Liouville property of fractional Lane-Emden equation in general unbounded domain

Ying Wang; Yuanhong Wei

doi:10.1515/anona-2020-0147

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Liouville property of fractional Lane-Emden equation in general unbounded domain

Ying Wang and Yuanhong Wei

Published/Copyright: August 22, 2020

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From the journal Advances in Nonlinear Analysis Volume 10 Issue 1

Abstract

Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation

(−Δ)αu=upinΩ,u=0inRN∖Ω,

where α ∈ (0, 1), N ≥ 1, p > 0 and Ω ⊂ ℝ^N–1 × [0, +∞) is an unbounded domain satisfying that Ω_t := {x′ ∈ ℝ^N–1 : (x′, t) ∈ Ω} with t ≥ 0 has increasing monotonicity, that is, Ω_t ⊂ Ω_t′ for t′ ≥ t. The shape of Ω_∞ := lim_t→∞ Ω_t in ℝ^N–1 plays an important role to obtain the nonexistence of positive solutions for the fractional Lane-Emden equation.

Keywords: Fractional Laplacian; Lane-Emden equation; Nonexistence

MSC 2010: 35J60; 35B53

1 Introduction

In this paper, we consider Liouville property for the fractional Lane-Emden equation

(−Δ)αu=upinΩ,u=0inRN∖Ω, (1.1)

where α ∈ (0, 1), p > 0, Ω is an unbounded domain in ℝ^N with N ≥ 1, and (–Δ)^α with α ∈ (0, 1) is the fractional Laplacian defined in the principle value sense,

(−Δ)αu(x)=cN,αlimϵ→0+∫RN∖Bϵ(0)u(x)−u(x+z)|z|N+2αdz,

here B_ϵ(0) is the ball with radius ϵ centered at the origin and c_N,α > 0 is the normalized constant. We say that u is a bounded solution of (1.1) if u ∈ C(ℝ^N) ∩ L^∞(ℝ^N) and u satisfies (1.1) pointwisely.

As an important property, the Liouville theorem for Lane-Emden equation has attracted a lot of attentions by many mathematician by the application in the derivation of uniform bound via blowing-up analysis. Note that the nonexistence of stable solution is studied in [1] by finite morse index with restrictions on the boundary and at infinity. Without the zero Dirichlet boundary condition, Liouville results could be obtained by Hadamard property in [2, 3], by iterating the decaying rate at infinity in [4] and by Hardy estimates in [5].

It is known that the Leray-Schauder degree theory is a very useful method for deriving solutions of elliptic equations on bounded domains. The essential step is to obtain a uniform bound by considering a sequence of solutions {u_n}_n such that u_n(x_n) = ∥u_n∥_L^∞ = M_n → +∞ as n → +∞ for some {x_n}_n ⊂ Ω. Then let Ω_n = {x ∈ ℝ^N : 1Mnκ x + x_n ∈ Ω} and vn(x)=1Mnun(1Mnκx+xn) for some κ > 0, then {v_n}_n is uniformly bounded, and the limit v_∞ of {v_n}_n is a solution of related limit semilinear equation in the limit domain Ω^∞. Note that for C² domain Ω, the limit domain of Ω_n is either Ω^∞ = ℝ^N or Ω^∞ = ℝ^N–1 × (0, ∞). While if the original domain contains a cone point on the boundary, then the limit domain has the third possibility that Ω^∞ is a cone. As a consequence, the nonexistence of positive solutions to the limit equation in a cone has to be involved additionally. As far as we know, the nonexistence of elliptic equation depends on the shape of limit domain at infinity in some direction. Our concern in this article is to consider the non-existence of elliptic equations in one type of unbounded domain.

Recently, qualitative properties of solutions for nonlocal elliptic equations have been studied extensively, such as the existence of weak solutions or very weak solutions in [6, 7] by variational methods, a survey in [8] on variational methods, large solution [9] by Perron’s method, the regularities in [10], Pohozaev’s identity [11] and Liouville results [12, 13, 14]. In [15] it develops the method of moving plane in fractional setting to obtain the classification of critical elliptic equations in an integral form and then it is used to obtain nonexistence of bounded solutions for semilinear elliptic equations in half space [12, 13, 16] subject to various boundary type conditions. In particular, Fall and Weth in [13] obtained the nonexistence of positive bounded solution of (1.1) when Ω=R+N and p<N−1+2αN−1−2α, by applying the method of moving plane, via some interesting estimates of Green’s kernel in R+N . It is worth noting that star-shaped domain with respect to infinity is involved in obtaining the nonexistence of fractional Lane-Emden equation in [14] and it is an important notation in our derivation of nonexistence to (1.1).

Before stating our main result, we introduce the following notations.

(𝓓) Given 𝓞 ⊂ ℝ^M × [0, +∞) and t ∈ [0, +∞), with M ≥ 1 being an integer, denote

Ot:={x′∈RM:(x′,t)∈O}.

For t ≥ 0, 𝓞_t is nonempty, bounded and the mapping t ↦ 𝓞_t is increasing in the following sense

Ot⊂Ot′if0≤t≤t′<+∞.

Denote

O∞=⋃t∈[0,∞)Ot. (1.2)

Now we introduce two types domains:

star-shaped domain 𝓞 ⊂ ℝ^M, if there exists e ∈ 𝓞 such that for every point x ∈ ∂ 𝓞, the segment {te + (1 – t)(x – e): t ∈ [0, 1]} is contained in 𝓞;
star-shaped domain 𝓞 ⊂ ℝ^M with respect to infinity, if there exists a point e ∈ ℝ^M ∖ 𝓞̄ such that for every point x ∈ 𝓞, the half-line {e + t(x – e): t ≥ 1} is contained in 𝓞.

The main results state as follows.

Theorem 1.1

Assume that α ∈ (0, 1), Ω ⊂ ℝ^N–1 × [0, +∞) is a C^0,1 domain verifying (𝓓) and Ω_∞ is given as (1.2) with 𝓞 = Ω. Then problem (1.1) has no nonnegative, nontrivial and bounded solutions if one of the following holds:

Ω_∞ = ℝ^N–1 and 0 1 + 2α, otherwise for p > 0, 1 ≤ N ≤ 1 + 2α;
Ω_∞ ⊂ ℝ^N–1 is a star-shaped domain with respect to infinity and 1 ≤ p ≤ N−1+2αN−1−2α with N > 1 + 2α;
Ω_∞ = ℝ^N–2 × [0, +∞) and 1 2 + 2α, otherwise for p > 1, 1 ≤ N ≤ 2 + 2α;
Ω_∞ is bounded, C² star-shape in ℝ^N–1 and N > 1 + 2α, p ≥ N−1+2αN−1−2α .

Our basic tool is the traditional method of moving plane, involved by [17] in the fractional setting, we develop this traditional method of moving planes to obtain the increasing monotonicity in the direction x_N and reduce problem (1.1) into

(−Δ)RN−1αu=upinΩ∞, (1.3)

subject to zero Dirichlet boundary condition when Ω_∞ ≠ ℝ^N–1, where (−Δ)RN−1α is the fractional Laplacian in ℝ^N–1. Then the nonexistence results could be obtained for (1.3) as in [13, 14].

Remark 1.1

In the particular case that Ω is a cone such as {x = (x′, x_N) ∈ ℝ^N : x_N > θ|x′|} for some θ > 0, problem (1.1) has no nonnegative, nontrivial and bounded solutions.

If Ω_∞ also verifies the similar assumptions of Theorem 1.1, we can repeat our above procedure and derive the following corollary directly:

Corollary 1.1

Under the assumptions of Theorem 1.1, we assume that the domain Ω_∞ ⊂ ℝ^N–2 × [0, +∞) is a C² domain verifying (𝓓) and Ω_∞,∞ is given by (1.2) with 𝓞_∞ = Ω_∞,∞. Then problem (1.1) does not admit nonnegative, nontrivial and bounded solutions if one of the following holds:

Ω_∞,∞ = ℝ^N–2 and 0 2 + 2α, otherwise for p > 0, 1 ≤ N ≤ 1 + 2α;
Ω_∞,∞ ⊂ ℝ^N–2 is a star-shaped domain with respect to infinity and 1 ≤ p ≤ N−2+2αN−2−2α with N > 2 + 2α;
Ω_∞,∞ = ℝ^N–3 × [0, +∞) and 1 3 + 2α, otherwise for p > 1, 1 ≤ N ≤ 3 + 2α;
Ω_∞,∞ is bounded, star-shape C² domain in ℝ^N–2 and N > 2 + 2α, p ≥ N−2+2αN−2−2α .

2 The proof of nonexistence results

For the domain Ω verifying (𝓓), we shall prove that the solution of (1.1) has the x_N-increasing property by using the method of moving planes. To this end, we introduce the following notations. For λ > 0, denote

Σλ={x=(x′,xN)∈RN∩Ω|xN<λ}, (2.1)

Tλ={x=(x′,xN)∈RN|xN=λ}, (2.2)

uλ(x)=u(xλ)andwλ(x)=uλ(x)−u(x), (2.3)

where x_λ = (x′, 2λ – x_N) for x = (x′, x_N) ∈ ℝ^N. For any subset A of ℝ^N, we write A_λ = {x_λ : x ∈ A} the reflection of A with regard to T_λ. Since Ω_t is bounded for t ≥ 0, then the domain Σ_λ is always bounded for any λ > 0.

Proposition 2.1

Under the hypotheses of Theorem 1.1, let u be a nonnegative nontrivial solution of (1.1), then

u(x′,t)≤u(x′,s)fors≥t,∀x′∈RN−1. (2.4)

Proof

We divide the proof into two steps.

By the assumption of Ω ⊂ R+N , we may assume

λ0=inf{λ≥0:wλ≥0inΣλ}.

The purpose of this step is to show that if λ > λ₀ is close to λ₀, then w_λ > 0 in Σ_λ. To this end, let Σλ− = {x ∈ Σ_λ | w_λ(x) < 0}, we first claim that if λ > λ₀ is close to λ₀, then

Σλ−=∅. (2.5)

By contradiction, we assume (2.5) is not true, that is Σλ− ≠ ∅. We denote

wλ+(x)=wλ(x),x∈Σλ−,wλ+(x)=0,x∈RN∖Σλ−, (2.6)

wλ−(x)=0,x∈Σλ−,wλ−(x)=wλ(x),x∈RN∖Σλ− (2.7)

It is obvious that wλ+(x)=wλ(x)−wλ−(x) for all x ∈ ℝ^N. By direct computation, for x ∈ Σλ− , we have

(−Δ)αwλ−(x) = −∫(Ω∖Ωλ)∪(Ωλ∖Ω)wλ(z)|x−z|N+2αdz−∫(Σλ∖Σλ−)∪(Σλ∖Σλ−)λwλ(z)|x−z|N+2αdz−∫(Σλ−)λwλ(z)|x−z|N+2αdz=−I1−I2−I3.

We look at each of these integrals separately. Since u = 0 in Ω_λ ∖ Ω and u_λ = 0 in Ω ∖ Ω_λ, then

I1 = ∫Ωλ∖Ωuλ(z)|x−z|N+2αdz−∫Ω∖Ωλu(z)|x−z|N+2αdz=∫Ωλ∖Ωuλ(z)(1|x−z|N+2α−1|x−zλ|N+2α)dz≥0,

by the fact that u_λ ≥ 0 and |x – z_λ| > |x – z| for all x ∈ Σλ− and z ∈ Ω_λ ∖ Ω.

In order to fix the sign of I₂, note that w_λ(z_λ) = –w_λ(z) for any z ∈ ℝ^N and then

I2 = ∫Σλ∖Σλ−wλ(z)|x−z|N+2αdz+∫Σλ∖Σλ−wλ(zλ)|x−zλ|N+2αdz=∫Σλ∖Σλ−wλ(z)(1|x−z|N+2α−1|x−zλ|N+2α)dz≥0,

since w_λ ≥ 0 in Σ_λ ∖ Σλ− and |x – z_λ| > |x – z| for all x ∈ Σλ− and z ∈ Σ_λ ∖ Σλ− . Finally, since w_λ(z) < 0 for z ∈ Σλ− , we have

I3=∫Σλ−wλ(zλ)|x−zλ|N+2αdz=−∫Σλ−wλ(z)|x−zλ|N+2αdz≥0.

Hence, we obtain that for all λ > λ₀,

(−Δ)αwλ−(x)≤0,∀x∈Σλ− (2.8)

and then for x ∈ Σλ− ,

(−Δ)αwλ+(x)≥(−Δ)αwλ(x)=(−Δ)αuλ(x)−(−Δ)αu(x). (2.9)

Combining (1.1) with (2.9) and (2.6), we have that

(−Δ)αwλ+(x)≥(−Δ)αuλ(x)−(−Δ)αu(x)=uλp(x)−up(x)=−φ(x)wλ+(x),x∈Σλ−,

where

φ(x)=−uλp(x)−up(x)uλ(x)−u(x),|φ|≤2p[uλp−1+up−1]≤2p∥u∥L∞p−1.

Hence, we have

Δαwλ+(x)≤φ(x)wλ+(x),x∈Σλ− (2.10)

and observe that wλ+ = 0 in (Σλ−)c, then we have that

supx∈Σλ−wλ+(x)≤cR(Σλ−)2α∥φwλ+∥L∞(Σλ−)≤cR(Σλ−)2α∥φ∥L∞(Σλ−)supx∈Σλ−wλ+(x),

that is,

1≤c2pR(Σλ−)2α∥u∥L∞p−1, (2.11)

where c is a positive constant independent of Σλ− and

R(Σλ−)=infr>0:|Br(x)∖Σλ−|≥12|Br(x)|,∀x∈Σλ−.

Choosing λ > λ₀ close enough to λ₀, R( Σλ− ) is small, a contraction is derived from (2.11) and then

wλ=wλ+≥0inΣλ−.

But this is a contradiction with the definition of Σλ− , so we have that w_λ ≥ 0 in Σ_λ.

We next claim that if w_λ ≥ 0 and w_λ ≢ 0 in Σ_λ, then w_λ > 0 in Σ_λ. Assuming this claim is true, we complete the proof, since the function u is positive in Ω and u = 0 on ∂Ω, so that w_λ is positive on ∂Ω ∩ ∂Σ_λ and then, by continuity w_λ ≢ 0 in Σ_λ.

Now we prove above claim. In fact, assume there exists x₀ ∈ Σ_λ such that w_λ(x₀) = 0, that is, u_λ(x₀) = u(x₀). One hand we have that

(−Δ)αwλ(x0)=(−Δ)αuλ(x0)−(−Δ)αu(x0)=uλp(x0)−up(x0)=0. (2.12)

On the other hand, let A_λ = {(x₁, x′) ∈ ℝ^N | x₁ > λ}, since w_λ(z_λ) = –w_λ(z) for any z ∈ ℝ^N and w_λ(x₀) = 0, we find

(−Δ)αwλ(x0) = −∫Aλwλ(z)|x0−z|N+2αdz−∫Aλwλ(zλ)|x0−zλ|N+2αdz=−∫Aλwλ(z)(1|x0−z|N+2α−1|x0−zλ|N+2α)dz.

Since |x₀ – z_λ| > |x₀ – z| for z ∈ A_λ, w_λ(z) ≥ 0 and w_λ(z) ≢ 0 in A_λ, then

(−Δ)αwλ(x0)<0, (2.13)

which contradicts (2.12).
Our purpose of this step is to move the planes forward for any λ > λ₀ up to

λ1:=sup{λ>λ0|wμ>0inΣμforanyμ∈(λ0,λ)}=+∞.

By contradiction, we assume that λ₁ < +∞. Since for any λ ∈ (0, λ₁), we have that w_λ > 0 in Σ_λ, by the continuity, we derive that w_λ₁ ≥ 0 in Σ_λ₁ and w_λ₁ ≢ 0 in Σ_λ₁. Thus, by the claim just proved above, we have w_λ₁ > 0 in Σ_λ₁.

Now we claim that if w_λ > 0 in Σ_λ₁, then there exists ϵ > 0 such that w_λ > 0 in Σ_λ for all λ ∈ [λ₁, λ₁ + ϵ). This claim directly implies that λ₁ = +∞, completing Step 2.

In fact, if λ₁ < +∞, then under our hypotheses, we have that Σ_λ is compact. Denote

Dμ={x∈Σλ|dist(x,∂Σλ)≥μ}

for μ > 0 small. Since w_λ₁ > 0 in Σ_λ₁ and D_μ is compact, then there exists μ₀ > 0 such that w_λ₁ ≥ μ₀ in D_μ. By continuity of w_λ(x), for ϵ > 0 small enough and any λ ∈ (λ₁, λ₁ + ϵ), we have that

wλ≥0inDμ.

As a consequence,

Σλϵ−⊂Σλϵ∖Dμ

and R( Σλϵ− ) is small if ϵ and μ are small. Using (2.8) and proceeding as in Step 1, we have that for all x ∈ Σλϵ− ,

(−Δ)αwλϵ+(x) = (−Δ)αuλϵ(x)−(−Δ)αu(x)−(−Δ)αwλϵ−(x)≥(−Δ)αuλϵ(x)−(−Δ)αu(x)=uλϵp(x)−up(x)=−φ(x)wλϵ+(x),

where φ(x)=−uλϵp(x)−up(x)uλϵ(x)−u(x) is bounded. Since wλϵ+ = 0 in (Σλϵ−)c and | Σλϵ− | is small, for ϵ and μ small, from the analysis Step 1, we obtain that w_{λ_ϵ} ≥ 0 in Σ_{λ_ϵ}. Since λ_ϵ > 0 and w_{λ_ϵ} ≢ 0 in Σ_{λ_ϵ}, as before we have w_{λ_ϵ} > 0 in Σ_{λ_ϵ}, completing the proof of the claim. The proof ends.□

Proof of Theorem 1.1

We prove this argument by contradiction. Assume that u ≥ 0 is nontrivial solution of (1.1). When Ω verifies (𝓓), then by Proposition 2.1, we have that u satisfies (2.4). Let

um(x)=u(x′,xN+m),∀x∈RN,

then {u_m}_m is an increasing and bounded sequence of functions and satisfies

(−Δ)αu=upinΩ−m.

Note that Ω_t ⊂ Ω_t′ for 0 ≤ t ≤ t′ < +∞,

Ω−m⊂Ω−kifk>m,limm→+∞Ω−m=RN,

from the regularity results in [9, Theorem 2.1] and the stability property [9, Theorem 2.4], we obtain that

u∞:=limm→+∞um

is a bounded classical solution to

(−Δ)αu=upinΩ∞×R,u=0inRN∖(Ω∞×R)ifΩ∞≠RN−1.

By the increasing property of {u_m}_m, we have that

u∞(x′,xN)=u∞(x′,yN)foranyxN,yN∈R,

which implies that u_∞ is x_N-independent. Letting v_∞(x′) = u_∞(x′, x_N), by the standard argument, we have that

(−Δ)RN−1αv∞(x′)=cN(−Δ)αu∞(x),∀x∈Ω∞×R,

then v_∞ is a positive, bounded and classical solution for

(−Δ)RN−1αv∞=cNv∞pinΩ∞, (2.14)

subject to v_∞ = 0 in ℝ^N–1 ∖ Ω_∞ if Ω_∞ ≠ ℝ^N–1.

A contradiction is derived by the fact that problem (2.14) has no positive bounded classical solution when

Ω_∞ = ℝ^N–1 and 0 1 + 2α, otherwise for p > 0, 1 ≤ N ≤ 1 + 2α by [13, Theorem 1.2];
Ω_∞ ⊂ ℝ^N–1 is a star-shaped domain with respect to infinity and 1 ≤ p ≤ N−1+2αN−1−2α with N > 1 + 2α by [14, Theorem 1.5];
Ω_∞ = ℝ^N–2 × [0, +∞) and 1 2 + 2α, otherwise for p > 1, 1 ≤ N ≤ 2 + 2α by [14, Theorem 1.2];
Ω_∞ is bounded, C² and star-shape in ℝ^N–1, N > 1 + 2α, and p ≥ N−1+2αN−1−2α . by [14, Corollary 1.2].

As a consequence, we obtain the nonexistence results of (1.1) in Theorem 1.1.□

Proof of Corollary 1.1

When problem (1.1) has a positive solution u, then Proposition 2.1 and regularity results guarantee that v_∞(x′) = lim_m→+∞u(x′, x_N + m) is a positive, bounded and classical solution for problem

(−Δ)RN−1αv∞=cNv∞pinΩ∞andv∞=0inRN−1∖Ω∞. (2.15)

Furthermore, we note that v_∞,∞(x″) = lim_m→+∞ v_∞(x″, x_N–1 + m) is a positive, bounded and classical solution of

(−Δ)RN−2αv∞,∞=cNv∞pinΩ∞,∞andv∞,∞=0inRN−2∖Ω∞,∞, (2.16)

subject to v_∞,∞ = 0 in ℝ^N–2 ∖ Ω_∞,∞ if Ω_∞,∞ ≠ℝ^N–2. Then the same conclusion is obtained as the proof of Theorem 1.1.□

Acknowledgements

Y. Wang is supported by NNSF of China, No: 11661045, by the Jiangxi Provincial Natural Science Foundation, No: 20202ACBL201001, 20202BAB201005, and by Key R&D plan of Jiangxi Province, No:20181ACE50029. Y. Wei is supported by NNSF of China, No: 11871242.

References

[1] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of ℝ^N, J. Math. Pures Appl. 87 (2007), 537-561.10.1016/j.matpur.2007.03.001Search in Google Scholar

[2] S. Alarcon, J. Garcia-Melian and A. Quaas, Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian, Ann. della Scuola Normale Super. di Pisa. 16 (2016), no. 1, 129-158.10.2422/2036-2145.201402_007Search in Google Scholar

[3] M. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math. 84 (2001), 1-49.10.1007/BF02788105Search in Google Scholar

[4] H. Chen, Liouville theorem for the fractional Lane-Emden equation in an unbounded domain, J. Math. Pures Appl. 111 (2018), 21-46.10.1016/j.matpur.2017.07.010Search in Google Scholar

[5] H. Chen, R. Peng and Zhou, Nonexistence of positive supersolution to a class of semilinear elliptic equations and systems in an exterior domain, Sci. China Math. 63, 1307-1322(2020).10.1007/s11425-018-9447-ySearch in Google Scholar

[6] M. Bhakta and P.T. Nguyen, On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures, Adv. Nonlinear Anal. 9 (2020), no. 1, 1480-1503.10.1515/anona-2020-0060Search in Google Scholar

[7] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237-1262.10.1017/S0308210511000746Search in Google Scholar

[8] G. Molica Bisci, V.D. Radulescu and R. Servadei, Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016.10.1017/CBO9781316282397Search in Google Scholar

[9] H. Chen, P. Felmer and A. Quaas, Large solution to elliptic equations involving fractional Laplacian, Ann. I. H. Poincare-Analyse Non Lineaire 32 (2015), 1199-1228.10.1016/j.anihpc.2014.08.001Search in Google Scholar

[10] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian, regularity up to the boundary, J. Math. Pures Appl. 101 (2014), no. 3, 275-302.10.1016/j.matpur.2013.06.003Search in Google Scholar

[11] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mecha. Anal. 213 (2014), no. 2, 587-628.10.1007/s00205-014-0740-2Search in Google Scholar

[12] W. Chen, Y. Fang, and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math. 274 (2015), 167-198.10.1016/j.aim.2014.12.013Search in Google Scholar

[13] M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Comm. Contem. Math. 18 (2016), no. 1, 1-25.10.1142/S0219199715500121Search in Google Scholar

[14] M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal. 263 (2012), 2205-2227.10.1016/j.jfa.2012.06.018Search in Google Scholar

[15] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330-343.10.1002/cpa.20116Search in Google Scholar

[16] A. Harrabi and B. Rahal, Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions, Adv. Nonlinear Anal. 8 (2019), no. 1, 193-202.10.1515/anona-2016-0168Search in Google Scholar

[17] P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Contem. Math. 16 (2013), no. 1, 1-24.10.1142/S0219199713500235Search in Google Scholar

Received: 2019-04-26

Accepted: 2020-07-08

Published Online: 2020-08-22

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Keywords for this article

Fractional Laplacian; Lane-Emden equation; Nonexistence

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