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Existence and nonexistence of solutions to elliptic equations involving the Hardy potential
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J. Math. Anal. Appl. ••• (••••) •••–•••
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Existence and nonexistence of solutions to elliptic equations involving the Hardy potential Ying Wang Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 9 June 2016 Available online xxxx Submitted by C. Gutierrez
The purpose of this paper is to study the nonexistence of nonnegative super solutions to the problem (−Δ)α u +
Keywords: Hardy potential Fractional Laplacian Singular solution Liouville theorem
μ u ≥ Qup |x|2α
in
RN \ K,
(0.1)
where α ∈ (0, 1], μ ∈ R, p > 0, K is a compact set in RN with N ≥ 1 and Q is a potential in RN \ K satisfying that lim inf |x|→+∞ Q(x)|x|γ > 0 for some γ < 2α. When α = 1, (−Δ)α is the Laplacian operator, and when α ∈ (0, 1), it is the fractional Laplacian which is a typical nonlocal operator. In this paper, we find the critical exponent p∗ > 1 depending on α, μ and γ such that problem (0.1) has no nontrivial nonnegative super solutions for 0 < p < p∗ . Furthermore, we also consider the existence and nonexistence of isolated singular solutions to the equation ⎧ α ⎪ ⎨ (−Δ) u + ⎪ ⎩
lim
μ u = Qup |x|2α
|x|→+∞
in
RN \ {0},
u(x) = 0,
where μ > 0, p > 0 and Q(x) = (1 + |x|)−γ with γ ∈ (0, 2α). © 2017 Elsevier Inc. All rights reserved.
1. Introduction We are concerned with the nonexistence of nontrivial nonnegative super solutions to the problem (−Δ)α u +
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2017.07.002 0022-247X/© 2017 Elsevier Inc. All rights reserved.
μ u ≥ Qup |x|2α
in
RN \ K,
(1.1)
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where α ∈ (0, 1], μ ∈ R, p > 0, K is a compact set in RN with N ≥ 1 and Q is a potential in RN \K satisfying that lim inf |x|→+∞ Q(x)|x|γ > 0 for some γ < 2α. When α = 1, the operator (−Δ)α is the Laplacian, and when α ∈ (0, 1), it is the fractional Laplacian defined in the principle value sense as (−Δ)α u(x) = cN,α lim+ →0
RN \B (0)
u(x) − u(x + z) dz, |z|N +2α
where B (0) is the ball centered at the origin with radius , cN,α is the normalized constant cN,α = 22α απ − 2
N
Γ( N +2α 2 ) Γ(1 − α)
and Γ is the Gamma function. The fractional Laplacian is a nonlocal operator, so if Lebesgue measure |K| = 0, we have to assume moreover that u ≥ 0 a.e. in K. The semilinear elliptic equations involving the fractional Laplacian and the related Sobolev spaces have been studied extensively, see [1,5,10–12,20,21] and the references therein. It is known that the fundamental solution and Comparison Principle play an important role in the obtention of the nonexistence of solutions to semilinear elliptic equations. In the Laplacian case, the authors in [2,6] used the fundamental solution of Laplacian and Comparison Principle to obtain the nonexistence of positive solutions to the problem −Δu = Qup
in RN \ K.
In the fractional case, i.e. α ∈ (0, 1), [14] shows the nonexistence results of (1.1) when μ = 0, K = ∅, Q = 1 N , by using the fundamental solution of the fractional Laplacian and Comparison Principle. and p ≤ N −2α To study the nonexistence of nonnegative nontrivial super solutions of (1.1), we first clarify the fundamental solution of (−Δ)α + |x|μ2α as follows. Proposition 1.1. Assume that α ∈ (0, 1] and N ∈ N. (i) When N > 2α, let us denote
τ¯ = −
N − 2α 2
and
μ0 =
⎧ 2 ⎨ − (N −2) 4
2 N +2α ) 4 ⎩ −22α−1 cN,α Γ2 ( N −2α Γ ( 4 )
if
α = 1,
if
α ∈ (0, 1),
(1.2)
then (−Δ)α |x|τ¯ + μ0 |x|τ¯−2α = 0,
∀ x ∈ RN \ {0}.
For μ > μ0 , there exists a unique τα (μ) ∈ (−N, τ¯) such that φτα (μ) (x) := |x|τα (μ) is a fundamental solution of (−Δ)α +
μ |x|2α ,
(1.3)
i.e.
(−Δ)α φτα (μ) +
μ φτ (μ) = 0 |x|2α α
in RN \ {0}.
(1.4)
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(ii) When N ≤ 2α, then for μ > 0, there exists a unique τα (μ) ∈
(−∞, 0)
if
α = 1,
(−N, 0)
if
α ∈ (0, 1)
such that φτα (μ) defined in (1.3) is a fundamental solution of (−Δ)α + |x|μ2α . Furthermore, the mapping μ → τα (μ) is strictly decreasing in (μ0 , +∞) if N > 2α and in (0, +∞) if N ≤ 2α, and lim τα (μ) =
−∞
if
α = 1,
−N
if
α ∈ (0, 1).
μ→+∞
Note that for α = 1, τα (μ) has the explicit formula τα (μ) = −
(N − 2)+ +
[(N − 2)+ ]2 + 4μ . 2
For α ∈ (0, 1) and N > 2α, denote e1 = (1, 0, · · · , 0) ∈ RN , cα (τ ) = −
cN,α 2
RN
|x − e1 |τ + |x + e1 |τ − 2 dx, |x|N +2α
(1.5)
then the function cα (·) + μ has two zero points for μ ∈ (μ0 , 0), one zero point when μ = μ0 , and τα (μ) is the smaller zero point. μ0 is the best constant of the fractional Hardy inequalities, see the references [4,15]. Moreover, Lemma 3.1 in [13] provides an explicit expression of cα (τ ), cα (τ ) = 22α
Γ( N 2+τ )Γ( 2α−τ 2 ) . −τ N +τ −2α Γ( 2 )Γ( ) 2
(1.6)
Then the nonexistence of super solutions to (1.1) states as follows. Theorem 1.1. Suppose that α ∈ (0, 1], N ∈ N and Q is a nonnegative function satisfying lim inf Q(x)|x|γ > 0
|x|→+∞
for some γ < 2α. Then problem (1.1) has no nontrivial nonnegative super solution for 0 < p < p∗μ,γ , where
p∗μ,γ
⎧ 2α−γ ⎪ 1 + −τ ⎪ α (μ) ⎨ 2α−γ = 1 + N −2α ⎪ ⎪ ⎩ +∞
We remark that if μ > 0, the mapping μ →
if
μ > 0,
if
μ≤0
and
N > 2α,
if
μ≤0
and
N ≤ 2α.
2α−γ −τα (μ)
2α − γ = lim μ→+∞ −τα (μ)
(1.7)
is decreasing and
0
if
α = 1,
2α−γ N
if
α ∈ (0, 1).
2α−γ 2α−γ When N > 2α and μ ∈ (μ0 , 0], we observe that 1 + −τ > 1+ N −2α , but we do not know the nonexistence α (μ)
2α−γ 2α−γ α results when p ∈ 1 + N −2α , 1 + −τα (μ) due to the lack of comparison principle for the operator (−Δ) +
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μ |x|2α
in unbounded domains. To prove the nonexistence results, an essential tool is the Hadamard property derived by the fundamental solution and the Comparison Principle, see [2,6,14]. In this paper, we employ a new method to show the nonexistence. Our idea is to obtain an initial decay at infinity from the fundamental solution by applying the Comparison Principle, then an iterating technique is used to improve the power of the decay for p < p∗μ,γ until it makes the solution blow up everywhere. It is worthy to point out that for μ ∈ (μ0 , 0), the initial decay is not |x|τα (μ) , but |x|2α−N . This leads to the independence of the critical exponent p∗μ,γ with the parameter μ in (μ0 , 0). Our second aim of this paper is to show that p∗μ,γ is sharp for the nonexistence when μ > 0. To this end, we take K = {0}, 0 < γ < 2α, Q(x) = (1 + |x|)−γ ,
∀x ∈ RN \ {0},
(1.8)
and to consider solutions of the problem ⎧ α ⎪ ⎨ (−Δ) u + ⎪ ⎩
μ u = Qup |x|2α
lim
|x|→+∞
in
RN \ {0}, (1.9)
u(x) = 0.
Theorem 1.2. Let α ∈ (0, 1], μ > 0 and the function Q satisfy (1.8) with 0 < γ < 2α. (i) When 0 < p < p∗μ,γ ,
(1.10)
then problem (1.9) has no nontrivial nonnegative solution. (ii) When p∗μ,γ ≤ p < p∗μ,0 ,
(1.11)
then there exists k∗ > 0 such that for any k ∈ (0, k∗ ), problem (1.9) has a positive solution u satisfying lim u(x)|x|−τα (μ) = k.
(1.12)
|x|→0+
Note that p∗μ,γ < p∗μ,0 = 1 + −τ2α . When p < p∗μ,γ , the nonexistence result in Theorem 1.2 (i) follows α (μ) by Theorem 1.1 directly and for the existence, we have to mention Lions’ work [18], where an equivalence is built between the elliptic problem −Δu = up
in
Ω \ {0},
u = 0 in ∂Ω
(1.13)
and the one involving the Dirac mass at the origin −Δu = up + kδ0
in Ω,
where k > 0 and Ω is a bounded, smooth domain containing the origin. Then for 1 < p < singular solutions of (1.13) could be obtained by iterating the sequence of v0 = G1 [kδ0 ]
(1.14) N N −2α ,
positive
p and vn = G1 [vn−1 ] for n = 1, 2, · · · ,
where k > 0, G1 is the Green operator of −Δ. When k > 0 suitably small, a barrier function of (1.14) could be constructed and then the limit of {vn }n is the desired solution. More general second order problems with
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Hardy potentials in bounded domain considered in [3,16,17] and the singularities of semilinear fractional equations in bounded domain have been studied in [7,8]. However, for μ > 0, the singularity of (1.9) could not be expressed by the Dirac mass due to the Hardy potential, i.e. the term |x|μ2α u is no longer in L1loc (RN ). Another difficulty is the unbounded domain RN \{0}. To overcome these difficulties, we consider the sequence {vn }n , the solution of equation ⎧ ⎨ ⎩
(−Δ)α vn + lim
|x|→+∞
μ |x|2α vn
p = Qvn−1
in
RN \ {0},
lim vn (x)|x|−τα (μ) = k,
vn (x) = 0 and
|x|→0+
where v0 = k|x|τα (μ) , then we approximate the singular solutions of (1.9) by {vn }n as n → ∞. An upper bound has to be constructed to control the singularity at the origin and the decay at infinity. The rest of the paper is organized as follows. In Section 2, we clarify the fundamental solutions of (1.4) and provide a version of Comparison Principle. Section 3 is devoted to prove the nonexistence of nontrivial nonnegative solutions to problem (1.1). Finally, we prove the existence of isolated singular solutions of (1.9) in Section 4. 2. Preliminary This section is devoted to clarify the fundamental solution of (−Δ)α + |x|μ2α and the Comparison Principle. For convenience, let us denote φτ (x) = |x|τ ,
x ∈ RN \ {0},
(2.1)
where τ < 0. By direct computation, we have that −Δφτ (x) = −τ (τ + N − 2)|x|τ −2 ,
∀ x ∈ RN \ {0},
and the mapping τ → −τ (τ + N − 2) is strictly concave in (−∞, 0) and lim [−τ (τ + N − 2)] = −∞.
τ →−∞
When α ∈ (0, 1), by the definition of the fractional Laplacian, we have that cN,α (−Δ) φτ (x) = − 2
α
RN
|x + y|τ + |x − y|τ − 2|x|τ dy |y|N +2α
cN,α τ −2α |x| =− 2
RN
|ex + z|τ + |ex − z|τ − 2 dz, |z|N +2α
τ τ x x −z| −2 where ex = |x| . We know that RN |ex +z||z|+|e dz is independent of x and so we may replace ex by N +2α e1 = (1, 0, · · · , 0), combining with the formula of cα (τ ) in (1.5), we have that (−Δ)α φτ (x) = cα (τ )|x|τ −2α , From (1.6), we have the following estimates. Lemma 2.1. Suppose that α ∈ (0, 1). (i) If N ≤ 2α, then cα (τ ) < 0 for τ ∈ (−N, 0).
x ∈ RN \ {0}.
(2.2)
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(ii) If N > 2α, then ⎧ < 0, ⎪ ⎪ ⎨ cα (τ ) = 0, ⎪ ⎪ ⎩ > 0,
τ ∈ (−N, 2α − N ), τ = 2α − N,
(2.3)
τ ∈ (2α − N, 0).
Moreover, cα (·) is strictly concave in (−N, 0), lim
τ →(−N )+
cα (τ ) = −∞
and
lim cα (τ ) = 0.
τ →0−
(2.4)
Proof. From (1.6), we have that cα (τ ) = 22α
Γ( N 2+τ )Γ( 2α−τ 2 ) , −τ N +τ −2α Γ( 2 )Γ( ) 2
−τ N +τ −2α where Γ( N 2+τ ), Γ( 2α−τ ). When τ → 0− , 2 ), Γ( 2 ) > 0, the signs of cα (·) are decided by the ones of Γ( 2 −τ N +τ + we have that Γ( 2 ) → +∞, the others keep bounded and when τ → (−N ) , Γ( 2 ) → +∞, the others keep bounded. For the convexity, by directly calculus, we have that
cα (τ ) = −
cN,α 2
RN
|e1 − x|τ log |e1 − x| + |e1 + x|τ log |e1 + x| dx |x|N +2α
and cα (τ )
cN,α =− 2
RN
|e1 − x|τ (log |e1 − x|)2 + |e1 + x|τ (log |e1 + x|)2 dx < 0. |x|N +2α
The proof ends. 2 Proposition 2.1. Assume that α ∈ (0, 1), N > 2α, φτ is given by (2.1) and τ¯ = −
N − 2α , 2
μ0 = −22α−1 cN,α
Γ2 ( N +2α 4 ) . N −2α 2 Γ ( 4 )
Then (−Δ)α ϕτ¯ (x) + μ0 |x|τ¯−2α = 0,
x ∈ RN \ {0}
(2.5)
and μ0 = −
sup τ ∈(−N, 0)
cα (τ ).
(2.6)
Proof. From Lemma 3.1 in [13], we have that (2.5) holds true. To obtain (2.6), on the one hand, for any τ ∈ (−N, 0) and u ∈ Cc∞ (RN ), we have that [u(x) − u(y)]2 + u(x)2
u(y) 2 u(x) φτ (y) − φτ (x) φτ (x) − wτ (y) + u(y)2 = φτ (x)φτ (y)[ − ] ≥ 0, φτ (x) φτ (y) φτ (x) φτ (y)
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then RN RN
⎞ ⎞ ⎛ ⎛ u(x)2 ⎝ φτ (x) − φτ (y) ⎠ u(y)2 ⎝ φτ (y) − φτ (x) ⎠ dy dx + dx dy φτ (x) |x − y|N +2α φτ (y) |x − y|N +2α
≥ RN
=
=
[u(x) − u(y)]2 dxdy |x − y|N −2α
2 cN,α
RN
u(x)2 RN
2cα (τ ) cN,α
RN
RN
RN
(−Δ)α φτ (x) dx φτ (x)
u(x)2 dx. |x|2α
2 On the other hand, Yafaev in [22] indicated that − cN,α μ0 is the sharp constant in the Hardy–Rellich inequality as
RN RN
2 |u(x) − u(y)|2 dxdy ≥ − μ0 N +2α |x − y| cN,α
RN
u(x)2 dx. |x|2α
Then μ0 ≤ −
sup τ ∈(−N,0)
cα (τ ),
which, combining with the fact that cα (¯ τ ) = −μ0 , implies that cα (¯ τ ) ≥ supτ ∈(−N,0) cα (τ ). As a consequence, (2.6) holds true. 2 Proof of Proposition 1.1. When α = 1, the statements in Proposition 1.1 are clear. For α ∈ (0, 1), by Proposition 2.1, we have that cα (¯ τ ) = −μ0 for τ¯ = − N −2α 2 . Since cα (·) is concave and strictly increasing in (−N, τ¯). Then for any μ ∈ [μ0 , +∞), there exists a unique τα (μ) ∈ (−N, τ¯) such that μ = −cα (τα (μ)). The proof ends. 2 Remark 2.1. For μ ∈ (μ0 , 0), there exists a unique ςα (μ) ∈ (¯ τ , 0) such that cα (ςα (μ)) + μ = 0. It is obvious that ςα (μ0 ) = τα (μ0 ) and μ → ςα (μ) is increasing for μ ∈ (μ0 , 0). The following Comparison Principle plays an important role in the obtention of nonexistence results for (1.1). Lemma 2.2. Let α ∈ (0, 1), μ ≥ 0, D be a C 2 domain such that 0 ∈ / D, functions f1 , f2 ∈ C(D) satisfy that f2 ≥ f1 in D and g1 , g2 ∈ L1 (RN \ D, 1+|x|dxN +2α ) satisfy that g2 ≥ g1 a.e. in RN \ D. Assume more that u1 is a super solution of
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μ |x|2α
(−Δ)α u +
u = fi
in
D,
u = gi
in
RN \ D
(2.7)
with i = 1 and u2 is a sub solution of (2.7) with i = 2. Suppose that lim inf (u2 − u1 )(x) ≥ 0, x→∂D
(2.8)
in addition, if D is an unbounded domain, u1 and u2 satisfy that lim inf (u2 − u1 )(x) ≥ 0,
|x|→+∞
then u 2 ≥ u1
in D.
Proof. If inf x∈D (u2 − u1 )(x) < 0, then there exists a point x0 ∈ D such that (u2 − u1 )(x0 ) = inf (u2 − u1 )(x) = essinf x∈RN (u2 − u1 )(x) < 0, x∈D
which implies that
μ |x0 |2α (u2
− u1 )(x0 ) < 0 and
(−Δ) (u2 − u1 )(x0 ) = α
RN
(u2 − u1 )(x0 ) − (u2 − u1 )(z) dz < 0. |x0 − z|N +2α
Thus, (−Δ)α (u2 − u1 )(x0 ) +
μ (u2 − u1 )(x0 ) < 0, |x|2α
combining with (2.7), this contradicts f2 (x0 ) − f1 (x0 ) ≥ 0. 2 Remark 2.2. Lemma 2.2 holds for α = 1 by replacing the boundary type condition in (2.7) by u = gi
on ∂D.
Corollary 2.1. Let α ∈ (0, 1], μ ≥ 0, functions f1 , f2 ∈ C(RN \ {0}) satisfy that f2 ≥ f1 in RN \ {0}. Assume that u1 is a positive super solution of (−Δ)α u +
μ u = fi |x|2α
in
RN \ {0}
with i = 1 and u2 is a positive sub solution of (2.9) with i = 2. Suppose that lim inf
x=0,x→0
u2 (x) > 1, u1 (x)
and lim inf u2 (x) ≥ lim sup u1 (x).
|x|→+∞
|x|→+∞
Then u2 ≥ u1
in RN \ {0}.
(2.9)
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Remark 2.3. If α ∈ (0, 1] and μ ∈ (μ0 , 0), the above Comparison Principle fails. We give a counterexample as follows. Let ∈ (0, |μ|), u1 (x) = |x|ςα (μ+) and u2 (x) = |x|τα (μ) , where ςα (μ) is given by Remark 2.1, we observe that lim inf
x=0,x→0
u2 (x) > 1, u1 (x)
lim inf u2 (x) ≥ lim sup u1 (x),
|x|→+∞
|x|→+∞
and (−Δ)α u2 +
μ − μ u2 = 0 > u1 (x) = (−Δ)α u1 + 2α u1 |x|2α |x|2α |x|
in RN \ {0}.
But it is not true that u2 ≥ u1 in RN \ {0}. 3. Nonexistence We prove the nonexistence of nontrivial nonnegative solutions of (1.1) by contradiction. Let u be a nonnegative nontrivial solution of problem (1.1), then we will obtain a contradiction from the decay of u at infinity. Without loss generality, we may assume that K ⊂ B1 (0) and Q(x) ≥ q0 |x|−γ
for |x| > 4,
(3.1)
where q0 > 0. Proposition 3.1. Assume that α ∈ (0, 1], γ < 2α, p > 0 and u is a nonnegative nontrivial solution of (1.1). Then for some b0 > 0, (i) if μ > 0, we have that u(x) ≥ b0 |x|τα (μ) ,
∀ |x| > 4;
(3.2)
u(x) ≥ b0 |x|2α−N ,
∀ |x| > 4;
(3.3)
(ii) if N > 2α and μ ≤ 0, we have that
(ii) if N ≤ 2α and μ ≤ 0, for any τ < 0, we have that u(x) ≥ b0 |x|τ ,
∀ |x| > 4.
(3.4)
Proof. For α = 1, the proof follows the procedure of the case of α ∈ (0, 1). So we concentrate on the case α ∈ (0, 1). We first deal with part (i). To this end, let us denote ⎧ t|x|τα (μ) ⎪ ⎪ ⎨ wt,ν (x) = t(|x|τα (μ) + ν) ⎪ ⎪ ⎩ 0 where t, ν > 0 will be chosen later. We observe that
for |x| ≥ 3, for 2 < |x| < 3, for |x| ≤ 2,
(3.5)
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for |x| > 4
(−Δ)α wt,ν (x) = t(−Δ)α w1,ν (x) and it follows by Proposition 1.1 that for |x| > 4, μ (−Δ) w1,ν (x) + 2α w1,ν (x) = cN,α |x|
|y|τα (μ) dy − cN,α ν |x − y|N +2α
α
B2 (0)
⎡
⎢ ≤ 2−N −2α cN,α ⎣
B3 (0)\B2 (0)
1 dy |x − y|N +2α ⎤
⎥ |y|τα (μ) dy − ν|B3 (0) \ B2 (0)|⎦
B2 (0)
≤ 0,
if we choose that ν = ν0 = that
B2 (0)
|y|τα (μ) dy
|B3 (0)\B2 (0)|
. Since u is continuous in RN \ B1 (0), there exists t0 > 0 such
u ≥ t0 w1,ν0 = wt0 ,ν0
in
B4 (0) \ B2 (0).
Observe that (−Δ)α u(x) +
μ u(x) ≥ Q(x)u(x)p ≥ 0 and |x|2α
lim sup wt,ν (x) = 0,
|x|→+∞
then it follows by Lemma 2.2 that u ≥ t0 w1,ν0 , which implies (3.2). We next prove part (ii). Re-define the barrier function wt,ν replacing τα (μ) by 2α − N . By direct computation, we have that (−Δ)α w1,ν0 (x) ≤ 0 ≤
where ν0 =
B2 (0)
|x|2α−N dx
|B3 (0)\B2 (0)|
−μ u(x) + Q(x)u(x)p = (−Δ)α u(x), |x|2α
∀x ∈ RN \ B4 (0),
. Taking t = t0 > 0 small, we have that u ≥ wt0 ,ν0
in B4 (0) \ B2 (0),
so (3.3) holds true for μ ≤ 0. Finally, we deal with part (iii). In this case, the power τα (μ) of the barrier function wt,ν could be replaced by any τ ∈ (−N, 0) and we may have that (−Δ)α w1,ν0 (x) ≤ 0. We omit the rest of the proof. 2 The next step is to improve the decay of u at infinity. To this end, we introduce some notations. Let τ0 =
τα (μ)
if μ > 0,
2α − N
if
μ ≤ 0 and N > 2α
(3.6)
and {τj }j be the sequence generated by τj = 2α − γ + pτj−1
for j = 1, 2, 3 · · · .
(3.7)
Lemma 3.1. Assume that μ > 0 with N ≥ 1 or μ ≤ 0 with N > 2α, p∈
γ − 2α 0, 1 + τ0
,
(3.8)
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then {τj }j is a strictly increasing sequence and there exists j0 ∈ N such that τ j0 ≥ 0 Proof. For p ∈ (0, 1 +
γ−2α τ0 ),
and
τj0 −1 < 0.
(3.9)
we have that τ1 − τ0 = 2α − γ + (p − 1)τ0 > 0
and τj − τj−1 = p(τj−1 − τj−2 ) = pj−1 (τ1 − τ0 ),
(3.10)
which imply that the sequence {τj }j is increasing. If p ≥ 1, our conclusions are obvious. If p ∈ (0, 1), we have that in the case that τ1 ≥ 0, we are done, and in the case that τ1 < 0, we deduce from (3.10) that 1 − pj (τ1 − τ0 ) + τ0 1−p 1 2α − γ (τ1 − τ0 ) + τ0 = >0 → 1−p 1−p
τj =
then there exists j0 > 0 satisfying (3.9).
as j → +∞,
2
Remark 3.1. We note that p∗μ,γ = 1 +
γ − 2α . τ0
From the strictly increasing monotonicity of the sequence {τj }j , we have that τj−1 < τj = 2α − γ + pτj−1 , that is, τj−1 p − γ > τj−1 − 2α.
(3.11)
Proposition 3.2. Let τ0 and {τj }j be defined by (3.6) and (3.7) respectively, and u be a nonnegative solution of (1.1) satisfying u(x) ≥ cj |x|τj ,
∀ |x| > 4
for some cj > 0 and j ≤ j0 − 2, where j0 is from Lemma 3.1. Then for p ∈ (0, 1 + cj+1 > 0 such that u(x) ≥ cj+1 |x|τj+1 ,
∀ |x| > 4.
Proof. Case 1. μ > 0. Let ⎧ t|x|τj+1 ⎪ ⎪ ⎨ vt,ν (x) = t(|x|τj+1 + ν) ⎪ ⎪ ⎩ 0
for |x| ≥ 3, for 2 < |x| < 3, for |x| ≤ 2,
(3.12) γ−2α τ0 ),
there exists
(3.13)
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where t, ν > 0 will be chosen later. We observe that (−Δ)α vt,ν (x) = t(−Δ)α v1,ν (x) for |x| > 4 and it follows by Proposition 1.1 that for |x| > 4, (−Δ)α v1,ν (x) +
μ v1,ν (x) |x|2α
= (−Δ)α |x|τj+1 +
μ |x|τj+1 + cN,α |x|2α
|y|τj+1 dy − cN,α ν |x − y|N +2α
B2 (0)
⎡
⎢ ≤ (cα τj+1 + μ)|x|τj+1 −2α + 2−N −2α cN,α ⎣
1 dy |x − y|N +2α
B3 (0)\B2 (0)
⎤
⎥ |x|τj+1 dx − ν|B3 (0) \ B2 (0)|⎦
B2 (0)
≤ (cα τj+1 + μ)|x|τj+1 −2α ,
if we choose that ν = νj =
B2 (0)
|x|τj+1 dx
|B3 (0)\B2 (0)|
. From (3.12), we have that
Q(x)u(x)p ≥ q0 cpj |x|pτj −γ , and then there exists tj1 > 0 such that for t ∈ (0, tj1 ], (−Δ)α vt,ν (x) +
μ μ vt,ν (x) ≤ (−Δ)α u(x) + 2α u(x) for |x| > 4. |x|2α |x|
Furthermore, since u is continuous in RN \ B1 (0), there exists tj ≤ tj1 such that u ≥ vtj ,νj = tj v1,νj
in
B4 (0) \ B2 (0).
Applying Lemma 2.2, we have that u ≥ tj v1,νj , which implies (3.13) with cj+1 = tj . Case 2. μ < 0 with N > 2α. Since τj > τ0 = 2α − N , we have that (−Δ)α v1,νj (x) ≤ cα (τj+1 )|x|τj+1 −2α . By (3.11), we have that (−Δ)α u(x) = Q(x)u(x)p +
−μ u(x) ≥ cpj |x|pτj −γ |x|2α
and choosing t = tj > 0 small, we have that (−Δ)α u ≥ (−Δ)α vtj ,νj
in
RN \ B4 (0),
u ≥ vtj ,νj
in
by Lemma 2.2, we have that u ≥ tj v1,νj , which implies (3.13) with cj+1 = tj .
B4 (0) \ B2 (0), 2
Proof of Theorem 1.1. By contradiction, we assume that problem (1.1) has a nonnegative nontrivial super solution u ≥ 0. We first claim that u > 0 in RN \ B1 (0). For α = 1, the positivity of u follows by the strong maximum principle. For α ∈ (0, 1), since u is nonnegative nontrivial, then
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∅. x ∈ RN \ B1 (0) : u(x) > 0 =
If there exists x0 ∈ RN \ B1 (0) such that u(x0 ) = 0, then (−Δ) u(x0 ) = − α
RN
u(z) dz < 0, |x0 − z|N −2α
which implies that (−Δ)α u(x0 ) +
μ u(x0 ) < 0 = Q(x0 )u(x0 )p , |x0 |2α
which is impossible from (1.1). Thus, u > 0 in RN \ B1 (0). In the case that μ > 0 or μ ≤ 0 with N > 2α, from Proposition 3.1, there exists c0 > 0 such that u(x) ≥ c0 |x|τ0 ,
∀ |x| > 4,
(3.14)
where τ0 is defined by (3.6). For p ∈ (0, p∗μ,γ ), we use Proposition 3.2 to iterate, then we obtain that for any j ≤ j0 − 1, u(x) ≥ cj |x|τj ,
∀ |x| > 4,
(3.15)
where {τj }j is defined by (3.7) and cj > 0. In case that μ ≤ 0 with N ≤ 2α, (3.15) also holds when we take τ = τj0 −1 = − 2α−γ by Proposition 3.1 2p (iii). Next we use the decay of (3.15) with j = j0 − 1 to derive that u blows up everywhere. From (3.15) and (3.11) with j = j0 − 1, there exists r0 > 4 such that for |x| > r0 , (−Δ)α u(x) ≥ Q(x)u(x)p −
μ u(x) |x|2α
≥ q0 cpj0 −1 |x|τj0 −1 p−γ − cj0 −1 μ|x|τj0 −1 −2α ≥
q0 p c |x|τj0 −1 p−γ . 2 j0 −1
We note that if μ ≤ 0, the above inequality holds directly. In order to obtain a contradiction, we introduce some auxiliary functions vr with r > 8r0 , which is the solution of problem ⎧ (−Δ)α vr (x) = fr (x), ⎪ ⎪ ⎨ vr (x) = 0, ⎪ ⎪ ⎩ lim vr (x) = 0,
∀x ∈ RN \ Br0 (0), ∀x ∈ Br0 (0),
(3.16)
|x|→+∞
where fr (x) =
q0 p c |x|τj0 −1 p−γ χBr (0)\B4r0 (0) (x) 2 j0 −1
and χBr (0)\B4r0 (0) is the characteristic function of the set Br (0) \ B4r0 (0). By Lemma 2.2, for all r > 8r0 , we have that u(x) ≥ vr (x),
∀ x ∈ RN .
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Then for any x ∈ B6r0 (0) \ B4r0 (0), z ∈ Br (0) \ B8r0 (0), we have that |x − z| ≤ 2|z| and
|z|τj0 −1 p−γ dz |x − z|N −2α
u(x) ≥ vr (x) ≥ c¯cpj0 −1 Br (0)\B8r0 (0)
≥ c˜
|z|2α−N +τj0 −1 p−γ dz
Br (0)\B8r0 (0)
≥
⎧ τ ⎨ c[r j0 − (8r0 )τj0 ],
if τj0 > 0
⎩ c[log r − log(8r0 )]
if τj0 = 0
→ +∞
as r → +∞,
which contradicts that u satisfies (1.1). The proof ends. 2 Proof of Theorem 1.2 (i). If problem (1.9) has a nontrivial nonnegative solution u, then u satisfies (1.1), which contradicts Theorem 1.1 with 0 < p < p∗μ,γ . 2 4. Existence In this section, we are concerned with the existence of positive singular solutions of problem (1.9) when p∗μ,γ ≤ p < p∗μ,0 . Lemma 4.1. Assume that α ∈ (0, 1], μ > 0, Q satisfies (1.8) with 0 < γ < 2α, f is a nonnegative function in C 1 (RN \ {0}) satisfying f (x)|x|−τ +2α < +∞
sup
(4.1)
x∈RN \{0}
for τ ∈ (τα (μ), τα (μ) + γ]. Then problem ⎧ ⎨ (−Δ)α u + ⎩
lim
μ |x|2α u
|x|→+∞
= Qf
in
RN \ {0}, (4.2)
u(x) = 0
has a minimal positive solution uf such that lim sup uf (x)|x|−τ < +∞ |x|→0+
and
uf (x) ≤ c|x|τα (μ) ,
x ∈ RN \ {0},
for some c > 0. Moreover, the mapping f → uf is increasing. Proof. Let η0 : [0 + ∞) → [0, 1] be a C ∞ nondecreasing function such that η0 = 1 in [2, +∞) and η0 = 0
in [0, 1],
(4.3)
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denote ηn (t) = η0 (nt)[1 − η0 (n−1 t)]
for n ∈ N.
(4.4)
We observe that the problem ⎧ ⎨ (−Δ)α u + ⎩
μ |x|2α u
lim
|x|→+∞
= Qf ηn
in RN \ {0}, (4.5)
u(x) = 0
has a unique solution wn ≥ 0 satisfying lim sup wn (x)|x|−τα (μ) = 0. |x|→0+
We first claim that {wn }n is increasing. By contradiction, we may assume that there exist n and x0 = 0 such that wn−1 (x0 ) > wn (x0 ). In fact, {f ηn }n is an increasing sequence, then w := wn−1 − wn satisfying μ w≤0 |x|2α
(−Δ)α w + and
lim sup w(x)|x|−τα (μ) = 0, |x|→0+
then, for any ε > 0, by Corollary 2.1, we have that w(x) ≤ ε|x|τα (μ) ,
x ∈ RN \ {0},
thus, ε≥
wn−1 (x0 ) − wn (x0 ) , |x0 |τα (μ)
which contradicts the arbitrary of ε > 0. Then {wn }n is an increasing sequence. Next we construct a uniform bound for {wn }n . By (4.1), there exists c > 0 such that Q(x)f (x) ≤ c|x|τ −2α (1 + |x|)−γ ≤ c|x|τ −2α ,
x ∈ RN \ {0}.
We observe that (−Δ)α |x|τ +
μ |x|τ = (cα (τ ) + μ)|x|τ −2α , |x|2α
where cα (τ ) + μ > 0 by the fact that τ > τα (μ). Therefore, Corollary 2.1, we have that for any n, wn (x) ≤
c |x|τ , cα (τ ) + μ
c τ cα (τ )+μ |x|
is a super solution of (4.2). By
x ∈ RN \ {0}.
Taking uf = limn→+∞ wn , for any x0 ∈ RN \ {0}, [9, Lemma 3.1] implies that for some β ∈ (0, α) wn C β (B |x0 | (x0 )) ≤ c, 4
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where c > 0 is independent of n, thus, from [19, Corollary 2.4], we have that
wn C 2α+β (B |x0 | (x0 )) ≤ c wn C β (B |x0 | (x0 )) + Qf L∞ (B |x0 | (x0 )) 8
≤ c,
2
4
where C 2α+β is the space of C [2α+β],(2α+β)−[2α+β] and [2α + β] is integer part. Therefore, by stability theorem [9, Theorem 2.2] (also see Theorem 2.2 in [7]), uf is a classical solution of (4.2) and uf (x) ≤
c |x|τ , cα (τ ) + μ
x ∈ RN \ {0}.
(4.6)
In order to get a better decay estimate of uf at infinity, we construct a new upper barrier function. Since τ − γ ∈ (τα (μ) − γ, τα (μ)), let us fix τ∞ ∈ (τ − γ, τα (μ)) and denote vt,s (x) = t|x|τα (μ) − s|x|τ∞ χRN \B1 (0) (x),
x ∈ RN \ {0},
where t > s > 0 will be chosen later. We observe that for |x| > 2, (−Δ)α vt,s (x) +
μ vt,s (x) ≥ −scτ∞ |x|τ∞ −2α + cN,α s(|x| + 1)−N −2α |x|2α ≥ −scτ∞ |x|τ∞ −2α ,
where −cτ∞ > 0 by the fact that τ∞ < τα (μ). Since τ − γ − 2α < τ∞ − 2α and Q(x)f (x) ≤ c|x|τ −γ−2α , so we may fix s=
1 sup Q(x)f (x)|x|−τ∞ +2α < +∞ −cτ∞ |x|>1
and choose t > s such that vt,s (x) ≥
c |x|τ ≥ uf (x) for 0 < |x| ≤ 2, cα (τ ) + μ
then by Corollary 2.1, we have that uf ≤ vt,s
in RN \ {0},
which implies that uf (x) ≤ c|x|τα (μ) The estimates (4.6) and (4.7) imply (4.3).
2
for |x| > 2.
(4.7)
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Corollary 4.1. Assume that μ > 0, p satisfies (1.11), Q satisfies (1.8) with 0 < γ < 2α. Then ⎧ ⎨ (−Δ)α u + ⎩
μ |x|2α u
lim
|x|→+∞
= Q|x|τα (μ)p
in
RN \ {0}, (4.8)
u(x) = 0
has a minimal positive solution v such that lim sup v(x)|x|−τα (μ)p−2α < +∞. |x|→0+
Furthermore, we have that v(x) ≤ c|x|τα (μ)
for
x ∈ RN \ {0}
(4.9)
for some c > 0. Proof. We apply Lemma 4.1 with f (x) = |x|τα (μ)p and τ = τα (μ)p + 2α. From (1.11), we have that τ ∈ (τα (μ), τα (μ) + γ] and then (4.9) follows by (4.3). 2 Proof of Theorem 1.2 (ii). Here we only provide the proof when α ∈ (0, 1). For α = 1, the proof is very similar, so we omit it. For k > 0, we define v0 (x) = k|x|τα (μ)
for x ∈ RN \ {0}
and vn = v0 + wn , where wn is the minimal positive solution of ⎧ ⎨ (−Δ)α wn + ⎩
lim
μ |x|2α wn
|x|→+∞
p = Qvn−1
in
RN \ {0}, (4.10)
vn (x) = 0.
Note that wn (x) ≤ c|x|τα (μ)p+2α , where τα (μ)p + 2α > τα (μ) for p ∈ [p∗μ,γ , p∗μ,0 ). By Lemma 4.1, we have that v1 ≥ v0 . By iterative argument, we assume that vN −1 ≥ vN −2 , then p p N QvN −1 ≥ QvN −2 and Lemma 4.1 implies that wn ≥ wn−1 , so vn ≥ vn−1 in R \ {0}, that is, the sequence {vn }n is an increasing sequence with respect to n. We next build an upper bound for the sequence {vn }n . For t > 0, denote w ¯t (x) = tkp w1 (x) + k|x|τα (μ) ≤ (ctkp + k)|x|τα (μ) , where c > 0 is from Corollary 4.1, then Q(x)w ¯t (x)p ≤ Q(x)(ctkp + k)p |x|τα (μ)p ≤ tkp Q(x)|x|τα (μ)p = (−Δ)α w ¯t (x) +
μ w ¯t (x), |x|2α
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if (ctkp + k)p ≤ tkp , that is, (ctkp−1 + 1)p ≤ t.
(4.11)
Note that the convex function hk (t) = (ctkp−1 + 1)p can intersect the line g(t) = t, if ck
Let k∗ =
1 cp
1
p−1
p−1 p ,
p−1
1 ≤ p
p−1 p
p−1 .
(4.12)
then if k ≤ k∗ , it always hold that hk (tp ) ≤ tp for tp = Qw ¯tpp
definition of w ¯tp , we have that w ¯tp ≥ v0 and implies Qw ¯tpp ≥ Qv1p . Inductively, we obtain that
≥
Qv0p ,
p p−1
p . Hence, by the
by Corollary 2.1, we have that v1 ≤ w ¯tp , which
vn ≤ w ¯ tp
(4.13)
for all n ∈ N. Therefore, the sequence {vn }n converges. Let us denote uk = limn→∞ vn . Since {vn }n is uniformly locally bounded and w ¯tp ∈ L1 (RN , 1+|x|1N +2α dx), then for any x0 ∈ RN \ {0}, [9, Lemma 3.1] implies that
wn C β (B |x0 | (x0 )) ≤ c w ¯tp L∞ (B |x0 | (x0 ))) + 2
4
Qw ¯tpp L∞ (B |x0 | (x0 )))
+ w ¯tp L1 (RN ,
2
1 1+|x|N +2α
dx)
,
thus, from [19, Corollary 2.4], wn C 2α+β (B |x0 | (x0 )) ≤ c wn C β (B |x0 | (x0 )) + Qw ¯tpp L∞ (B |x0 | (x0 )) 8
2
4
≤ c w ¯tp L∞ (B |x0 | (x0 )) + 2
Qw ¯tpp L∞ (B |x0 | (x0 ))
+ w ¯tp L1 (RN ,
2
1 1+|x|N +2α
dx)
.
By stability theorem [9, Theorem 2.2] (also see Theorem 2.2 in [7]), we have that uk is a classical solution of (1.9). From (4.13) and the fact that uk ≥ v0 , we have that v0 ≤ uk ≤ w ¯tp , which implies (1.12). 2 Acknowledgments Y. Wang is supported by National Sciences Foundation of China, No. 11526102. References [1] B. Abdellaoui, R. Bentifour, Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications, J. Funct. Anal. 272 (2017) 3998–4029. [2] S. Armstrong, B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011) 2011–2047.
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[3] M. Bidaut-Véron, G. Hoang, Q. Nguyen, L. Véron, An elliptic semilinear equation with source term and boundary measure data: the supercritical case, J. Funct. Anal. 269 (7) (2015) 1995–2017. [4] K. Bogdan, B. Dyda, The best constant in a fractional Hardy inequality, Math. Nachr. 284 (2011) 629–638. [5] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) 1245–1260. [6] H. Chen, P. Felmer, On the Liouville property for fully nonlinear elliptic equations with gradient term, J. Differential Equations 255 (2013) 2167–2195. [7] H. Chen, P. Felmer, A. Quaas, Large solution to elliptic equations involving fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015) 1199–1228. [8] H. Chen, L. Véron, Semilinear fractional elliptic equations involving measures, J. Differential Equations 257 (2014) 1457–1486. [9] H. Chen, L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations, Asymptot. Anal. 88 (2014) 165–184. [10] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012) 521–573. [11] S. Dipierro, L. Montoro, I. Peral, B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy–Leray potential, Calc. Var. Partial Differential Equations 55 (4) (2016) 99. [12] S. Dipierro, E. Valdinoci, A density property for fractional weighted Sobolev spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015) 397–422. [13] M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential, arXiv:1109.5530, 2012. [14] P. Felmer, A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math. 226 (2011) 2712–2738. [15] R. Frack, E. Lieb, R. Seiringer, Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc. 21 (4) (2008) 925–950. [16] K. Gkikas, L. Véron, Boundary singuarities of solutions of semilinear elliptic equations with critical hardy potentials, Nonlinear Anal. 121 (2015) 469–540. [17] K. Gkikas, L. Véron, Measure boundary value problem for semilinear elliptic equations with critical hardy potentials, C. R. Acad. Sci. Paris 353 (2015) 315–320. [18] P. Lions, Isolated singularities in semilinear problems, J. Differential Equations 38 (3) (1980) 441–450. [19] X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. 101 (2014) 275–302. [20] X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014) 587–628. [21] J. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in: Nonlinear Partial Differential Equations, 2012, pp. 271–298. [22] D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal. 168 (1999) 121–144.
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J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Existence and nonexistence of solutions to elliptic equations involving the Hardy potential Ying Wang Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 9 June 2016 Available online xxxx Submitted by C. Gutierrez
The purpose of this paper is to study the nonexistence of nonnegative super solutions to the problem (−Δ)α u +
Keywords: Hardy potential Fractional Laplacian Singular solution Liouville theorem
μ u ≥ Qup |x|2α
in
RN \ K,
(0.1)
where α ∈ (0, 1], μ ∈ R, p > 0, K is a compact set in RN with N ≥ 1 and Q is a potential in RN \ K satisfying that lim inf |x|→+∞ Q(x)|x|γ > 0 for some γ < 2α. When α = 1, (−Δ)α is the Laplacian operator, and when α ∈ (0, 1), it is the fractional Laplacian which is a typical nonlocal operator. In this paper, we find the critical exponent p∗ > 1 depending on α, μ and γ such that problem (0.1) has no nontrivial nonnegative super solutions for 0 < p < p∗ . Furthermore, we also consider the existence and nonexistence of isolated singular solutions to the equation ⎧ α ⎪ ⎨ (−Δ) u + ⎪ ⎩
lim
μ u = Qup |x|2α
|x|→+∞
in
RN \ {0},
u(x) = 0,
where μ > 0, p > 0 and Q(x) = (1 + |x|)−γ with γ ∈ (0, 2α). © 2017 Elsevier Inc. All rights reserved.
1. Introduction We are concerned with the nonexistence of nontrivial nonnegative super solutions to the problem (−Δ)α u +
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2017.07.002 0022-247X/© 2017 Elsevier Inc. All rights reserved.
μ u ≥ Qup |x|2α
in
RN \ K,
(1.1)
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where α ∈ (0, 1], μ ∈ R, p > 0, K is a compact set in RN with N ≥ 1 and Q is a potential in RN \K satisfying that lim inf |x|→+∞ Q(x)|x|γ > 0 for some γ < 2α. When α = 1, the operator (−Δ)α is the Laplacian, and when α ∈ (0, 1), it is the fractional Laplacian defined in the principle value sense as (−Δ)α u(x) = cN,α lim+ →0
RN \B (0)
u(x) − u(x + z) dz, |z|N +2α
where B (0) is the ball centered at the origin with radius , cN,α is the normalized constant cN,α = 22α απ − 2
N
Γ( N +2α 2 ) Γ(1 − α)
and Γ is the Gamma function. The fractional Laplacian is a nonlocal operator, so if Lebesgue measure |K| = 0, we have to assume moreover that u ≥ 0 a.e. in K. The semilinear elliptic equations involving the fractional Laplacian and the related Sobolev spaces have been studied extensively, see [1,5,10–12,20,21] and the references therein. It is known that the fundamental solution and Comparison Principle play an important role in the obtention of the nonexistence of solutions to semilinear elliptic equations. In the Laplacian case, the authors in [2,6] used the fundamental solution of Laplacian and Comparison Principle to obtain the nonexistence of positive solutions to the problem −Δu = Qup
in RN \ K.
In the fractional case, i.e. α ∈ (0, 1), [14] shows the nonexistence results of (1.1) when μ = 0, K = ∅, Q = 1 N , by using the fundamental solution of the fractional Laplacian and Comparison Principle. and p ≤ N −2α To study the nonexistence of nonnegative nontrivial super solutions of (1.1), we first clarify the fundamental solution of (−Δ)α + |x|μ2α as follows. Proposition 1.1. Assume that α ∈ (0, 1] and N ∈ N. (i) When N > 2α, let us denote
τ¯ = −
N − 2α 2
and
μ0 =
⎧ 2 ⎨ − (N −2) 4
2 N +2α ) 4 ⎩ −22α−1 cN,α Γ2 ( N −2α Γ ( 4 )
if
α = 1,
if
α ∈ (0, 1),
(1.2)
then (−Δ)α |x|τ¯ + μ0 |x|τ¯−2α = 0,
∀ x ∈ RN \ {0}.
For μ > μ0 , there exists a unique τα (μ) ∈ (−N, τ¯) such that φτα (μ) (x) := |x|τα (μ) is a fundamental solution of (−Δ)α +
μ |x|2α ,
(1.3)
i.e.
(−Δ)α φτα (μ) +
μ φτ (μ) = 0 |x|2α α
in RN \ {0}.
(1.4)
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(ii) When N ≤ 2α, then for μ > 0, there exists a unique τα (μ) ∈
(−∞, 0)
if
α = 1,
(−N, 0)
if
α ∈ (0, 1)
such that φτα (μ) defined in (1.3) is a fundamental solution of (−Δ)α + |x|μ2α . Furthermore, the mapping μ → τα (μ) is strictly decreasing in (μ0 , +∞) if N > 2α and in (0, +∞) if N ≤ 2α, and lim τα (μ) =
−∞
if
α = 1,
−N
if
α ∈ (0, 1).
μ→+∞
Note that for α = 1, τα (μ) has the explicit formula τα (μ) = −
(N − 2)+ +
[(N − 2)+ ]2 + 4μ . 2
For α ∈ (0, 1) and N > 2α, denote e1 = (1, 0, · · · , 0) ∈ RN , cα (τ ) = −
cN,α 2
RN
|x − e1 |τ + |x + e1 |τ − 2 dx, |x|N +2α
(1.5)
then the function cα (·) + μ has two zero points for μ ∈ (μ0 , 0), one zero point when μ = μ0 , and τα (μ) is the smaller zero point. μ0 is the best constant of the fractional Hardy inequalities, see the references [4,15]. Moreover, Lemma 3.1 in [13] provides an explicit expression of cα (τ ), cα (τ ) = 22α
Γ( N 2+τ )Γ( 2α−τ 2 ) . −τ N +τ −2α Γ( 2 )Γ( ) 2
(1.6)
Then the nonexistence of super solutions to (1.1) states as follows. Theorem 1.1. Suppose that α ∈ (0, 1], N ∈ N and Q is a nonnegative function satisfying lim inf Q(x)|x|γ > 0
|x|→+∞
for some γ < 2α. Then problem (1.1) has no nontrivial nonnegative super solution for 0 < p < p∗μ,γ , where
p∗μ,γ
⎧ 2α−γ ⎪ 1 + −τ ⎪ α (μ) ⎨ 2α−γ = 1 + N −2α ⎪ ⎪ ⎩ +∞
We remark that if μ > 0, the mapping μ →
if
μ > 0,
if
μ≤0
and
N > 2α,
if
μ≤0
and
N ≤ 2α.
2α−γ −τα (μ)
2α − γ = lim μ→+∞ −τα (μ)
(1.7)
is decreasing and
0
if
α = 1,
2α−γ N
if
α ∈ (0, 1).
2α−γ 2α−γ When N > 2α and μ ∈ (μ0 , 0], we observe that 1 + −τ > 1+ N −2α , but we do not know the nonexistence α (μ)
2α−γ 2α−γ α results when p ∈ 1 + N −2α , 1 + −τα (μ) due to the lack of comparison principle for the operator (−Δ) +
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μ |x|2α
in unbounded domains. To prove the nonexistence results, an essential tool is the Hadamard property derived by the fundamental solution and the Comparison Principle, see [2,6,14]. In this paper, we employ a new method to show the nonexistence. Our idea is to obtain an initial decay at infinity from the fundamental solution by applying the Comparison Principle, then an iterating technique is used to improve the power of the decay for p < p∗μ,γ until it makes the solution blow up everywhere. It is worthy to point out that for μ ∈ (μ0 , 0), the initial decay is not |x|τα (μ) , but |x|2α−N . This leads to the independence of the critical exponent p∗μ,γ with the parameter μ in (μ0 , 0). Our second aim of this paper is to show that p∗μ,γ is sharp for the nonexistence when μ > 0. To this end, we take K = {0}, 0 < γ < 2α, Q(x) = (1 + |x|)−γ ,
∀x ∈ RN \ {0},
(1.8)
and to consider solutions of the problem ⎧ α ⎪ ⎨ (−Δ) u + ⎪ ⎩
μ u = Qup |x|2α
lim
|x|→+∞
in
RN \ {0}, (1.9)
u(x) = 0.
Theorem 1.2. Let α ∈ (0, 1], μ > 0 and the function Q satisfy (1.8) with 0 < γ < 2α. (i) When 0 < p < p∗μ,γ ,
(1.10)
then problem (1.9) has no nontrivial nonnegative solution. (ii) When p∗μ,γ ≤ p < p∗μ,0 ,
(1.11)
then there exists k∗ > 0 such that for any k ∈ (0, k∗ ), problem (1.9) has a positive solution u satisfying lim u(x)|x|−τα (μ) = k.
(1.12)
|x|→0+
Note that p∗μ,γ < p∗μ,0 = 1 + −τ2α . When p < p∗μ,γ , the nonexistence result in Theorem 1.2 (i) follows α (μ) by Theorem 1.1 directly and for the existence, we have to mention Lions’ work [18], where an equivalence is built between the elliptic problem −Δu = up
in
Ω \ {0},
u = 0 in ∂Ω
(1.13)
and the one involving the Dirac mass at the origin −Δu = up + kδ0
in Ω,
where k > 0 and Ω is a bounded, smooth domain containing the origin. Then for 1 < p < singular solutions of (1.13) could be obtained by iterating the sequence of v0 = G1 [kδ0 ]
(1.14) N N −2α ,
positive
p and vn = G1 [vn−1 ] for n = 1, 2, · · · ,
where k > 0, G1 is the Green operator of −Δ. When k > 0 suitably small, a barrier function of (1.14) could be constructed and then the limit of {vn }n is the desired solution. More general second order problems with
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Hardy potentials in bounded domain considered in [3,16,17] and the singularities of semilinear fractional equations in bounded domain have been studied in [7,8]. However, for μ > 0, the singularity of (1.9) could not be expressed by the Dirac mass due to the Hardy potential, i.e. the term |x|μ2α u is no longer in L1loc (RN ). Another difficulty is the unbounded domain RN \{0}. To overcome these difficulties, we consider the sequence {vn }n , the solution of equation ⎧ ⎨ ⎩
(−Δ)α vn + lim
|x|→+∞
μ |x|2α vn
p = Qvn−1
in
RN \ {0},
lim vn (x)|x|−τα (μ) = k,
vn (x) = 0 and
|x|→0+
where v0 = k|x|τα (μ) , then we approximate the singular solutions of (1.9) by {vn }n as n → ∞. An upper bound has to be constructed to control the singularity at the origin and the decay at infinity. The rest of the paper is organized as follows. In Section 2, we clarify the fundamental solutions of (1.4) and provide a version of Comparison Principle. Section 3 is devoted to prove the nonexistence of nontrivial nonnegative solutions to problem (1.1). Finally, we prove the existence of isolated singular solutions of (1.9) in Section 4. 2. Preliminary This section is devoted to clarify the fundamental solution of (−Δ)α + |x|μ2α and the Comparison Principle. For convenience, let us denote φτ (x) = |x|τ ,
x ∈ RN \ {0},
(2.1)
where τ < 0. By direct computation, we have that −Δφτ (x) = −τ (τ + N − 2)|x|τ −2 ,
∀ x ∈ RN \ {0},
and the mapping τ → −τ (τ + N − 2) is strictly concave in (−∞, 0) and lim [−τ (τ + N − 2)] = −∞.
τ →−∞
When α ∈ (0, 1), by the definition of the fractional Laplacian, we have that cN,α (−Δ) φτ (x) = − 2
α
RN
|x + y|τ + |x − y|τ − 2|x|τ dy |y|N +2α
cN,α τ −2α |x| =− 2
RN
|ex + z|τ + |ex − z|τ − 2 dz, |z|N +2α
τ τ x x −z| −2 where ex = |x| . We know that RN |ex +z||z|+|e dz is independent of x and so we may replace ex by N +2α e1 = (1, 0, · · · , 0), combining with the formula of cα (τ ) in (1.5), we have that (−Δ)α φτ (x) = cα (τ )|x|τ −2α , From (1.6), we have the following estimates. Lemma 2.1. Suppose that α ∈ (0, 1). (i) If N ≤ 2α, then cα (τ ) < 0 for τ ∈ (−N, 0).
x ∈ RN \ {0}.
(2.2)
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(ii) If N > 2α, then ⎧ < 0, ⎪ ⎪ ⎨ cα (τ ) = 0, ⎪ ⎪ ⎩ > 0,
τ ∈ (−N, 2α − N ), τ = 2α − N,
(2.3)
τ ∈ (2α − N, 0).
Moreover, cα (·) is strictly concave in (−N, 0), lim
τ →(−N )+
cα (τ ) = −∞
and
lim cα (τ ) = 0.
τ →0−
(2.4)
Proof. From (1.6), we have that cα (τ ) = 22α
Γ( N 2+τ )Γ( 2α−τ 2 ) , −τ N +τ −2α Γ( 2 )Γ( ) 2
−τ N +τ −2α where Γ( N 2+τ ), Γ( 2α−τ ). When τ → 0− , 2 ), Γ( 2 ) > 0, the signs of cα (·) are decided by the ones of Γ( 2 −τ N +τ + we have that Γ( 2 ) → +∞, the others keep bounded and when τ → (−N ) , Γ( 2 ) → +∞, the others keep bounded. For the convexity, by directly calculus, we have that
cα (τ ) = −
cN,α 2
RN
|e1 − x|τ log |e1 − x| + |e1 + x|τ log |e1 + x| dx |x|N +2α
and cα (τ )
cN,α =− 2
RN
|e1 − x|τ (log |e1 − x|)2 + |e1 + x|τ (log |e1 + x|)2 dx < 0. |x|N +2α
The proof ends. 2 Proposition 2.1. Assume that α ∈ (0, 1), N > 2α, φτ is given by (2.1) and τ¯ = −
N − 2α , 2
μ0 = −22α−1 cN,α
Γ2 ( N +2α 4 ) . N −2α 2 Γ ( 4 )
Then (−Δ)α ϕτ¯ (x) + μ0 |x|τ¯−2α = 0,
x ∈ RN \ {0}
(2.5)
and μ0 = −
sup τ ∈(−N, 0)
cα (τ ).
(2.6)
Proof. From Lemma 3.1 in [13], we have that (2.5) holds true. To obtain (2.6), on the one hand, for any τ ∈ (−N, 0) and u ∈ Cc∞ (RN ), we have that [u(x) − u(y)]2 + u(x)2
u(y) 2 u(x) φτ (y) − φτ (x) φτ (x) − wτ (y) + u(y)2 = φτ (x)φτ (y)[ − ] ≥ 0, φτ (x) φτ (y) φτ (x) φτ (y)
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then RN RN
⎞ ⎞ ⎛ ⎛ u(x)2 ⎝ φτ (x) − φτ (y) ⎠ u(y)2 ⎝ φτ (y) − φτ (x) ⎠ dy dx + dx dy φτ (x) |x − y|N +2α φτ (y) |x − y|N +2α
≥ RN
=
=
[u(x) − u(y)]2 dxdy |x − y|N −2α
2 cN,α
RN
u(x)2 RN
2cα (τ ) cN,α
RN
RN
RN
(−Δ)α φτ (x) dx φτ (x)
u(x)2 dx. |x|2α
2 On the other hand, Yafaev in [22] indicated that − cN,α μ0 is the sharp constant in the Hardy–Rellich inequality as
RN RN
2 |u(x) − u(y)|2 dxdy ≥ − μ0 N +2α |x − y| cN,α
RN
u(x)2 dx. |x|2α
Then μ0 ≤ −
sup τ ∈(−N,0)
cα (τ ),
which, combining with the fact that cα (¯ τ ) = −μ0 , implies that cα (¯ τ ) ≥ supτ ∈(−N,0) cα (τ ). As a consequence, (2.6) holds true. 2 Proof of Proposition 1.1. When α = 1, the statements in Proposition 1.1 are clear. For α ∈ (0, 1), by Proposition 2.1, we have that cα (¯ τ ) = −μ0 for τ¯ = − N −2α 2 . Since cα (·) is concave and strictly increasing in (−N, τ¯). Then for any μ ∈ [μ0 , +∞), there exists a unique τα (μ) ∈ (−N, τ¯) such that μ = −cα (τα (μ)). The proof ends. 2 Remark 2.1. For μ ∈ (μ0 , 0), there exists a unique ςα (μ) ∈ (¯ τ , 0) such that cα (ςα (μ)) + μ = 0. It is obvious that ςα (μ0 ) = τα (μ0 ) and μ → ςα (μ) is increasing for μ ∈ (μ0 , 0). The following Comparison Principle plays an important role in the obtention of nonexistence results for (1.1). Lemma 2.2. Let α ∈ (0, 1), μ ≥ 0, D be a C 2 domain such that 0 ∈ / D, functions f1 , f2 ∈ C(D) satisfy that f2 ≥ f1 in D and g1 , g2 ∈ L1 (RN \ D, 1+|x|dxN +2α ) satisfy that g2 ≥ g1 a.e. in RN \ D. Assume more that u1 is a super solution of
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μ |x|2α
(−Δ)α u +
u = fi
in
D,
u = gi
in
RN \ D
(2.7)
with i = 1 and u2 is a sub solution of (2.7) with i = 2. Suppose that lim inf (u2 − u1 )(x) ≥ 0, x→∂D
(2.8)
in addition, if D is an unbounded domain, u1 and u2 satisfy that lim inf (u2 − u1 )(x) ≥ 0,
|x|→+∞
then u 2 ≥ u1
in D.
Proof. If inf x∈D (u2 − u1 )(x) < 0, then there exists a point x0 ∈ D such that (u2 − u1 )(x0 ) = inf (u2 − u1 )(x) = essinf x∈RN (u2 − u1 )(x) < 0, x∈D
which implies that
μ |x0 |2α (u2
− u1 )(x0 ) < 0 and
(−Δ) (u2 − u1 )(x0 ) = α
RN
(u2 − u1 )(x0 ) − (u2 − u1 )(z) dz < 0. |x0 − z|N +2α
Thus, (−Δ)α (u2 − u1 )(x0 ) +
μ (u2 − u1 )(x0 ) < 0, |x|2α
combining with (2.7), this contradicts f2 (x0 ) − f1 (x0 ) ≥ 0. 2 Remark 2.2. Lemma 2.2 holds for α = 1 by replacing the boundary type condition in (2.7) by u = gi
on ∂D.
Corollary 2.1. Let α ∈ (0, 1], μ ≥ 0, functions f1 , f2 ∈ C(RN \ {0}) satisfy that f2 ≥ f1 in RN \ {0}. Assume that u1 is a positive super solution of (−Δ)α u +
μ u = fi |x|2α
in
RN \ {0}
with i = 1 and u2 is a positive sub solution of (2.9) with i = 2. Suppose that lim inf
x=0,x→0
u2 (x) > 1, u1 (x)
and lim inf u2 (x) ≥ lim sup u1 (x).
|x|→+∞
|x|→+∞
Then u2 ≥ u1
in RN \ {0}.
(2.9)
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Remark 2.3. If α ∈ (0, 1] and μ ∈ (μ0 , 0), the above Comparison Principle fails. We give a counterexample as follows. Let ∈ (0, |μ|), u1 (x) = |x|ςα (μ+) and u2 (x) = |x|τα (μ) , where ςα (μ) is given by Remark 2.1, we observe that lim inf
x=0,x→0
u2 (x) > 1, u1 (x)
lim inf u2 (x) ≥ lim sup u1 (x),
|x|→+∞
|x|→+∞
and (−Δ)α u2 +
μ − μ u2 = 0 > u1 (x) = (−Δ)α u1 + 2α u1 |x|2α |x|2α |x|
in RN \ {0}.
But it is not true that u2 ≥ u1 in RN \ {0}. 3. Nonexistence We prove the nonexistence of nontrivial nonnegative solutions of (1.1) by contradiction. Let u be a nonnegative nontrivial solution of problem (1.1), then we will obtain a contradiction from the decay of u at infinity. Without loss generality, we may assume that K ⊂ B1 (0) and Q(x) ≥ q0 |x|−γ
for |x| > 4,
(3.1)
where q0 > 0. Proposition 3.1. Assume that α ∈ (0, 1], γ < 2α, p > 0 and u is a nonnegative nontrivial solution of (1.1). Then for some b0 > 0, (i) if μ > 0, we have that u(x) ≥ b0 |x|τα (μ) ,
∀ |x| > 4;
(3.2)
u(x) ≥ b0 |x|2α−N ,
∀ |x| > 4;
(3.3)
(ii) if N > 2α and μ ≤ 0, we have that
(ii) if N ≤ 2α and μ ≤ 0, for any τ < 0, we have that u(x) ≥ b0 |x|τ ,
∀ |x| > 4.
(3.4)
Proof. For α = 1, the proof follows the procedure of the case of α ∈ (0, 1). So we concentrate on the case α ∈ (0, 1). We first deal with part (i). To this end, let us denote ⎧ t|x|τα (μ) ⎪ ⎪ ⎨ wt,ν (x) = t(|x|τα (μ) + ν) ⎪ ⎪ ⎩ 0 where t, ν > 0 will be chosen later. We observe that
for |x| ≥ 3, for 2 < |x| < 3, for |x| ≤ 2,
(3.5)
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for |x| > 4
(−Δ)α wt,ν (x) = t(−Δ)α w1,ν (x) and it follows by Proposition 1.1 that for |x| > 4, μ (−Δ) w1,ν (x) + 2α w1,ν (x) = cN,α |x|
|y|τα (μ) dy − cN,α ν |x − y|N +2α
α
B2 (0)
⎡
⎢ ≤ 2−N −2α cN,α ⎣
B3 (0)\B2 (0)
1 dy |x − y|N +2α ⎤
⎥ |y|τα (μ) dy − ν|B3 (0) \ B2 (0)|⎦
B2 (0)
≤ 0,
if we choose that ν = ν0 = that
B2 (0)
|y|τα (μ) dy
|B3 (0)\B2 (0)|
. Since u is continuous in RN \ B1 (0), there exists t0 > 0 such
u ≥ t0 w1,ν0 = wt0 ,ν0
in
B4 (0) \ B2 (0).
Observe that (−Δ)α u(x) +
μ u(x) ≥ Q(x)u(x)p ≥ 0 and |x|2α
lim sup wt,ν (x) = 0,
|x|→+∞
then it follows by Lemma 2.2 that u ≥ t0 w1,ν0 , which implies (3.2). We next prove part (ii). Re-define the barrier function wt,ν replacing τα (μ) by 2α − N . By direct computation, we have that (−Δ)α w1,ν0 (x) ≤ 0 ≤
where ν0 =
B2 (0)
|x|2α−N dx
|B3 (0)\B2 (0)|
−μ u(x) + Q(x)u(x)p = (−Δ)α u(x), |x|2α
∀x ∈ RN \ B4 (0),
. Taking t = t0 > 0 small, we have that u ≥ wt0 ,ν0
in B4 (0) \ B2 (0),
so (3.3) holds true for μ ≤ 0. Finally, we deal with part (iii). In this case, the power τα (μ) of the barrier function wt,ν could be replaced by any τ ∈ (−N, 0) and we may have that (−Δ)α w1,ν0 (x) ≤ 0. We omit the rest of the proof. 2 The next step is to improve the decay of u at infinity. To this end, we introduce some notations. Let τ0 =
τα (μ)
if μ > 0,
2α − N
if
μ ≤ 0 and N > 2α
(3.6)
and {τj }j be the sequence generated by τj = 2α − γ + pτj−1
for j = 1, 2, 3 · · · .
(3.7)
Lemma 3.1. Assume that μ > 0 with N ≥ 1 or μ ≤ 0 with N > 2α, p∈
γ − 2α 0, 1 + τ0
,
(3.8)
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then {τj }j is a strictly increasing sequence and there exists j0 ∈ N such that τ j0 ≥ 0 Proof. For p ∈ (0, 1 +
γ−2α τ0 ),
and
τj0 −1 < 0.
(3.9)
we have that τ1 − τ0 = 2α − γ + (p − 1)τ0 > 0
and τj − τj−1 = p(τj−1 − τj−2 ) = pj−1 (τ1 − τ0 ),
(3.10)
which imply that the sequence {τj }j is increasing. If p ≥ 1, our conclusions are obvious. If p ∈ (0, 1), we have that in the case that τ1 ≥ 0, we are done, and in the case that τ1 < 0, we deduce from (3.10) that 1 − pj (τ1 − τ0 ) + τ0 1−p 1 2α − γ (τ1 − τ0 ) + τ0 = >0 → 1−p 1−p
τj =
then there exists j0 > 0 satisfying (3.9).
as j → +∞,
2
Remark 3.1. We note that p∗μ,γ = 1 +
γ − 2α . τ0
From the strictly increasing monotonicity of the sequence {τj }j , we have that τj−1 < τj = 2α − γ + pτj−1 , that is, τj−1 p − γ > τj−1 − 2α.
(3.11)
Proposition 3.2. Let τ0 and {τj }j be defined by (3.6) and (3.7) respectively, and u be a nonnegative solution of (1.1) satisfying u(x) ≥ cj |x|τj ,
∀ |x| > 4
for some cj > 0 and j ≤ j0 − 2, where j0 is from Lemma 3.1. Then for p ∈ (0, 1 + cj+1 > 0 such that u(x) ≥ cj+1 |x|τj+1 ,
∀ |x| > 4.
Proof. Case 1. μ > 0. Let ⎧ t|x|τj+1 ⎪ ⎪ ⎨ vt,ν (x) = t(|x|τj+1 + ν) ⎪ ⎪ ⎩ 0
for |x| ≥ 3, for 2 < |x| < 3, for |x| ≤ 2,
(3.12) γ−2α τ0 ),
there exists
(3.13)
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where t, ν > 0 will be chosen later. We observe that (−Δ)α vt,ν (x) = t(−Δ)α v1,ν (x) for |x| > 4 and it follows by Proposition 1.1 that for |x| > 4, (−Δ)α v1,ν (x) +
μ v1,ν (x) |x|2α
= (−Δ)α |x|τj+1 +
μ |x|τj+1 + cN,α |x|2α
|y|τj+1 dy − cN,α ν |x − y|N +2α
B2 (0)
⎡
⎢ ≤ (cα τj+1 + μ)|x|τj+1 −2α + 2−N −2α cN,α ⎣
1 dy |x − y|N +2α
B3 (0)\B2 (0)
⎤
⎥ |x|τj+1 dx − ν|B3 (0) \ B2 (0)|⎦
B2 (0)
≤ (cα τj+1 + μ)|x|τj+1 −2α ,
if we choose that ν = νj =
B2 (0)
|x|τj+1 dx
|B3 (0)\B2 (0)|
. From (3.12), we have that
Q(x)u(x)p ≥ q0 cpj |x|pτj −γ , and then there exists tj1 > 0 such that for t ∈ (0, tj1 ], (−Δ)α vt,ν (x) +
μ μ vt,ν (x) ≤ (−Δ)α u(x) + 2α u(x) for |x| > 4. |x|2α |x|
Furthermore, since u is continuous in RN \ B1 (0), there exists tj ≤ tj1 such that u ≥ vtj ,νj = tj v1,νj
in
B4 (0) \ B2 (0).
Applying Lemma 2.2, we have that u ≥ tj v1,νj , which implies (3.13) with cj+1 = tj . Case 2. μ < 0 with N > 2α. Since τj > τ0 = 2α − N , we have that (−Δ)α v1,νj (x) ≤ cα (τj+1 )|x|τj+1 −2α . By (3.11), we have that (−Δ)α u(x) = Q(x)u(x)p +
−μ u(x) ≥ cpj |x|pτj −γ |x|2α
and choosing t = tj > 0 small, we have that (−Δ)α u ≥ (−Δ)α vtj ,νj
in
RN \ B4 (0),
u ≥ vtj ,νj
in
by Lemma 2.2, we have that u ≥ tj v1,νj , which implies (3.13) with cj+1 = tj .
B4 (0) \ B2 (0), 2
Proof of Theorem 1.1. By contradiction, we assume that problem (1.1) has a nonnegative nontrivial super solution u ≥ 0. We first claim that u > 0 in RN \ B1 (0). For α = 1, the positivity of u follows by the strong maximum principle. For α ∈ (0, 1), since u is nonnegative nontrivial, then
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∅. x ∈ RN \ B1 (0) : u(x) > 0 =
If there exists x0 ∈ RN \ B1 (0) such that u(x0 ) = 0, then (−Δ) u(x0 ) = − α
RN
u(z) dz < 0, |x0 − z|N −2α
which implies that (−Δ)α u(x0 ) +
μ u(x0 ) < 0 = Q(x0 )u(x0 )p , |x0 |2α
which is impossible from (1.1). Thus, u > 0 in RN \ B1 (0). In the case that μ > 0 or μ ≤ 0 with N > 2α, from Proposition 3.1, there exists c0 > 0 such that u(x) ≥ c0 |x|τ0 ,
∀ |x| > 4,
(3.14)
where τ0 is defined by (3.6). For p ∈ (0, p∗μ,γ ), we use Proposition 3.2 to iterate, then we obtain that for any j ≤ j0 − 1, u(x) ≥ cj |x|τj ,
∀ |x| > 4,
(3.15)
where {τj }j is defined by (3.7) and cj > 0. In case that μ ≤ 0 with N ≤ 2α, (3.15) also holds when we take τ = τj0 −1 = − 2α−γ by Proposition 3.1 2p (iii). Next we use the decay of (3.15) with j = j0 − 1 to derive that u blows up everywhere. From (3.15) and (3.11) with j = j0 − 1, there exists r0 > 4 such that for |x| > r0 , (−Δ)α u(x) ≥ Q(x)u(x)p −
μ u(x) |x|2α
≥ q0 cpj0 −1 |x|τj0 −1 p−γ − cj0 −1 μ|x|τj0 −1 −2α ≥
q0 p c |x|τj0 −1 p−γ . 2 j0 −1
We note that if μ ≤ 0, the above inequality holds directly. In order to obtain a contradiction, we introduce some auxiliary functions vr with r > 8r0 , which is the solution of problem ⎧ (−Δ)α vr (x) = fr (x), ⎪ ⎪ ⎨ vr (x) = 0, ⎪ ⎪ ⎩ lim vr (x) = 0,
∀x ∈ RN \ Br0 (0), ∀x ∈ Br0 (0),
(3.16)
|x|→+∞
where fr (x) =
q0 p c |x|τj0 −1 p−γ χBr (0)\B4r0 (0) (x) 2 j0 −1
and χBr (0)\B4r0 (0) is the characteristic function of the set Br (0) \ B4r0 (0). By Lemma 2.2, for all r > 8r0 , we have that u(x) ≥ vr (x),
∀ x ∈ RN .
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Then for any x ∈ B6r0 (0) \ B4r0 (0), z ∈ Br (0) \ B8r0 (0), we have that |x − z| ≤ 2|z| and
|z|τj0 −1 p−γ dz |x − z|N −2α
u(x) ≥ vr (x) ≥ c¯cpj0 −1 Br (0)\B8r0 (0)
≥ c˜
|z|2α−N +τj0 −1 p−γ dz
Br (0)\B8r0 (0)
≥
⎧ τ ⎨ c[r j0 − (8r0 )τj0 ],
if τj0 > 0
⎩ c[log r − log(8r0 )]
if τj0 = 0
→ +∞
as r → +∞,
which contradicts that u satisfies (1.1). The proof ends. 2 Proof of Theorem 1.2 (i). If problem (1.9) has a nontrivial nonnegative solution u, then u satisfies (1.1), which contradicts Theorem 1.1 with 0 < p < p∗μ,γ . 2 4. Existence In this section, we are concerned with the existence of positive singular solutions of problem (1.9) when p∗μ,γ ≤ p < p∗μ,0 . Lemma 4.1. Assume that α ∈ (0, 1], μ > 0, Q satisfies (1.8) with 0 < γ < 2α, f is a nonnegative function in C 1 (RN \ {0}) satisfying f (x)|x|−τ +2α < +∞
sup
(4.1)
x∈RN \{0}
for τ ∈ (τα (μ), τα (μ) + γ]. Then problem ⎧ ⎨ (−Δ)α u + ⎩
lim
μ |x|2α u
|x|→+∞
= Qf
in
RN \ {0}, (4.2)
u(x) = 0
has a minimal positive solution uf such that lim sup uf (x)|x|−τ < +∞ |x|→0+
and
uf (x) ≤ c|x|τα (μ) ,
x ∈ RN \ {0},
for some c > 0. Moreover, the mapping f → uf is increasing. Proof. Let η0 : [0 + ∞) → [0, 1] be a C ∞ nondecreasing function such that η0 = 1 in [2, +∞) and η0 = 0
in [0, 1],
(4.3)
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denote ηn (t) = η0 (nt)[1 − η0 (n−1 t)]
for n ∈ N.
(4.4)
We observe that the problem ⎧ ⎨ (−Δ)α u + ⎩
μ |x|2α u
lim
|x|→+∞
= Qf ηn
in RN \ {0}, (4.5)
u(x) = 0
has a unique solution wn ≥ 0 satisfying lim sup wn (x)|x|−τα (μ) = 0. |x|→0+
We first claim that {wn }n is increasing. By contradiction, we may assume that there exist n and x0 = 0 such that wn−1 (x0 ) > wn (x0 ). In fact, {f ηn }n is an increasing sequence, then w := wn−1 − wn satisfying μ w≤0 |x|2α
(−Δ)α w + and
lim sup w(x)|x|−τα (μ) = 0, |x|→0+
then, for any ε > 0, by Corollary 2.1, we have that w(x) ≤ ε|x|τα (μ) ,
x ∈ RN \ {0},
thus, ε≥
wn−1 (x0 ) − wn (x0 ) , |x0 |τα (μ)
which contradicts the arbitrary of ε > 0. Then {wn }n is an increasing sequence. Next we construct a uniform bound for {wn }n . By (4.1), there exists c > 0 such that Q(x)f (x) ≤ c|x|τ −2α (1 + |x|)−γ ≤ c|x|τ −2α ,
x ∈ RN \ {0}.
We observe that (−Δ)α |x|τ +
μ |x|τ = (cα (τ ) + μ)|x|τ −2α , |x|2α
where cα (τ ) + μ > 0 by the fact that τ > τα (μ). Therefore, Corollary 2.1, we have that for any n, wn (x) ≤
c |x|τ , cα (τ ) + μ
c τ cα (τ )+μ |x|
is a super solution of (4.2). By
x ∈ RN \ {0}.
Taking uf = limn→+∞ wn , for any x0 ∈ RN \ {0}, [9, Lemma 3.1] implies that for some β ∈ (0, α) wn C β (B |x0 | (x0 )) ≤ c, 4
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where c > 0 is independent of n, thus, from [19, Corollary 2.4], we have that
wn C 2α+β (B |x0 | (x0 )) ≤ c wn C β (B |x0 | (x0 )) + Qf L∞ (B |x0 | (x0 )) 8
≤ c,
2
4
where C 2α+β is the space of C [2α+β],(2α+β)−[2α+β] and [2α + β] is integer part. Therefore, by stability theorem [9, Theorem 2.2] (also see Theorem 2.2 in [7]), uf is a classical solution of (4.2) and uf (x) ≤
c |x|τ , cα (τ ) + μ
x ∈ RN \ {0}.
(4.6)
In order to get a better decay estimate of uf at infinity, we construct a new upper barrier function. Since τ − γ ∈ (τα (μ) − γ, τα (μ)), let us fix τ∞ ∈ (τ − γ, τα (μ)) and denote vt,s (x) = t|x|τα (μ) − s|x|τ∞ χRN \B1 (0) (x),
x ∈ RN \ {0},
where t > s > 0 will be chosen later. We observe that for |x| > 2, (−Δ)α vt,s (x) +
μ vt,s (x) ≥ −scτ∞ |x|τ∞ −2α + cN,α s(|x| + 1)−N −2α |x|2α ≥ −scτ∞ |x|τ∞ −2α ,
where −cτ∞ > 0 by the fact that τ∞ < τα (μ). Since τ − γ − 2α < τ∞ − 2α and Q(x)f (x) ≤ c|x|τ −γ−2α , so we may fix s=
1 sup Q(x)f (x)|x|−τ∞ +2α < +∞ −cτ∞ |x|>1
and choose t > s such that vt,s (x) ≥
c |x|τ ≥ uf (x) for 0 < |x| ≤ 2, cα (τ ) + μ
then by Corollary 2.1, we have that uf ≤ vt,s
in RN \ {0},
which implies that uf (x) ≤ c|x|τα (μ) The estimates (4.6) and (4.7) imply (4.3).
2
for |x| > 2.
(4.7)
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Corollary 4.1. Assume that μ > 0, p satisfies (1.11), Q satisfies (1.8) with 0 < γ < 2α. Then ⎧ ⎨ (−Δ)α u + ⎩
μ |x|2α u
lim
|x|→+∞
= Q|x|τα (μ)p
in
RN \ {0}, (4.8)
u(x) = 0
has a minimal positive solution v such that lim sup v(x)|x|−τα (μ)p−2α < +∞. |x|→0+
Furthermore, we have that v(x) ≤ c|x|τα (μ)
for
x ∈ RN \ {0}
(4.9)
for some c > 0. Proof. We apply Lemma 4.1 with f (x) = |x|τα (μ)p and τ = τα (μ)p + 2α. From (1.11), we have that τ ∈ (τα (μ), τα (μ) + γ] and then (4.9) follows by (4.3). 2 Proof of Theorem 1.2 (ii). Here we only provide the proof when α ∈ (0, 1). For α = 1, the proof is very similar, so we omit it. For k > 0, we define v0 (x) = k|x|τα (μ)
for x ∈ RN \ {0}
and vn = v0 + wn , where wn is the minimal positive solution of ⎧ ⎨ (−Δ)α wn + ⎩
lim
μ |x|2α wn
|x|→+∞
p = Qvn−1
in
RN \ {0}, (4.10)
vn (x) = 0.
Note that wn (x) ≤ c|x|τα (μ)p+2α , where τα (μ)p + 2α > τα (μ) for p ∈ [p∗μ,γ , p∗μ,0 ). By Lemma 4.1, we have that v1 ≥ v0 . By iterative argument, we assume that vN −1 ≥ vN −2 , then p p N QvN −1 ≥ QvN −2 and Lemma 4.1 implies that wn ≥ wn−1 , so vn ≥ vn−1 in R \ {0}, that is, the sequence {vn }n is an increasing sequence with respect to n. We next build an upper bound for the sequence {vn }n . For t > 0, denote w ¯t (x) = tkp w1 (x) + k|x|τα (μ) ≤ (ctkp + k)|x|τα (μ) , where c > 0 is from Corollary 4.1, then Q(x)w ¯t (x)p ≤ Q(x)(ctkp + k)p |x|τα (μ)p ≤ tkp Q(x)|x|τα (μ)p = (−Δ)α w ¯t (x) +
μ w ¯t (x), |x|2α
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if (ctkp + k)p ≤ tkp , that is, (ctkp−1 + 1)p ≤ t.
(4.11)
Note that the convex function hk (t) = (ctkp−1 + 1)p can intersect the line g(t) = t, if ck
Let k∗ =
1 cp
1
p−1
p−1 p ,
p−1
1 ≤ p
p−1 p
p−1 .
(4.12)
then if k ≤ k∗ , it always hold that hk (tp ) ≤ tp for tp = Qw ¯tpp
definition of w ¯tp , we have that w ¯tp ≥ v0 and implies Qw ¯tpp ≥ Qv1p . Inductively, we obtain that
≥
Qv0p ,
p p−1
p . Hence, by the
by Corollary 2.1, we have that v1 ≤ w ¯tp , which
vn ≤ w ¯ tp
(4.13)
for all n ∈ N. Therefore, the sequence {vn }n converges. Let us denote uk = limn→∞ vn . Since {vn }n is uniformly locally bounded and w ¯tp ∈ L1 (RN , 1+|x|1N +2α dx), then for any x0 ∈ RN \ {0}, [9, Lemma 3.1] implies that
wn C β (B |x0 | (x0 )) ≤ c w ¯tp L∞ (B |x0 | (x0 ))) + 2
4
Qw ¯tpp L∞ (B |x0 | (x0 )))
+ w ¯tp L1 (RN ,
2
1 1+|x|N +2α
dx)
,
thus, from [19, Corollary 2.4], wn C 2α+β (B |x0 | (x0 )) ≤ c wn C β (B |x0 | (x0 )) + Qw ¯tpp L∞ (B |x0 | (x0 )) 8
2
4
≤ c w ¯tp L∞ (B |x0 | (x0 )) + 2
Qw ¯tpp L∞ (B |x0 | (x0 ))
+ w ¯tp L1 (RN ,
2
1 1+|x|N +2α
dx)
.
By stability theorem [9, Theorem 2.2] (also see Theorem 2.2 in [7]), we have that uk is a classical solution of (1.9). From (4.13) and the fact that uk ≥ v0 , we have that v0 ≤ uk ≤ w ¯tp , which implies (1.12). 2 Acknowledgments Y. Wang is supported by National Sciences Foundation of China, No. 11526102. References [1] B. Abdellaoui, R. Bentifour, Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications, J. Funct. Anal. 272 (2017) 3998–4029. [2] S. Armstrong, B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011) 2011–2047.
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