x)∈L1Rn.

(EK) 2< r<min2s⋆,q,infBRE(x)≥EBR>0 for each R > 0, K(x) ∈ L^∞(Ω) for every subset Ω⊊Rn with 𝔏(Ω) < ∞, and Lx∈Rn:K(x)E−2s∗−r2s∗−2(x)≥c<∞ for all c > 0.

Since the celebrated paper of Ambrosetti, Brézis and Cerami [3] on the concave and convex model problem, there has been a lot of studies on this topic. In 1996, Alama and Tarantello [1] considered an elliptic problem of different type that is the origin of all the research on equation (1.6). It seems to the author that Chabrowski [20] was the first person extending a result in [1] to ℝⁿ; ever since, other important results appeared such as the renowned papers of Pucci and Rădulescu [42, 44] where for the first time the multiplicity results were discussed; Autuori and Pucci [8] later extended this research to the nonlocal setting with the same equation as (1.6) here (see also the closely related paper [7]). In this sense, our study continues their work, as we further utilize the compact embedding results for the functions E (x), K (x), V(x), not merely those for K(x) and/or V(x). See [31] for more details and references.

Of course, the authors in [7, 8] used Hardy’s potential for E(x); here, we shall fully employ E (x) by inscribing it into the compact embedding results with K(x), V(x), which include the one where β=0,κ=rq−r and ψ2s⋆0,rq−r=qq−r in [7, 8] as a special case. Other two closely related important work are from Pucci and Zhang [43], and Rădulescu, Xiang and Zhang [45], where a weaker condition corresponding to β = 0, κ=r−2q−r⋅n2sandψ2s⋆0,r−2q−r⋅n2s=q−2q−r⋅n2s was used. Notice both conditions come from those in [1]. In addition, our condition (EK) here is broader than any of those appeared in the associated earlier work.

Now, our second main result can be formulated as follows.

Theorem 1.2

Assume that n ≥ 2, 0 < s < 1, 2 < r < q < ∞ both possibly being (super-)critical, and E(x), K(x), V(x) > 0 : ℝⁿ → ℝ satisfy conditions (EKV2-1) or (EKV2-2). Then, there is a λ₁ ≥ 0 such that equation (1.6) has at least a positive solution for all λ > λ₁. In addition to condition (EKV2-1), when condition (EK) is true, then λ₁ > 0 and equation (1.6) has at least a positive solution if and only if λ ≥ λ₁; moreover, there exists a λ₂(≥ λ₁ > 0) such that equation (1.6) has at least two distinct positive solutions for all λ > λ₂.

Note in problems (1.1) and (1.6), we are interested in the situation where E(x), K(x), V(x) vanish at infinity, as initiated by Ambrosetti, Felli and Malchiodi in their celebrated work [4].

The paper is organized as follows: Section 2 analyzes a family of general fractional Sobolev-Slobodeckij spaces M^s^;q,p(ℝⁿ), Section 3 describes a version of the fractional Moser-Trudinger inequality when sp = n, Section 4 discusses some compact embedding results that are particularly useful for our studies here of equations (1.1) and (1.2), while Sections 5 and 6 are devoted to the detailed proofs of Theorems 1.1 and 1.2 respectively.

An excellent reference for the approaches involved in this paper is the recent monograph of Papageorgiou, Rădulescu and Repovš [40] for nonlinear analysis in large.

It is worthwhile to remark finally that our conditions fully utilize the (compact) embedding results both in the upper-triangle case and the lower-triangle case (see Section 4 for definition) in each individual problem, which seems new even in the Laplacian setting.

2 Function space analysis

Recently, nonlocal equations and the associated fractional Sobolev function spaces W^s^,p(ℝⁿ) attracted great attentions due to their impressive applications to real-world problems in various disciplines as briefed, for example, in the introduction from Di Nezza, Palatucci and Valdinoci [23]; see also Demengel and Demengel [22, Chapter 4], or Molica Bisci, Rădulescu and Servadei [37, Chapters 1-2] for some concise, self-contained treatment of this matter.

In his classical monograph [36, Section 5.1.1], Maz’ya defined a class of general Sobolev function spaces Wp,q1Rn; using the symbol M^q^,p(ℝⁿ), the author analyzed some relations among those spaces and their compact embedding results in [28, 29, 30, 31] hoping to provide a complement to Lemma I.1 of Lions [34] in the function space setting.

In the desire of furnishing a bit more insight regarding the similarity between W^1,p(ℝⁿ) and W^s^,p(ℝⁿ) as seen in [22, 23, 37], Sections 2-4 here are devoted to the description of a family of general fractional Sobolev-Slobodeckij spaces M^s^;q,p(ℝⁿ) when n ≥ 2, their relations and their compact embedding results that are akin to those observed for M^q^,p(ℝⁿ).

In the sequel, we always assume n ≥ 2, 1 ≤ p, q ≤ ∞and 0 < s < 1.

Denote by Ω a bounded domain in ℝⁿ having a compact, Lipschitz boundary ∂Ω, or Ω = ℝⁿ. Define M^s^;q,p(Ω) to be the Banach space as the completion of the set C1(Ω‾)ifΩ is bounded, or as that of the set CC1RnforΩ=Rn, with respect to the norm

(2.1) ∥u∥Ms;q,p(Ω):=[u]S,p,Ω+∥u∥q,Ω,

where [u]s,p,Ωp:=∫Ω∫Ω|u(x)−u(y)|p|x−y|n+spdxdyand∥u∥q,Ωq:=∫Ω|u|qdx.

Note Ws,p(Ω)=Ms;p,p(Ω),andMs;q,∞(Ω)=C0,s(Ω) independent of q. From now on, we shall write both continuous embedding of function spaces and convergence of functions by “→”, compact embedding of function spaces by “↦”, and weak convergence of functions by “⇀”. Other notations will be clarified when appropriate.

Below, we will embark on our detailed analysis for the spaces M^s^;q,p(Ω).

2.1 Ω is a bounded domain having a compact, Lipschitz boundary ∂Ω

In this case, it is trivial to see M^s^;q,p(Ω) → L¹(Ω); so, via the fractional Poincaré’s inequality of Bellido and Mora-Corral [11, Lemma 3.1], we have M^s^;q,p(Ω) → W^s^,p(Ω) through Minkowski’s inequality. This, together with [22, Corollary 4.53], leads to the conclusions as follows.

(1) When sp < n, we have, with ps⋆:=npn−sp,

(2.2) Ms;q,p(Ω)=Ws,p(Ω) for 1≤q≤ps⋆,Ms;q1,p(Ω)→Ms;q2,p(Ω) for ps⋆≤q2≤q1≤∞.

Also, there exists a constant C_p_,q > 0, depending on n, p, q, s, Ω, such that

(2.3) ∥u∥Ws,p(Ω)≤Cp,q∥u∥Ms;q,p(Ω),∀u∈Ms;q,p(Ω)andq∈[1,∞].

(2) When sp = n, we have

(2.4) Ms;∞,p(Ω)→Ms;q,p(Ω)=Ws,p(Ω)for1≤q<∞,

and M^s^;q,p(Ω) in general cannot be a proper subspace of L^∞(Ω) unless q = ∞. The fractional Moser-Trudinger inequality of Parini and Ruf [41] says

(2.5) sup[u]s,p,Rn≤1∫Ωeα|u|nn−sdx≤CΩ,∀u∈W˜0s,p(Ω) and α∈0,α⋆,

Here, α_*, C Ω > 0 are constants of n, p, s, Ω, and W˜0S,p(Ω), as a proper subspace of W^s^,p(Ω), is the completion of CC1(Ω) with respect to the Gagliardo semi-norm [u]_s_,p,^Rn which is equivalent to (2.1) on ℝⁿ for CC1(Ω) by Brasco, Lindgren and Parini [16, Remark 2.5].

(3) When sp > n, we have, with γs⋆:=sp−np,

(2.6) Ms;q,p(Ω)=Ws,p(Ω)→C0,γ(Ω) for 0≤γ≤γs⋆ and 1≤q≤∞.

Next, by [22, Theorem 4.54] or [23, Theorem 7.1], one easily verifies the result below.

Proposition 2.1

Assume n ≥ 2,1 ≤ p, q ≤ ∞ and 0 < s < 1. When sp < n, the embedding M^s^;q,p(Ω) → L^r(Ω) is continuous if 1≤r≤maxps⋆,q and compact if 1≤r<maxps⋆,q . When sp ≥ n, the embedding M^s^;q,p(Ω) ➨ L^r(Ω) is compact if 1 ≤ r < ∞.

Proof. Viewing (2.2)-(2.4) and (2.6), we only need to consider the case sp < n following almost identically the proofs of [22, Theorem 4.54] or [23, Theorem 7.1]. Actually, noticing that Ω is an extension domain, for any cube Q ( ⊋ Ω ) and each u ∈ M^s^;q,p(Ω), denote the extension of u to W^s^,p(ℝⁿ) by u˜; then, for some constants CΩ′,Cp,q′>0 depending on n, p, q, s, Ω,

∥u˜∥Ws,p(Q)≤∥u˜∥Ws,pRn≤CΩ′∥u∥Ws,p(Ω)≤Cp,q′∥u∥Ms;q,p(Ω)

holds. So, our proof is finished using [22, Theorem 4.54] or [23, Theorem 7.1].

Finally, recall for sp > n, in view of (2.6), one actually has

Ms;q,p(Ω)↔C0,γ(Ω) for 0≤γ<γs⋆ and 1≤q≤∞.

On the other hand, from our preceding analysis and the remark at page 197 of [22], we have an equivalent definition of the space M^s^;q,p(Ω), when p, q ≠ ∞, such as

Ms;q,p(Ω):=u:Ω→R measurable with [u]s,p,Ω+∥u∥q,Ω<∞.

2.2 Ω ⁼ ℝⁿ

Recall M^s^;q,p(ℝⁿ) denotes the Banach space as the completion of the set Cc1Rn in terms of the norm (2.1). Now, following Lieb and Loss [33, Sections 3.2 and 4.3], a function u∈Lloc1Rn is said to vanish at infinity provided 𝔏({x ∈ ℝⁿ : |u(x)| ≥ c}) < ∞ for all positive constants c > 0, where 𝔏 represents the Lebesgue measure.

(1) When sp < n, denote by D^s^,p(ℝⁿ) the space of functions u∈Lloc1Rn where u vanish at infinity and [u]_s_,p,^Rn < ∞. Then, a careful check for [23, Lemma 6.3 and Theorem 6.5] implies that u∈Lps⋆Rn, and the fractional Sobolev inequality reads as follows

(2.7) ∫Rn|u|ps∗dxn−spn≤C1∫Rn∫Rn|u(x)−u(y)|p|X−y|n+spdxdy,∀u∈Ds,pRn.

Here, C₁ > 0 is an absolute constant depending on n, p, s for 1 ≤ p < ∞.

By virtue of Dipierro and Valdinoci [24] with a = 0, D^s^,p(ℝⁿ) can be equivalently defined to be the completion of Cc1Rn through [u]s,p,Rn; so, Ds,pRn=Ms;ps⋆,pRn and one recognizes an equivalent definition of the space M^s^;q,p(ℝⁿ), for q ≠ ∞, such as

Ms;q,pRn:=u:Rn→R measurable with [u]s,p,Rn+∥u∥q,Rn<∞.

From Chebyshev’s inequality and the standard interpolation inequality, one has

(2.8) either 1≤q1≤q2≤ps⋆Ms;q1,pRn→Ms;q2,pRn→Ds,pRnfor or ps⋆≤q2≤q1<∞.

If one would like to include q = ∞, then u needs to be compactly supported.

The best fractional critical Sobolev embedding constant 𝒮_p is defined to be

(2.9) Sp:=infu∈Ds,pRn∖{0}[u]s,p,Rnp∥u∥ps⋆,Rnp≥C1−1>0.

As common practice, we write D^s(ℝⁿ) := D^s^,2(ℝⁿ) and 𝒮 := 𝒮₂ in the sequel.

(2) When sp = n, the fractional Gagliardo-Nirenberg inequality of Nguyen and Squassina [38, Lemma 2.1] says

(2.10) ∥u∥r,Rn≤C2∥u∥q,Rnς[u]s,p,Rn1−ς,∀u∈Ms;q,pRn.

Here, we only write a tailored version for our purpose with 1 ≤ q ≤ r < ∞and ς:=qr∈ (0, 1], and C₂ > 0 is an absolute constant depending on n, p, q, r, s. Thus, we have

(2.11) Ms;q1,pRn→Ms;q2,pRn for 1≤q1≤q2<∞.

Notice here q = ∞does not contribute to this continuous embedding (2.11).

(3) When sp > n, a careful check for Theorem 8.2 in [23], especially their estimates (8.3) and (8.9), suggests that Ms;q,pRn→C0,γs∗Rn→L∞Rn. Therefore, one has

(2.12) Ms;q1,pRn→Ms;q2,pRn→C0,γs∗Rn for 1≤q1≤q2≤∞.

Note M^s^;q,∞(ℝⁿ)→ C^0,s(ℝⁿ) which is consistent with the observation at page 565 of [23], and M^s^;∞,p(ℝⁿ) is the largest possible function space for fixed p, s in (2.12).

Summarizing all the preceding analysis yields the result below.

Proposition 2.2

Assume n ≥ 2,1 ≤ p, q < ∞and0 < s < 1.When sp < n, the embedding M^s^;q,p(ℝⁿ)→ L^r(ℝⁿ) is continuous if minps⋆,q≤r≤maxps⋆,q. When sp = n or sp > n, the embedding M^s^;q,p(ℝⁿ)→ L^r(ℝⁿ) is continuous if q ≤ r < ∞or q ≤ r ≤ ∞.

3 A fractional Moser-Trudinger inequality

In this section, we utilize [41] to provide an extension of the inequality (2.5) to ℝⁿ for functions in M ^s^;q,p(ℝⁿ) when sp = n. The proof presented here follows essentially from do Ó [25, Lemma 1] and Ruf [46, Proposition 2.1]. Consult also Li and Ruf [32], Han [30, Section 2.2], and several other important contributions in ℝⁿ that cannot be exhaustively listed here.

Below, we review some fundamental facts about Schwarz symmetrization based on Berestycki and Lions [13, Appendix III], and Lieb and Loss [33, Section 3.3]. Assume f (x) is a measurable function vanishing at infinity in the measure-theoretic sense of Lieb and Loss, and let f^*(x) ≥ 0 be its Schwarz symmetrization or spherical rearrangement through Lχ∈Rn:f⋆(x)≥c=Lx∈Rn:|f(x)|≥c for all positive constants c > 0. Then, f^* is unique, radial, decreasing in |x|, and lower semi-continuous (and therefore, measurable). Also, we have

(3.1) ∫RnΦf⋆dx=∫RnΦ(|f|)dx

for all continuous functions Φ with Φ(|f |) integrable; so, f⋆q,Rn=∥f∥q,Rniff∈LqRnfor1≤q≤∞. We recall the important Theorem 3 of Beckner [10] and observe

(3.2) ∫Rn∫Rnf⋆(x)−f⋆(y)p|X−y|n+spdxdy≤∫Rn∫Rn|f(x)−f(y)|p|X−y|n+spdxdy,

while moreover the Berestycki-Lions’ radial lemma [13, Lemma A.IV] leads to

(3.3) f⋆(x)≤|x|−nqnωn−11qf⋆q,Rn,∀f⋆∈LqRn,

with x ≠0, where ω_n₋₁ represents the surface area of the unit sphere in ℝⁿ.

It is notable that neither in [13, Appendix III] or [34, Lemma I.1], nor in [33, Section 3.3] or [10, Theorem 3] was the condition f being integrable needed.

Now, we characterize an inequality of fractional Moser-Trudinger type in the function space M^s^;q,p(ℝⁿ) using its full norm, instead of the Gagliardo semi-norm [u]s,p,Rn.

Theorem 3.1

Let n ≥ 2, 1 ≤ p, q < ∞, 0 < s < 1, sp = n, 0 ≤ α < α_*, and u ∈ M^s^;q,p(ℝⁿ). Then, we have

(3.4) sup∥u∥Ms;q,p(Rn≤1∫RnΨα,q,s(|u|)dx≤C(α,q,s).

Here, C(α, q, s) > 0 is an absolute constant that depends on α, n, p, q, s, and for v ≥ 0 and the least positive integer l₀ with l0nn−s≥q≥1, we set Ψα,q,s(v):=∑l=l0∞αll!vlnn−s.

Proof. To save notation, without loss of generality, take u ≥ 0. Recalling (3.1) and (3.2), and noticing that Ψα,q,su⋆≥0 for u^*, we may simply consider u = u^* and decompose

∫RnΨα,q,s(u)dx=∫BR0Ψα,q,s(u)dx+∫BR0cΨα,q,s(u)dx.

Here, and hereafter, BR⊊Rn is the ball of radius R centered at the origin and BRc:=Rn∖BR, while R₀ > 0 is a sufficiently large absolute constant to be determined later.

To estimate the integral over BR0, write v(x) := max{u(x) − u₀, 0} for u₀ := u(R₀) > 0, a constant. Then, v≡0onBR0c and v∈W˜0s,pBR0. In fact, as in [16, Lemma 2.4], for the ball BR0x0⊊BR0c of radius R₀ centered at a point x₀ with |x₀| = 4R₀, one has

vp(x)=|v(x)−v(y)|p|X−y|2n|x−y|2n for x∈BR0 and y∈BR0x0,

which further yields

ωn−1R0nnvp(x)≤diamBR0x0∪BR02n∫BR0x0|v(x)−v(y)|p|χ−y|2ndy.

Thus, it follows that

ω n − 1 n 36 R 0 n ∫ B R 0 v p ( x ) d x ≤ ∫ B R 0 B R 0 ∫ x 0 | v ( x ) − v ( y ) | p | x − y | 2 n d y d x ≤ ∫ B R 0 ∫ B R 0 | v ( x ) − v ( y ) | p | x − y | 2 n d x d y + 2 ∫ B R 0 ∫ B R 0 c | v ( x ) − v ( y ) | p | x − y | 2 n d x d y = ∫ B R 0 ∫ B R 0 | u ( x ) − u ( y ) | p | x − y | 2 n d x d y + 2 ∫ B R 0 ∫ B R 0 c 0 u ( x ) − u 0 p | X − y | 2 n d x d y ;

in view of u(y)≤u0onBR0c, we have

ωn−1n36R0n∥v∥p,BR0p≤[v]S,p,Rnp≤[u]S,p,Rnp≤1.

Recall [25, Estimate (5)] says (1+t)nn−s≤tnn−s+C3tsn−s+1 for all t ∈ (0,∞) with C₃ > 0 an absolute constant. Actually, for

f(t):=(1+t)nn−s−tnn−s−1tsn−s

continuous on (0,∞), one observes limt→0+f(t)=0 as n≥2>2s and lim→∞f(t)=nn−sthrought˜:=1t. Hence, for t=v(x)u0, it follows that

unn−s≤v+u0nn−s≤vnn−s+C3vsn−su0+u0nn−s.

Apply Young’s inequality to C3vsn−su0 to get, for an absolute constant C3′>0,

(3.5) unn−s≤vnn−s1+u0ns+C3′+u0nn−s.

Set u˜:=v1+u0nsn−sn∈W˜0s,pBR0 and recall [v]S,p,Rnp≤[u]S,p,Rnp to see

[u˜]s,p,Rnp≤1+u0nsn−ss[u]s,p,Rnp≤1+u0nsn−ss1−∥u∥q,Rnns.

From (3.3), one has u0ns≤σ∥u∥q,Rnns for σ:=nωn−1R0nnqs. Define

g(t):=1+σtnsn−ss(1−t)ns−1

with g(0) = 0 and g(1) = −1, and note

g′(t)=nS1+σtnsn−2ss(1−t)n−ssn−ssσtn−ss−nsσtns−1.

Write

h(t):=n−sSσtn−ss−nSσtns−1

with h′t0=0att0:=n−sn2. Thus, when R₀ is so large that σ≤nn−S2n−ss, one deduces h(t) ≤ 0 and hence g_′(t) ≤ 0 on [0, 1]. That is, g(t) ≤ 0 for 0 ≤ t ≤ 1, which clearly implies that [u˜]s,p,Rn≤1 provided R0n≥nωn−1n−snq(2n−s)n.

Thus, one may combine the preceding analysis with (2.5) and (3.5) to derive

∫BR0Ψα,q,s(u)dx≤∫BR0eαunn−sdx≤eαC3′+αu0nn−s∫BR0eαu˜nn−sdx≤C1(α,q,s).

Here, C₁(α, q, s) > 0 is an absolute constant depending on α, n, p, q, s.

To estimate the integral over BR0C, we use (3.3) with ∥u∥Ms;q,pRn≤1 to see

∫ B R 0 c u ln n − s d x ≤ ω n − 1 n ω n − 1 ln q ( n − s ) ∫ R 0 ∞ t − ln 2 q ( n − s ) + n − 1 d t ≤ R 0 n ω n − 1 n ω n − 1 R 0 n l n q ( n − s ) q ( n − s ) ln 2 − n q ( n − s ) = R 0 n ω n − 1 σ 1 s n − s q ( n − s ) ln 2 − n q ( n − s ) .

As l0nn−S≥q, we apply Proposition 2.2 to the first term of Ψ_α_,q,s(u) and observe

∫BR0cΨα,q,s(u)dx≤αl0l0!∫BR0cul0nn−sdx+q(n−s)n2R0nωn−1eασsn−s≤C2(α,q,s).

Here, C₂(α, q, s) > 0 is an absolute constant depending on α, n, p, q, s.

Therefore, (3.4) follows with C(α, q, s) := C₁(α, q, s) + C₂(α, q, s) > 0.

It is worth to remark that Parini and Ruf [41] found an upper bound

αs,n⋆:=n2ωn−12Γ(p+1)n!∑k=0∞(n+k−1)!(n+2k)pk!sn−s

for α_* utilizing renowned functions in analytic number theory with sophisticated calculations. Following Proposition 5.2 of [41], one observes a similar result as follows.

Proposition 3.2

Assume that n ≥ 2, 1 ≤ p, q < ∞, 0 < s < 1, sp = n, and αs,n⋆<α<∞. Then, we have

(3.6) sup∥u∥Ms;q,pRn≤1∫RnΨα,q,s(|u|)dx=∞.

Proof. Define the nonlocal/modified Moser function u_ε by |lnε|1−snif|x|≤ε,|ln|x||⋅|lnε|−snifε≤|x|≤1, and 0 if |x| ≥ 1. Then, limε→0+uεS,p,Rnp=αs,n⋆nn−ss as shown in Proposition 5.1 of [41] through delicate computations, and uεq,Rnq=O|lnε|−sqn which is infinitesimal when ε→0+. Set accordingly Vε:=uεuεMs;q,p(Rn≥0 with vεMs;q,pRn=1. When α>αs,n⋆ and ε is sufficiently small, αuεMs;q,pRn−nn−s≐αuεS,p,Rn−nn−s≐αnαs,n⋆> n. So, one derives

∫RnΨα,q,s(vε)dx≥∫BεΨα,q,s(vε)dx=∫Bεeα∥uε∥Ms;q,p(Rn)−nn−s|lnε|dx−∑l=0l0−1αll!∫Bε∥uε∥Ms;q,p(Rn)−lnn−s|lnε|ldx≥ωn−1n(εn−αnαs,n∗−O(εn|lnε)|l0)→∞asε→0+,

which certainly implies the desired estimate (3.6) and thus finishes the proof.

Not like in the local case, it is not clear if αS,n⋆ is optimal, yet, as pointed out in Remark 5.4 of [41], at least αs.2⋆ coincides with the optimal exponent α1,2⋆ up to the multiplicative constant appeared in the asymptotic behaviour of the Gagliardo semi-norm as s → 1⁻.

4 Some compact embedding results

In this section, we describe some compact embedding results from subspaces of M^s^;q,p(ℝⁿ) or D^s^,p(ℝⁿ) to LKrRn that are useful when studying problems (1.1) and (1.6).

Some closely related results on^1,p(ℝⁿ)may be found in Bartsch and Wang [9, Theorem2.1], Chiappinelli [21, Theorem 1], Schneider [47, Theorem 2.3], Bonheure and Van Schaftingen [15, Theorem 4], and Han [30, Section 4.1] and [31, Propositions 2.3-2.4]; note nevertheless the results presented below seem not known in the literature, even on W1,pRn.

Let n≥2,1≤p,q<∞,1≤p˜≤ps⋆, and 0 < s < 1 with sp < n. Suppose E(x), V(x) > 0 with infBRV(x)≥ VBR>0 for each R > 0 in the sequel. Denote by ME,Vs;q,pRn the Banach space as the completion of functions u∈Lloc1(Rn) that vanish at infinity (measure-theoretically) with [u]s,p,Rn<∞, or equivalently as that of u∈Cc1Rn, with respect to the norm

(4.1) ∥u∥ME,vs;a,pRn:=[u]s,p,Rn+∥u∥LEp˜Rn+∥u∥LVqRn

for ∥u∥LEp˜Rnp˜:=∫RnE|u|p˜dx and ∥u∥LVqRnq:=∫RnV|u|qdx;so,ME,Vs;q,pRn→Ds,pRn.

Theorem 4.1

Assume that n ≥ 2,1 ≤ p < ∞,0 < s < 1with sp < n, and1≤p˜,q≤r< ps⋆. Let E(x), K(x), V(x) > 0 : ℝⁿ → ℝ be Lebesgue measurable functions such that K(x) ∈ L^t(Ω) for some t∈ps⋆ps⋆−r,∞ on each set Ω of ℝⁿ having L(Ω)<∞ while K(x)E−ϕps⋆(τ)(x)V−τ(x) vanishes at infinity with ϕps⋆(τ):=ps∗−rps∗−p˜−τps∗−qps∗−p˜ for some τ∈0,ps⋆−rps∗−q; so, ϕps⋆(τ)∈0,ps⋆−rps∗−p˜. Then, the embedding ME,Vs;q,pRn→LKrRn is compact.

Proof. Without loss of generality, assume {u_l : l ≥ 1} is a sequence of functions in ME.Vs;q,pRn with u_l ⇀ 0 when l → ∞and ulME.Vs;q,pRn uniformly bounded. For each ε > 0, setWε:=x∈Rn:K(x)E−ϕps⋆(τ)(x)V−τ(x)≥ ε} andWεC:=Rn∖Wε to decompose

∫RnKulrdx=∫WεKulrdx+∫WεcKulrdx.

For the integral overWec, noting that ps⋆=r−p˜ϕps⋆(τ)−qτ1−ϕps⋆(τ)−τ, it is easy to get

(4.2) ∫WεcK|ul|rdx=∫WεcKE−ϕps∗(τ)V−τ(E|ul)|p˜ϕps∗(τ)(V|ul)|qτ|ul|r−p˜ϕps∗(τ)−qτdx≤ε(∫WεcE|ul|p˜dx)ϕps∗(τ)(∫WεcV|ul|qdx)τ(∫Wεc|ul|ps∗dx)1−ϕps∗(τ)−τ≤εC1′∥ul∥LEp˜(Rn)p˜ϕps∗(τ)∥ul∥LVq(Rn)qτ[ul]s,p,Rnr−p˜ϕps∗(τ)−qτ≤εC1′∥ul∥ME,Vs;q,p(Rn)r

using (2.7), with C1′>0 an absolute constant independent of u_l for all l ≥ 1.

For the integral over W_ε, as LWε<∞ and infBRV(x)≥VBR>0, one sees from (2.3) and Proposition 2.1 that ME,Vs;q,pBR→Ws,pBR↔Lrtt−1BR with rtt−1∈r,ps⋆, and thus

(4.3) ∫WεK|ul|rdx=∫BR∩WεK|ul|rdx+∫BRc∩WεK|ul|rdx≤(∫WεKtdx)1t{(∫BR|ul|rtt−1dx)t−1t+(∫Rn|ul|ps∗dx)rps∗(L)(BRc∩Wε)1−1t−rps∗}→0

as R → ∞and l → ∞for a subsequence {u_l : l ≥ 1} using the same notation.

So, (4.2)-(4.3) yields u l L K r R n → 0 as l → ∞and thus finishes the proof.

Note the embedding ME,Vs;q,pRn→LKrRn is continuous for ps⋆ps⋆−r≤t≤∞, and if K(x)∈L∞Rn, then the integrability requirement of K(x) is fulfilled automatically.

Theorem 4.2

Assume that n ≥ 2, 1 ≤ p < ∞, 0 < s < 1 with sp < n, and 1≤p˜≤r<minps⋆,q. Let E(x), K(x), V(x) > 0 : ℝⁿ → ℝ be such that K(x)∈Lloct(Rn) for some t∈maxps⋆,qmaxpc⋆,q−r,∞ and Kψps⋆(β,κ)(x)E−β(x)V−κ(x)∈L1Rn with ψps⋆(β,κ):=ps⋆+βps⋆−p˜+κps⋆−qps∗−r for some β ∈ [0,∞) and κ∈0,β(r−p˜)+rq−r. Then, the embedding ME,Vs;q,pRn→LKrRn is compact.

Proof. Define w:=β(ps∗−r)ps∗+β(ps∗−p˜)+κ(ps∗−q),x:=κ(ps∗−r)ps∗+β(ps∗−p˜)+κ(ps∗−q),y:=r+β(r−p˜)+κ(r−q)ps∗+β(ps∗−p˜)+κ(ps∗−q) and z:=ps∗−rps∗+β(ps∗−p˜)+κ(ps∗−q)=ψps∗−1(β,κ) with w+x+y+z=1. Then, for all u∈ME.Vs;q,pRn and on each set Ω of ℝⁿ, one has

(4.4) ∫ΩK|u|rdx=∫ΩKE−wV−x(E|u)|p˜w(V|u)|qx|u|r−p˜w−qxdx≤(∫ΩKψps∗(β,κ)E−βV−κdx)z(∫ΩE|u|p˜dx)w(∫ΩV|u|qdx)x(∫Ω|u|ps∗dx)y≤C1′∥Kψps∗(β,κ)E−βV−κ∥1,Ωψps∗−1(β,κ)∥u∥LEp˜(Ω)p˜w∥u∥LVq(Ω)qx[u]s,p,Rnps∗y

by Hölder’s inequality and (2.7), as p˜w+qx+ps⋆y=r, provided w,x,y,z∈(0,1).

Note that β ≥ 0 while β = ∞furnishes a version of Theorem 4.1. As w=βz and x=κX,to get z>0 , one has 0<κ<βps⋆−p˜+ps⋆q−ps⋆, while to get y>0, one has either 0<κ<β(r−p˜)+rq−r or κ>βps∗−p˜+ps⋆q−pc∗; so, 0<κ<β(r−p˜)+rq−r, When β=0=w, one simply ignores E(x) or treats E(x) as 0 with 00=1, while when κ=0=x, the same instead occurs to V(x). We may set 𝔶 = 0 to get κ=β(r−p˜)+ra−r without having p_s involved, and can simultaneously take w=x=0 corresponding to ME,Vs;q,pRn=Ds,pRn with E(x)≡V(x)≡0. So, MEVs;q,pRn→LKrRn, upon replacing Ω in (4.4) by ℝⁿ, is continuous without using K(x)∈LloctRn.

Finally, utilizing the same {u_l : l ≥ 1} as the one in Theorem 4.1, we derive

∫RnKulrdx=∫BRKulrdx+∫BRcKulrdx.

For the integral on B_R, as ME,Vs;q,pBR→Ms;q,pBR→Lrt−1BR with rtt−1∈r,maxps⋆,q,

(4.5) ∫BRKulrdx≤∫BRKtdx1t∫BRulrtt−1dxt−1t→0

when l → ∞ for a subsequence {u_l : l ≥ 1} by the same index l, via infBRV(χ)≥VBR>0, (2.2) and Proposition 2.1; for the integral on BRc, we apply (4.4) on Ω=BRc to deduce, as R →∞,

∫BRcKulγdx≤C1′Kψps⋆∗(β,κ)E−βV−κ1,BRcψps∗−1(β,κ)ulME,Vs;q,pRnr→0.

Hence, u_l → 0 in LKrRn for a subsequence relabeled using the same index l.

Theorem 4.3

Suppose n ≥ 2, 1 ≤ p < ∞, 0 < s < 1 with sp < n, and 1≤p˜≤ps⋆≤r< q. Let E(x),K(x),V(x)> 0 : ℝⁿ → ℝ be such that K(x)∈Lloct(Rn) for some t∈qq−r,∞ and Kψps⋆(β,κ)(x)E−β(x)V−κ(x)∈L1Rn for some β ∈ [0,∞) and κ∈β(r−p˜)+rq−r,∞. Then, the embedding ME.Vs;q,pRn→LKrRn is compact.

Proof. The proof is essentially the same as that of Theorem 4.2. Keep in mind q> r≥ps⋆≥p˜ when ensuring w,x,n,z∈(0,1). Actually, to have z>0, one sees κ>βps⋆−p˜+ps⋆a−ps⋆, while to get 𝔶 > 0, one has either 0 < κ < βps⋆−p˜+ps⋆a−p⋆ or κ>β(r−p˜)+ra−r; so,κ≥β(r−p˜)+ra−r after taking y=0. Recall ME.Vs;q,pRn→LKrRn is continuous with no need of K(x)∈LloctRn.

Several remarks are in order: first, Theorem 4.2 merged [21, Theorem 1] (the upper-triangle case r≥p˜) and [47, Theorem 2.3] (the lower-triangle case r < q) in one result, while β = ∞or κ = 0 yield versions of Theorem 4.1 via integrability conditions; second, Theorem 4.2 is useful particularly for Theorem 1.2, but the case β = 0 contributes to condition (f3) of Theorem 1.1; finally, Theorem 4.1 is intended for Theorem 1.1, but the case τ = 0 is used in condition (EK) of Theorem 1.2. As such, it is probably helpful to summarize them as follows.

Proposition 4.4

Assume that n≥2,p=p˜=2 and 0 < s < 1.

(i). Suppose further 1 ≤ r<min2s⋆,q, and K(x), V(x) > 0 satisfy K2s∗+K2⋆−q2s∗−r(x)V−κ(x)∈L1Rn for some κ∈0,rq−r; then, the embedding MVs;q,2(Rn)→LKr(Rn) is continuous, where MVs;q,2(Rn) is the function space defined via (4.1) without the term Eu².

(ii). Suppose further 2≤r<2s∗, infBRE(x)≥EBR>0 for each R > 0, K(x) ∈ L^t(Ω) for some t∈[ps∗ps∗−r,∞] on each set Ω of ℝⁿ with L(Ω)<∞, and K(x)E−2s∗−r2s∗−2(x) vanishes at infinity; then, the embedding HEs(Rn)→LKr(Rn) is continuous, where HEs(Rn) is the Hilbert function space defined through (4.1) with the term Vu^q dropped.

We can take p=p˜ and V(x) ≡ 0 to haveWEs,p(Rn)↪LKp(Rn) or WEs,p(Rn)↪LK1(Rn), or take E(x) ≡ V(x) ≡ 0 to have Ds,p(Rn)↪LKp(Rn), for suitable E(x) and K (x), which seems to provide a complement to [14, Condition (h₁)] in terms of compact embedding.

Finally, following Servadei and Valdinoci [48, Section 3] or [49], or applying Auchmuty [6] (see also [37, Section 3.1]), one observes the eigenvalue problems

(4.6) (−Δ)sf+Ef=ϱKfand(−Δ)sg=ξKginRn,

when n ≥ 2, admit families {ϱ_k > 0 : k ≥ 1} and {ξ_l > 0 : l ≥ 1} of eigenvalues, and sequences {fk∈HEs(Rn): k ≥ 1} and {g_l ∈ D^s(ℝⁿ) : l ≥ 1} of associated eigenfunctions, respectively, in view of Theorem 4.1 (when taking V(x) ≡ 0 and p=p˜=r=2 to work in Hilbert spaces) and Theorem 4.2 (when taking E(x) ≡ V(x) ≡ 0 and p = r = 2 to work in Hilbert spaces).

ϱ₁, ξ₁ are simple, each ϱk,ξl are of finite multiplicities, and the sequences {ϱ_k : k ≥ 1} and {ξ_l : l ≥ 1} of eigenvalues increase without bound.

f₁, g₁ are nonnegative, and the sequences {f_k : k ≥ 1} and {g_l : l ≥ 1} provide orthogonal bases to the spaces HEs(Rn),LK2(Rn) and Ds(Rn),LK2(Rn), respectively.

The procedure is standard, with details left for the interested reader.

5 Proof of Theorem 1.1

First, recall for a weakly convergent sequence {uk:k≥1}⊊Ds(Rn) having the weak limit u, there are bounded measures μ, ν and constants μ_i , ν_i , μ_∞, ν_∞ ≥ 0 such that

(5.1) (∇suk)2dx⇀dμ≥(∇su)2dx+∑i∈Nμiδxi+μ∞δ∞and|uk|2s∗dx⇀dν=|u|2s∗dx+∑i∈Nνiδxi+ν∞δ∞

in the sense of weakly^* convergence of measures, where

(∇sv)2=(∇sv(x))2:=∫Rn(v(x)−v(y))2|x−y|n+2sdyfor eachv∈Ds(Rn),μ∞:=limR→∞lim supk→∞∫BRc(∇suk)2dxandν∞:=limR→∞lim supk→∞∫BRc|uk|2s∗dx,

δ_xi denotes the Dirac delta at xi∈Rn, and Sνi2/2s∗≤μi for i∈N and Sν∞2/2s∗≤μ∞.

We seek a positive solution to (1.1) as the critical point of its energy functional

(5.2) F(u):=12∫Rn∫Rn(u(x)−u(y))2|x−y|n+2sdxy+12∫RnEu2dx+1q∫RnV|u|qdx−∫RnKF+(u)dx−12s∗∫Rn(u+)2s∗dx,

where F+(u):=∫0uf+(v)dv≥0 for f₊(u) := f(u) if u ≥ 0 and f₊(u) := 0 otherwise through condition (f2). Then, u is a positive weak solution to equation (1.1), provided

(5.3) ∫Rn∫Rn(u(x)−u(y))(v(x)−v(y))|x−y|n+2sdxdy+∫RnEuvdx+∫RnV|u|q−2uvdx=∫{x:u(x)≥0}Kf+(u)vdx+∫Rn(u+)2s∗−1vdx,∀v∈ME,Vs;q,2(Rn).

To see this fact, take v = u⁻ := min{u, 0} ∈ ME,Vs;q,2Rn. Notice for each u, u = u⁺ + u⁻ with u⁺ := max{u, 0} ∈ ME,Vs;q,2Rn and u⁺ · u⁻ ≡ 0. Then, we get from (5.3) that

(5.4) 0=∫Rn∫Rn(u(x)−u(y))(u−(x)−u−(y))|x−y|n+2sdxdy+∫RnE(u−)2dx+∫RnV|u−|qdx=∫Rn∫Rn(u−(x))2+(u−(y))2−2u(x)u−(y)|x−y|n+2sdxdy+∫RnE(u−)2dx+∫RnV|u−|qdx=∫Rn∫Rn(u−(x)−u−(y))2−2u+(x)u−(y)|x−y|n+2sdxdy+∫RnE(u−)2dx+∫RnV|u−|qdx≥∫Rn∫Rn(u−(x)−u−(y))2|x−y|n+2sdxdy+∫RnE(u−)2dx+∫RnV|u−|qdx≥0,

which further implies ∥u−∥ME,Vs;q,2(Rn)=0; so, u=u+≥0 follows immediately.

Now, via condition (f1), f+(u)≤ε|u|q−1+aε|u|2s∗−1 for each ε > 0 and a sufficiently large constant a_ε > 0 depending on ε; thus, we have

∫RnKF+(u)dx≤εq∫RnK|u|qdx+aε2s∗∥K∥∞,Rn∫Rn|u|2s∗dx.

From condition (EKV1) and Theorem 4.1, one accordingly derives

F(u)≥12[u]s,2,Rn2−CKεq[u]s,2,Rnq+12∥u∥LE2(Rn)2−CKεq∥u∥LE2(Rn)q+1q∥u∥LVq(Rn)q−CKεq∥u∥LVq(Rn)q−12s∗(1+aε,K)∥u∥2s∗,Rn2s∗≥[u]s,2,Rn2(12−CKεq[u]s,2,Rnq−2−C1′1+aε,K2s∗[u]s,2,Rn2s∗−2)+∥u∥LE2(Rn)2(12−CKεq∥u∥LE2(Rn)q−2)+1q(1−εCK)∥u∥LVq(Rn)q,

so that F(u)≥ρ0>0 when 0<ρ2≤∥u∥ME,Vs;q,2(Rn)≤ρ for sufficiently small ε, where C_K > 0 depends on the embedding ME,Vs;q,2(Rn)→LKq(Rn) and aε,K:=aε∥K∥∞,Rn>0.

F(0)=0 and F(w0)<0 for a w0∈ME,Vs;q,2(Rn) with ∥w0∥ME,Vs;q,2(Rn)>ρ. So, the standard mountain pass structure is satisfied, and F attains a Palais-Smale sequence at level c for c=infh∈Hmaxt∈[0,1]F(h(t))>0 with H:={h∈C([0,1];ME,Vs;q,2(Rn)):h(0)=0,h(1)=w0}.

Below, we find an upper bound for c > 0 via the mountain pass solution v₀ > 0 to equation (1.3) seeing ME,Vs;q,2(Rn)↪LKϑ(Rn). For t ≥ 0, condition (f2) and (1.5) yield

(5.5) F(tv0)≤t22(∫Rn∫Rn(v0(x)−v0(y))2|x−y|n+2sdxdy+∫RnEv02dx)+tqq∫RnVv0qdx−Aϑtϑϑ∫RnKv0ϑdx≤{t22+tqq−Aϑtϑϑ}∫RnKv0ϑdx≤supt≥0{t2−Aϑtϑϑ}∫RnKv0ϑdx<s2nSn2s

using a relation similar to (5.3) fulfilled by u = v = v₀; thus, 0< c<s2nSn2s.

Now, take {u_k : k ≥ 1} to be a Palais-Smale sequence of F at level c in ME,Vs;q,2(Rn); that is, F(uk)→c and F′(uk)→0 in the dual space of ME,Vs;q,2(Rn) as k→∞. Then, one has

c+o(∥uk∥ME,Vs;q,2(Rn))+o(1)=F(uk)−1θF′(uk)(uk)≥(12−1θ)[uk]s,2,Rn2+(1q−1θ)∥uk∥LVq(Rn)q+(1θ−12s∗)∥uk∥2s∗,Rn2s∗−η1θ∫RnK1|uk|r1dx−η2θ∫RnK2|uk|r2dx+(12−1θ)∥uk∥LE2(Rn)2≥(12−1θ)[uk]s,2,Rn2+(1q−1θ)∥uk∥LVq(Rn)q+(1θ−12s∗)∥uk∥2s∗,Rn2s∗−η1θCK1∥uk∥2s∗,Rnr1−η2θCK2[uk]s,2,Rnr2−η2θCK2∥uk∥LVq(Rn)r2

from condition (f3) using Proposition 4.4 (i) ^[1], with C_K2 > 0 depending only on MVs;q,2(Rn)→LK2r2(Rn). Thus, {u_k : k ≥ 1} is bounded in ME,Vs;q,2(Rn)⊊Ds(Rn) and L2s∗(Rn), and there is a subsequence, still denoted by {u_k : k ≥ 1}, and a function u∈ME,Vs;q,2(Rn) such that both the estimates in (5.1) hold, and u_k → u in LKq(Rn) and LVq(BR) for each R > 0.

Fix an increasing function χ_R with χ_R ≡ 0 on |x| ≤ R, χ_R ≡ 1 on |x| ≥ R², and 1−χR∈Cc1(Rn) (see, for instance, [28, Appenidx I]). Next, from the discussions in [14, Section 2], and noting condition (f1) yields f+(u)≤bε|u|q−1+ε|u|2s∗−1 for each ε > 0 and a sufficiently large constant b_ε > 0 depending on ε, it follows that

(5.6) 0=limk→∞F′(uk)(ukχR)≥lim supk→∞{∫Rn∫Rn(uk(x)−uk(y))2|x−y|n+2sχR(x)dxdy+∫Rn∫Rn(uk(x)−uk(y))(χR(x)−χR(y))|x−y|n+2suk(y)dxdy+∫RnEuk2χRdx+∫RnV|uk|qχRdx−bε∫RnK|uk|qχRdx−(1+ε∥K)∥∞,Rn∫Rn|uk|2s∗χRdx}.

By virtue of Lemma 3.6 in [14], one has

∫Rn∫Rn(uk(x)−uk(y))2|x−y|n+2sχR(x)dxdy→μ∞,∫Rn|uk|2s∗χRdx→ν∞,and∫Rn∫Rn(uk(x)−uk(y))(χR(x)−χR(y))|x−y|n+2suk(y)dxdy→0whenR,k→∞;

moreover, ∫RnKukqχRdx→0 when R,k→∞ as ME.Vs;q,2Rn→LKqRn. So, one sees from (5.6) that (1 + ε∥K∥∞,Rnv∞≥μ∞+v∞≥μ∞ for v∞:=limR→∞limsupk→∞∫RnVukqχRdx≥0 ^[2]; a similar but simpler analysis shows 1+ε∥K∥∞,Rnvi≥μi for each x_i ∈ ℝⁿ. Thus, μi,μ∞≥Sn2s11+ε∥K∥∞,Rn22s∗−2 and vi,v∞≥Sn2s11+ε∥K∥∞,Rn2s⋆2s−2; so, μi,μ∞,vi,v∞≥Sn2s, if they are not zero, by the arbitrariness of ε, which however would lead to a contradiction as follows.

In fact, set ℓ:=μRn−[u]S,2,Rn2−∑i∈Nμi−μ∞≥0 to observe that

c=limk→∞{F(uk)−1θF′(uk)(uk)}≥(12−1θ)([u]s,2,Rn2+∑i∈Nμi+μ∞+ℓ)−η2θCK2([u]s,2,Rn2+∑i∈Nμi+μ∞+ℓ)r22+(1θ−12s∗)(∥u∥2s∗,Rn2s∗+∑i∈Nνi+ν∞)−η1θCK1(∥u∥2s∗,Rn2s∗+∑i∈Nνi+ν∞)r12s∗+(1q−1θ)(∥u∥LVq(Rn)q+υ∞)−η2θCK2(∥u∥LVq(Rn)q+υ∞)r2q,

which certainly implies only finitely many of μ i , μ ∞ , v i , v ∞ ≥ S n 2 s do not vanish. To illustrate the contradiction, assume without loss of generality that μ∞,v∞≥Sn2s. Then, we have

c=limk→∞{F(uk)−1θF′(uk)(uk)}≥56(12−1θ)μ∞+56(1θ−12s∗)ν∞+θ−26⋅2θ([u]s,2,Rn2+∑i∈Nμi+μ∞+ℓ)−η2θCK2([u]s,2,Rn2+∑i∈Nμi+μ∞+ℓ)r22+2s∗−θ6⋅2s∗θ(∥u∥2s∗,Rn2s∗+∑i∈Nνi+ν∞)−η1θCK1(∥u∥2s∗,Rn2s∗+∑i∈Nνi+ν∞)r12s∗+θ−qqθ(∥u∥LVq(Rn)q+υ∞)−η2θCK2(∥u∥LVq(Rn)q+υ∞)r2q≥56(12−12s∗)Sn2s−3⋅s9nSn2s=5s6nSn2s−s3nSn2s=s2nSn2s,

contradicting (5.5); parallel discussions yield contradictions again on μ_i , ν_i for all x_i ∈ ℝⁿ.

So, there occurs no evanescence phenomenon and we get uk→u∈ME,Vs;q,2Rn.

6 Proof of Theorem 1.2

In this section, we combine the strategies from [7, 8] and Han [31] to find nontrivial positive solutions to (1.6) as the critical points of its energy functional Fλ(u) defined by

(6.1) Fλ(u):=12∫Rn∫Rn(u(x)−u(y))2|x−y|n+2sdxdy+12∫RnEu2dx+1q∫RnV|u|qdx−λr∫RnK(u+)rdx.

Like the analysis in (5.4), u is a positive weak solution to equation (1.6) provided

(6.2) ∫Rn∫Rn(u(x)−u(y))(v(x)−v(y))|x−y|n+2sdxdy+∫RnEuvdx+∫RnV|u|q−2uvdx=λ∫RnK(u+)r−1vdx,∀v∈ME,Vs;q,2(Rn).

Lemma 6.1

Under the assumptions (EKV2-1) or (EKV2-2), the functional Fλ is C¹ and is coercive ME,Vs;q,2Rn, so that each sequence {u_k : k ≥ 1} of functions with Fλuk bounded has a weakly convergent subsequence in ME,Vs;q,2Rn Moreover, Fλ is sequentially weakly lower semi-continuous in ME,Vs;q,2Rn; that is, when u_k ⇀ u ∈ ME.Vs;q,2Rn, for a subsequence relabeled using the same notation, one has Fλ(u)≤liminfk→∞Fλuk.

Proof. The proof of showing FλC1 is standard ^[3], and the claim regarding the boundedness of Fλuk implying the existence of a weakly convergent subsequence in ME,Vs;q,2Rn follows from the coercivity of Fλ and the reflexivity of ME.Vs;q,2Rn; for the latter, see [7, 8] for details, since now ME,Vs;q,2Rn is a Banach subspace of the Hilbert function space HESRn.

Next, for r1:=w−1,r2:=x−1,r3:=22s⋆y−1 and r4:=11−1r1−1r2−1r3>0 (being true under our conditions (EKV2-1) or (EKV2-2)) in (4.4) of Theorem 4.2, one has

(6.3) λr∫RnK(u+)rdx≤λr∫RnK|u|rdx≤(r14∫RnEu2dx)1r1(r22q∫RnV|u|qdx)1r2(r34∫Rn∫Rn(u(x)−u(y))2|x−y|n+2sdxdy)1r3×{(λC1′r)r4(4r3)r4r3(2qr2)r4r2(4r1)r4r1(∫RnKψ2s∗(β,κ)E−βV−κdx)r4 ψ2s∗(β,κ)}1r4≤14∫Rn∫Rn(u(x)−u(y))2|x−y|n+2sdxdy+14∫RnEu2dx+12q∫RnV|u|qdx+λr4CEKV

by Young’s inequality for an absolute constant C_EKV > 0 independent of λ, u. So,

Fλ(u)≥C4∥u∥ME,Vs;q,2Rn−λr4CEKV−C4′

for some constants C4,C4′>0 as q>2, which implies the coercivity of F.

Since Fλ is coercive, each sequence {u_k : k ≥ 1} of functions in ME.Vs;q,2Rn with bounded Fλuk has a weakly convergent subsequence, written again as {u_k : k ≥ 1}, such that uk⇀u∈ME,Vs;q,2Rn. The weakly lower semi-continuity of norms in DSRn,LE2Rn and LVqRn, along with the compact embedding ME,Vs;q,2Rn→LKrRn by Theorems 4.2 and 4.3, leads to the sequentially weak lower semi-continuity of Fλ, which concludes the proof.

Define

(6.4) λ˜:=infu∈ME,Vs;q,2Rnu+LKrRn=1 r2∫Rn∫Rn(u(x)−u(y))2|X−y|n+2sdxdy+r2∫RnEu2dx+rq∫RnV|u|qdx.

Then, one has λ˜>0. Actually, if not, there would exist a sequence ul:l≥1⊊ME,Vs;q,2Rn with ul+LKrRn=1 such that

r2∫Rn∫Rnul(x)−ul(y)2|x−y|n+2sdxdy+r2∫RnEul2dx+rq∫RnVulqdx→0;

so, ulME.Vs;q,2Rn→0, contradicting the compact embedding ME,Vs;q,2Rn→LKrRn.

Define λ^* to be the supermum of λ such that equation (1.6) only has the trivial nonnegative solution for each ι < λ, and λ^** to be the infimum of λ such that equation (1.6) has at least a nontrivial positive solution at λ. Then, 0≤λ⋆=λ⋆⋆≤λ˜. As a matter of fact,

λ∫RnKvλ+rdx>r2∫Rn∫Rnvλ(x)−vλ(y)2|x−y|n+2sdxdy+r2∫RnEvλ2dx+rq∫RnVvλqdx

holds for all λ>λ˜ with some vλ∈ME,Vs;q,2Rn by homogeneity, and consequently Fλvλ<0, which along with Lemma 6.1 leads to Fλuλ=infu∈ME,Vs;q,2RnFλ(u)≤Fλvλ<0 for an u_λ ≥ 0 in ME,Vs;q,2Rn that is a nontrivial

positive solution to equation (1.6), since Fλuλ≤Fλuλ. So, we have λ⋆⋆≤λ˜. On the other hand, if λ^* > λ^**, we would find a λ′∈λ⋆⋆,λ⋆ such that problem (1.6) has at least a nontrivial positive solution at λ′ according to the definition of λ^**, which however is against the definition of λ^*; if λ^* < λ^**, we would find a λ′ ∈ (λ^*, λ^**] such that problem (1.6) has at least a nontrivial positive solution at some l′<λ′ according to the definition of λ^*, which however is against the definition of λ^**. Thus, λ^* = λ^**.

Write λ₁ := λ^* = λ^** in the sequel.

Proposition 6.2

Under our conditions (EKV2-1) or (EKV2-2), λ₁ ≥ 0 and equation (1.6) has a nontrivial positive solution u_λ ≥ 0 in ME,Vs;q,2Rn for all λ > λ₁.

Proof. By definition, λ₁ = λ^*; so, if u_λ is a nontrivial positive solution to (1.6) in ME,Vs;q,2Rn, then λ ≥ λ₁. Now, we show (1.6) has at least one nontrivial solution u_λ ≥ 0 in ME,Vs;q,2Rn for all λ > λ₁ combining Struwe [50, Theorem 2.4], [8, Theorem 4.2] and [31, Proposition 3.3].

By definition, λ₁ = λ^**; so, there is an ι ∈ [λ₁, λ) such that (1.6) has a nontrivial solution u_ι ≥ 0 in ME,Vs;q,2Rn which of course is a subsolution to (1.6) at λ. Consider the constrained minimization problem infu∈MFλ(u) for M:=u∈ME,Vs;q,2Rn:u≥ul≥0. Notice M is closed and convex, so that it is weakly closed in ME,Vs;q,2Rn. Thus, Lemma 6.1 ensures the attainment of a minimizer of Fλ in M; that is, there is an u_λ(≥ u_ι ≥ 0) in M with Fλuλ=infu∈MFλ(u). Take φ∈Cc1Rn, and set φ_ε := max{0, u_ι − u_λ + εφ} ≥ 0 and v_ε := φ_ε + u_λ − εφ(≥ u_ι ≥ 0) in M for all ε > 0. So, one has Fλ′uλuλ≤Fλ′uλvε that further implies

(6.5) Fλ′uλ(φ)≤1εFλ′uλφε.

Put Ωε:=x∈Rn:φε(x)>0=x∈Rn:uλ(x)−εφ(x)<ut(x)⊊suppφ+. Since u_ι is a subsolution to (1.6) at λ and φ_ε ≥ 0, Fλ′ulφε≤0 follows and one has ^[4]

Fλ′uλφε≤Fλ′uλφε−Fλ′utφε=∬Rn×Rn(ω(χ)−ω(y))φε(x)−φε(y)|χ−y|n+2sdxdy+∫RnEuλ−ulφεdx+∫RnVuλq−1−ulq−1φεdx−λ∫RnKuλr−1−utr−1φεdx=∬Ωεc×Ωεcc(ω(x)−ω(y))φε(x)−φε(y)|x−y|n+2sdxdy+∫ΩεEuλ−utφεdx+∫ΩεVuλq−1−utq−1φεdx−λ∫ΩεKuλr−1−ulr−1φεdx≤ε∬Ωεc×Ωεcc(ω(x)−ω(y))(φ(x)−φ(y))|x−y|n+2sdxdy−∬Ωεc×Ωεcc(ω(x)−ω(y))2|X−y|n+2sdxdy+∫ΩεEuλφεdx+∫ΩεEulφεdx+∫ΩεVuλq−1φεdx+∫ΩεVulq−1φεdx≤ε∬Ωε×Ωε(ω(x)−ω(y))(φ(x)−φ(y))|χ−y|n+2sdxdy+2ε∬Ωε×Ωεc(ω(x)−ω(y))(φ(x)−φ(y))|x−y|n+2sdxdy+ε∫ΩεEuλ|φ|dx+∫ΩεEuι|φ|dx+∫ΩεVuλq−1|φ|dx+∫ΩεVulq−1|φ|dx≤2ε∬Ωε×Rn(ω(x)−ω(y))(φ(x)−φ(y))|x−y|n+2sdxdy+ε∥φ∥LE2ΩεuλLE2Ωε+uιLE2Ωε+∥φ∥LVqΩεuλLVqΩεq−1+uιLVqΩεq−1=2ε∬Ωε×BR(ω(x)−ω(y))(φ(x)−φ(y))|x−y|n+2sdxdy+2ε∬Ωε×BRc(ω(x)−ω(y))(φ(x)−φ(y))|x−y|n+2sdxdy+ε∥φ∥LE2ΩεuλLE2Ωε+uℓLE2Ωε+∥φ∥LvqΩεuλLVqΩεq−1+utLVqΩεq−1≤2ε∬Ωε×BR(ω(x)−ω(y))(φ(x)−φ(y))|x−y|n+2sdxdy+2ε∬BR×BRc(ω(x)−ω(y))(φ(x)−φ(y))|x−y|n+2sdxdy

+ε∥φ∥LE2ΩεuλLE2Ωε+ulLE2Ωε+∥φ∥LVqΩεuλLVqΩεq−1+uℓLVqΩεq−1

with ω := u_λ − u_ι ≥ 0, seeing 0 < φ_ε ≤ ε|φ| on Ω_ε; that is, we have Fλ′uλφε→o(ε) when R → ∞ and ε → 0⁺. This combined with (6.5) yields Fλ′uλ(φ)≤0 for all φ∈Cc1Rn; so, we also get Fλ′uλ(−φ)≤0 via symmetry. By density, it follows that Fλ′uλ(v)=0 for each v∈MF.Vs;q,2Rn, and hence u_λ(≥ u_ι ≥ 0) is a nontrivial solution to (1.6) at λ.

Proposition 6.2 obviously provides the proof for the first part of Theorem 1.2. Now, we will proceed to discuss the second part of Theorem 1.2 by first giving some estimates for solutions to equation (1.6) in general, which are presented as the result below.

Lemma 6.3

Under the assumptions (EKV2-1) or (EKV2-2), each solution u_λ to equation (1.6) in ME,Vs;q,2Rn satisfies

(6.6) ∫Rn∫Rnuλ(x)−uλ(y)2|X−y|n+2sdxdy+∫RnEuλ2dx+∫RnVuλqdx≤λr4CEKV′.

where CEKV′>0 is an absolute constant independent of λ, u. Moreover, when the assumption (EK) also holds, there are absolute constants CEK,C˜EKV>0 such that λ1≥C˜EKV>0 and such that each nontrivial solution u_λ to equation (1.6) in ME,Vs;q,2Rn satisfies

(6.7) ∫RnKuλrdx≥λCEK2rr2−r.

Proof. First, (6.6) follows from the fact that each solution u_λ to (1.6) satisfies

(6.8) ∫Rn∫Rnuλ(x)−uλ(y)2|x−y|n+2sdxdy+∫RnEuλ2dx+∫RnVuλqdx=λ∫RnKuλrdx

and an estimate akin to (6.3), which is left as an easy exercise for the reader.

Now, when our condition (EK) also holds, from Proposition 4.4 (ii) and for u∈ME,Vs;q,2Rn with an absolute constant C_EK > 0, one has

(6.9) ∫RnK|u|rdx≤CEK∫Rn∫Rn(u(x)−u(y))2|X−y|n+2sdxdy+∫RnEu2dxr2.

Combining (6.8) and (6.9), one observes, for each nontrivial solution u_λ to (1.6),

∫RnKuλrdx2r≤λCEK2r∫RnKuλrdx,

which in turn yields (6.7). Also, by (6.6), (6.7) and (6.9), we get λCEK2r22−r≤λr4CEK2rCEKV′, so that λ≥C˜EKV>0 as 2 < r, which in particular leads to λ1≥C˜EKV>0.

Finally, we are ready to discuss the second twofold claim of Theorem 1.2.

Proposition 6.4

Under our conditions (EKV2-1) and (EK), we have λ₁ > 0 and equation (1.6) has a nontrivial positive solution u_λ ≥ 0 in ME,Vs;q,2Rn if and only if λ ≥ λ₁; besides, for λ2:=λ˜≥λ1>0 as given in (6.4), equation (1.6) has at least two distinct nontrivial positive solutions uλ,u˜λ≥0 in ME,Vs;q,2Rn for all λ > λ₂.

Proof. Note if u_λ ≥ 0 is a nontrivial positive solution to (1.6) in ME,Vs;q,2Rn, then λ ≥ λ₁ > 0; so, we are left to show (1.6) has a nontrivial positive solution at λ₁. Let {λ(k) > λ₁ : k ≥ 1} decrease to λ₁, with uλ(k)≥0:k≥1 an associated sequence of nontrivial solutions to (1.6) that is bounded in ME.Vs;q,2Rn by (6.6). Then, there exists a subsequence uλ(k)≥0:k≥1, using the same notation, such that uλ(k)⇀w in HESRn,uλ(k)⇀w in LVqRn,uλ(k)→w in LKΥRn, and uλ(k)→w a.e. on ℝⁿ for a function w ≥ 0 in ME,Vs;q,2Rn. Thus, one has

limk→∞∫RnKuλ(k)r−1vdx=∫RnKwr−1vdx,∀v∈ME,Vs;q,2Rn.

Keep in mind Fλ(k)′uλ(k)=0; that is, (6.2) holds for u = u_λ_(k) and λ = λ(k). Since Fλ′(w) is a continuous linear functional on ME,Vs;q,2Rn, it follows that

0←Fλ(k)′uλ(k)uλ(k)−w+λ(k)∫RnKuλ(k)r−1uλ(k)−wdx−Fλ′(w)uλ(k)−w−λ∫RnKwr−1uλ(k)−wdx=∫Rn∫Rn(ϖ(x)−ϖ(y))2|x−y|n+2sdxdy+∫RnEuλ(k)−w2dx+∫RnVuλ(k)q−1−wq−1uλ(k)−wdx≥uλ(k)−ws,2,Rn2+uλ(k)−wLE2Rn2+Cquλ(k)−wLVqRnq≥0 as k→∞

for ϖ:=uλ(k)−w and an absolute constant C_q > 0 (see Han [29, Lemma 3.2]); so, u_λ_(k) → w in HESRn and LVqRn. Therefore, (6.2) is satisfied by w and λ for all v∈ME,Vs;q,2Rn upon taking the limit k → ∞. Using (6.7), one sees ∫RnKwrdx≥λ(1)CEK2rr2−r>0 by virtue of the compact embedding MF.Vs;q,2Rn→LKrRn. So, w ≥ 0 is a nontrivial positive solution in MFVs;q,2Rn to equation (1.6); that is, w = u_λ1 in common practice notation.

On the other hand, for each λ > λ₂, there is a solution u_λ ≥ 0 to (1.6) with Fλuλ<0. Furthermore, it is easily seen from (6.9) that, for some constant CEK′>0,

Fλ(u)≥14∥u∥HEsRn2−λrCEK′∥u∥HEsRnr≥∥u∥HEsRn214−λrCEK′∥u∥HEsRnr−2,

and thus Fλ(u)≥ρ˜0>0 provided ∥ u ∥ H E s R n = ρ ~ < min u λ H E s R n , r 4 λ C E K ′ 1 r − 2 . So, via the mountain pass theorem of Candela and Palmieri [19, Theorem 2.5] (see also [7, Theorem A.3]), there exists a sequence {u_l : l ≥ 1} in ME,Vs;q,2Rn with Fλul→c>0 and Fλ′ul→0 in the dual space of ME,VS;q,2Rn when l → ∞ for c=infh∈Hλmaxt∈[0,1]Fλ(h(t))>0 with Hλ:={h∈C([0,1]; ME,Vs;q,2Rn:h(0)=0, h(1)=uλ, Lemma 6.1 provides a subsequence {u_l : l ≥ 1}, using the same notation, such that u1⇀w˜ in HESRn,u1⇀w˜ in LVqRn,u1→w˜ in LKrRn, and ul→w˜ a.e. on ℝⁿ for a function w˜∈MF.Vs;q,2Rn. Similar analysis then yields ul→w˜ in ME,Vs;q,2Rn; that is, u˜λ:=W˜≥0 is a nontrivial solution to (1.6) in ME,Vs;q,2Rn, as Fλ′u˜λ=0 and thus (6.2) is satisfied, which is distinct from u_λ since Fλu˜λ=c>0.

A Appendix

This section provides an additional analysis for Subsection 2.1. Recall Ω denotes a bounded domain having a compact, Lipschitz boundary ∂Ω. Note L^q(Ω) → L¹(Ω). Using Minkowski’s inequality and Lemma 3.1 of Bellido and Mora-Corral [11], we have

(A.1) ∥u∥p,Ω≤∥u−uˉ∥p,Ω+∥uˉ∥p,Ω≤C5[u]s,p,Ω+∥u∥1,Ω≤C5′[u]s,p,Ω+∥u∥q,Ω,∀u∈Ms;q,p(Ω) and 1≤q≤∞

with uˉ:=1L(Ω)∫Ωudx and C₅, C5′>0 some constants depending on n, p, q, s, Ω. Thus, (2.3) follows from (A.1) and (2.1) consequently; that is, M^s^;q,p(Ω) → W^s^,p(Ω).

On the other hand, seeing W^s^,p(Ω) → L^q(Ω) for 1≤q≤ps⋆ when sp < n, one accordingly has W^s^,p(Ω) → M^s^;q,p(Ω), which together with the preceding analysis yields (2.2). Moreover, noticing W^s^,p(Ω) → L^q(Ω) for 1 ≤ q < ∞or1 ≤ q ≤ ∞when sp = n or sp > n, the preceding analysis then leads to (2.4) and (2.6) respectively.

Dedicated to

In loving memory of my father Jiaxing Han (June 5, 1952–April 9, 2019).

Conflict of interest

Conflict of interest statement: There is no conflict of interest.

Acknowledgement

The author is greatly indebted to the anonymous referees for helpful suggestions. The author gratefully acknowledges the partial financial support of the Research Council Grant #21870120011 plus the 2017–2021 faculty research fellowships from the College of Arts and Sciences at Texas A&M University-San Antonio.

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Received: 2020-05-06

Accepted: 2021-04-07

Published Online: 2021-09-04

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

Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

Abstract

1 Introduction

Theorem 1.1

Theorem 1.2

2 Function space analysis

2.1 Ω is a bounded domain having a compact, Lipschitz boundary ∂Ω

Proposition 2.1

2.2 Ω ⁼ ℝⁿ

Proposition 2.2

3 A fractional Moser-Trudinger inequality

Theorem 3.1

Proposition 3.2

4 Some compact embedding results

Theorem 4.1

Theorem 4.2

Theorem 4.3

Proposition 4.4

5 Proof of Theorem 1.1

6 Proof of Theorem 1.2

Lemma 6.1

Proposition 6.2

Lemma 6.3

Proposition 6.4

A Appendix

Dedicated to

Acknowledgement

References

Journal and Issue

Articles in the same Issue

Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

Abstract

1 Introduction

Theorem 1.1

Theorem 1.2

2 Function space analysis

2.1 Ω is a bounded domain having a compact, Lipschitz boundary ∂Ω

Proposition 2.1

2.2 Ω = ℝn

Proposition 2.2

3 A fractional Moser-Trudinger inequality

Theorem 3.1

Proposition 3.2

4 Some compact embedding results

Theorem 4.1

Theorem 4.2

Theorem 4.3

Proposition 4.4

5 Proof of Theorem 1.1

6 Proof of Theorem 1.2

Lemma 6.1

Proposition 6.2

Lemma 6.3

Proposition 6.4

A Appendix

Dedicated to

Acknowledgement

References

Journal and Issue

Articles in the same Issue

2.2 Ω ⁼ ℝⁿ