Monotonicity formula and Liouville-type theorems of stable solution for the weighted elliptic system
aa r X i v : . [ m a t h . A P ] A ug Monotonicity formula and Liouville-type theorems ofstable solution for the weighted elliptic system ∗ Liang-Gen Hu † Department of Mathematics, Ningbo University, 315211, P.R. China
Abstract:
In this paper, we are concerned with the weighted elliptic system − ∆ u = | x | β v ϑ , − ∆ v = | x | α | u | p − u, in Ω , where Ω is a subset of R N , N ≥ α > −
4, 0 ≤ β ≤ N −
42 , p > ϑ = 1.We first apply Pohozaev identity to construct a monotonicity formula and revealtheir certain equivalence relation. By the use of Pohozaev identity , monotonicityformula of solutions together with a blowing down sequence, we prove Liouville-typetheorems of stable solutions (whether positive or sign-changing) for the weightedelliptic system in the higher dimension. Keywords:
Liouville-type theorem; stable solutions; Pohozaev identity; mono-tonicity formula; blowing down sequence
In this article, we examine the nonexistence of classical stable solutions of theweighted elliptic system given by − ∆ u = | x | β v ϑ , − ∆ v = | x | α | u | p − u, in Ω , (1.1)where Ω is a subset of R N , N ≥ α > −
4, 0 ≤ β ≤ N −
42 and pθ > ∗ The work was partially supported by NSFC of China (No. 11201248), K.C. Wong Fund of NingboUniversity and Ningbo Natural Science Foundation (No. 2014A610027). † email address: [email protected] u is stable if and only if its Morse index is equal to zero. In 2007,Farina considered the Lane-Emden equation − ∆ u = | u | p − u, (1.2)on bounded and unbounded domains of Ω ⊂ R N , with N ≥ p >
1. Based on adelicate application of the classical Moser’s iteration, he gave the complete classificationof finite Morse index solutions (positive or sign-changing) in his seminal paper [14].Hereafter, many experts utilized the Moser’s iterative method to discuss the stableand finite Morse index solutions of the harmonic and fourth-order elliptic equation andobtained many excellent results. We refer to [7, 29, 31, 32] and the reference therein.However, the classical Moser’s iterative technique does not completely classify finiteMorse index solutions of the biharmonic equation∆ u = | u | p − u, in Ω ⊂ R N . To solve the problem, D´avila et al. [9] have recently derived a monotonicity formulaof solutions and given the complete classification of stable and finite Morse index so-lutions for the biharmonic equation by the application of Pohozaev identity and themonotonicity formula. We note that many outstanding papers [8, 9, 20, 21, 30] utilize amonotonicity formula to study the partial regularity of stationary weak solution, stableand finite Morse index solutions for the harmonic and fourth-order equation.On the other hand, some experts were interesting in the Lane-Emden system andobtained some excellent results [3, 11–13]. In 2013, applying a iterative method and thepointwise estimate in [28], Cowan proved the following result.
Theorem A. ([3, Theorem 2])
Suppose that p > θ = 1 , α = β = 0 and N < p + 1) p − r pp + 1 + s pp + 1 − r pp + 1 . Then there is no positive stable solution of (1.1).
Adopting the same method as Cowan [3], Fazly obtained the following result.2 heorem B. ([12, Theorem 2.4])
Suppose that ( u, v ) is C ( R N ) nonnegative entiresemi-stable solution of − ∆ u = ρ (1 + | x | ) α v, − ∆ v = ̺ (1 + | x | ) α u p , with ρ, ̺ > in the dimension N < α + 8 + 4 αp − . Then, ( u, v ) is the trivial solution. We observe that the dimension
N < α + 8 + 4 αp − critical hyperbola , i.e., N = 4 + α + 8 + 4 αp − N < (cid:18) p ( β + 2) + α + 2 pθ − (cid:19) s pθ ( θ + 1) p + 1 + vuut pθ ( θ + 1) p + 1 − s pθ ( θ + 1) p + 1 . Clearly, if θ = 1 and α = β = 0 in (1.1), then their result is the same as Theorem A.Let us briefly recall the fact that Liouvile-type theorem of solutions for variousLane-Emden equations and systems is interesting and challenging for decades.First, Pohozaev identity shows that the Lane-Emden equation with the Dirichletboundary condition has no positive solution on a bounded star-shaped domain Ω ⊂ R N ,whenever p ≥ N + 2 N − < p < N + 2 N − ∞ , if N ≤ R N .In the case of the Lane-Emden systems (1.1) with α = β = 0, Pucci and Serrin[25] proved that if Np + 1 + Nθ + 1 ≤ N − R N , then there is no positive solution of (1.1) with the Dirichlet boundary conditions.Noting that the curve Np + 1 + Nθ + 1 = N − critical Sobolev hyperbola . Similarto the Lane-Emden equation, the following conjecture is interesting and challenging. Conjecture ( Lane-Emden Conjecture ) Suppose ( p, θ ) is under the critical Sobolev hy- erbola, i.e., Np + 1 + Nθ + 1 > N − . Then there is no positive solution for the elliptic system (1.1) with α = β = 0 . The case of radial solutions was solved by Mitidieri [18] in any dimension, and thepositive radial solutions on and above the critical Sobolev hyperbola was constructed by[18, 27], which is the optimal Liouville-type theorem for radial solutions. The conjecture (for non-radial solutions) seems difficult. In the dimension N = 3, Serrin and Zou [26]proved the conjecture for the polynomially bounded solutions, which the boundednesswas removed in [24]. In 2009, Souplet [28] solved the conjecture in N = 4 or a new regionfor N ≥
5. However, the weighted Lane-Emden system (1.1) is even less understood.For example, the paper [23] proved the conjecture for the equation − ∆ u = | x | α u p in N = 3; In 2012, Phan [22] solved the conjecture for the system (1.1) in two cases: case1 . N = 3 and bounded solutions; case 2 . N = 3 or 4 and α, β ≤ N ≥ α > −
4, 0 ≤ β ≤ N −
42 , p > θ = 1. Motivated by the ideas in [9, 10, 17], we will construct a monotonicityformula of solutions in the dimension 4 + β + 8 + 2 α + 2 βp − < N < N α,β ( p ) ( N α,β ( p )see below (3.1)) and get various integral estimates, and then use these results to studyLiouville-type theorems of stable solution for the weighted elliptic system (1.1). Theorem 1.1.
For any β + 8 + 2 α + 2 βp − < N < N α,β ( p ) , assume that u ∈ W , loc ( R N \{ } ) is a homogeneous, stable solution of (1.1), | x | α | u | p +1 ∈ L loc ( R N \{ } ) and | x | − β | ∆ u | ∈ L loc ( R N \{ } ) . Then u ≡ . Applying Theorem 1.1 and the properties of monotonicity formula (2.14), we get
Theorem 1.2. If u ∈ C ( R N ) is a stable solution of (1.1) in R N and ≤ N ≤ N α,β ( p ) ,then u ≡ . Remark 1.1. (1)
We apply Pohozaev identity to construct a monotonicity formula.From the process of the proof in Theorem 2.1, we can observe that Pohozaev iden-tity is equivalence to the certain derivative-type of the monotonicity formula. (2)
Let us note that for the dimensions β + 8 + 2 α + 2 βp − < N < N α,β ( p ) , weadopt a new method of monotonicity formula together with blowing down sequenceto investigate Liouville-type theorem. In addition, a difficulty stems from the fact hat the terms | x | α and | x | β in (1.1) leads to the singularity. For this reason, weuse a more delicate approach to derive improved integral estimates. (3) From the computation of N α,β ( p ) (in Section 3), we find the following relation: N , ( p ) > p + 1) p − r pp + 1 + s pp + 1 − r pp + 1 ! , if α = β = 0 ,N α,α ( p ) > α + 8 + 4 αp − , if α = β. Therefore, in contrast with
Theorem A and
Theorem B , we obtain Liouville-type theorem in the higher dimension.
Next, we list some definitions and notations. Let Ω be a subset of R N and f, g ∈ C (cid:0) R N +2 , Ω (cid:1) . Following Montenegro [19], we consider the general elliptic system( S f,g ) − ∆ u = f ( u, v, x ) , − ∆ v = g ( u, v, x ) , x ∈ Ω . A solution ( u, v ) ∈ C (Ω) × C (Ω) of ( S f,g ) is called stable , if the eigenvalue problem( E f,g ) − ∆ φ = f u ( u, v, x ) φ + f v ( u, v, x ) ψ + ηφ, − ∆ ψ = g u ( u, v, x ) φ + g v ( u, v, x ) ψ + ηψ, has a first positive eigenvalue η >
0, with corresponding positive smooth eigenvalue pair( φ, ψ ). A solution ( u, v ) is said to be semi-stable , if the first eigenvalue η is nonnegative.Inspired by the above definition, we give the integration-type definition of stability. Definition 1.1.
We recall that a critical point u ∈ C (Ω) of the energy function E ( u ) = Z Ω (cid:20) | ∆ u | | x | β − p + 1 | x | α | u | p +1 (cid:21) dx is said to be a stable solution of (1.1), if, for any ζ ∈ C (Ω) , we have p Z Ω | x | α | u | p − ζ dx ≤ Z Ω | ∆ ζ | | x | β dx. The definition is interesting and well-defined. In deed, if ( u, v ) is a semi-stablesolution, then there exist η ≥ φ, ψ ) such that − ∆ φ = | x | β ψ + ηφ, − ∆ ψ = p | x | α | u | p − φ + ηψ. ζ φ with ζ ∈ C (Ω) to get p Z Ω | x | α | u | p − ζ dx ≤ Z Ω − ∆ ψ ζ φ dx = Z Ω − ψ ∆ (cid:18) ζ φ (cid:19) dx = Z Ω | x | β (cid:20) − ηφ | x | β ψ + ηφ (cid:21) ∆ φ ∆ (cid:18) ζ φ (cid:19) dx. (1.3)A simple calculation leads to∆ (cid:18) ζ φ (cid:19) = 2 φ − |∇ ζ | + 2 ζφ − ∆ ζ − ζφ − ∇ ζ · ∇ φ + 2 ζ φ − |∇ φ | − ζ φ − ∆ φ. Then we find∆ φ ∆ (cid:18) ζ φ (cid:19) − | ∆ ζ | = 2 ζφ − ∆ ζ ∆ φ − ζ φ − | ∆ φ | − | ∆ ζ | + 2 φ − ∆ φ [ |∇ ζ | − ζφ − ∇ ζ · ∇ φ + ζ φ − |∇ φ | ]= − h ( ζφ − ∆ φ − ∆ ζ ) + 2( φ − | x | β ψ + η )( ∇ ζ − ζφ − ∇ φ ) i ≤ , implies Z Ω ∆ φ | x | β ∆ (cid:18) ζ φ (cid:19) dx ≤ Z Ω | ∆ ζ | | x | β dx. Therefore, combining the above inequality with (1.3), we obtain p Z Ω | x | α | u | p − ζ dx ≤ Z Ω | ∆ ζ | | x | β dx. Remark 1.2.
Since φ is a smooth function, ζ ∈ C (Ω) and β ≤ N − , then theintegration Z Ω | x | β dx is well defined. Notations.
Throughout this paper, B r ( x ) denotes the open ball of radius r centeredat x . If x = 0, we simply denote B r (0) by B r . C denotes various irrelevant positiveconstants.The rest of the paper is organized as follows. In Section 2, we derive various integralestimates and construct a monotonicity formula. In Section 3, we prove Liouville-typetheorem of homogeneous, stable solutions in the dimensions 4 + β + 8 + 2 α + 2 βp − < N Lemma 2.1. ([31, Lemma 2.2]) For any ζ ∈ C ( R N ) and η ∈ C ( R N ) , the identityholds ∆ ζ ∆ (cid:0) ζη (cid:1) = [∆( ζη )] − ∇ ζ · ∇ η ) − ζ | ∆ η | + 2 ζ ∆ ζ |∇ η | − ζ ∆ η ∇ ζ · ∇ η. Lemma 2.2. For any ζ ∈ C ( R N ) and η ∈ C ( R N ) , then the following equalities hold Z R N ∆ (cid:18) ∆ ζ | x | β (cid:19) ζη dx = Z R N [∆( ζη )] | x | β + Z R N | x | β h − ∇ ζ · ∇ η ) + 2 ζ ∆ ζ |∇ η | i dx + Z R N ζ | x | β h ∇ (∆ η ) · ∇ η + | ∆ η | − β | x | − ∆ η ( x · ∇ η ) i dx, (2.1) and Z R N |∇ ζ | |∇ η | | x | β dx = Z R N (cid:20) | x | β ζ ( − ∆ ζ ) |∇ η | + ζ | x | β ∆ (cid:0) |∇ η | (cid:1)(cid:21) dx + Z R N ζ | x | β +2 h β ( β + 2 − N ) |∇ η | − β (cid:0) x · ∇ (cid:0) |∇ η | (cid:1)(cid:1) i dx. (2.2) Proof. By the divergence theorem and integration by parts, we get − Z R N | x | β ζ ∆ η ∇ ζ · ∇ ηdx = − Z R N | x | β ∆ η ∇ ζ · ∇ ηdx = 2 Z R N ζ | x | β h ∇ (∆ η ) · ∇ η + | ∆ η | − β | x | − ∆ η ( x · ∇ η ) i dx. Combining with Lemma 2.1, it implies that the identity (2.1) holds.On the other hand, it is easy to see that12 ∆( ζ ) = ζ ∆ ζ + |∇ ζ | , then we obtain Z R N |∇ ζ | |∇ η | | x | β = Z R N ζ ( − ∆ ζ ) |∇ η | | x | β + 12 Z R N ζ ∆ (cid:18) |∇ η | | x | β (cid:19) dx. A direct computation yields∆ (cid:18) |∇ η | | x | β (cid:19) = 1 | x | β h β ( β + 2 − N ) | x | − |∇ η | − β | x | − ( x · ∇ ( |∇ η | )) + ∆( |∇ η | ) i . Substituting into the above identity, we get the identity (2.2).7 emma 2.3. Let u ∈ C ( R N ) be a stable solution of (1.1). Then we find Z B R ( x ) (cid:18) | ∆ u | | z | β + | z | α | u | p +1 (cid:19) dz ≤ CR − Z B R ( x ) \ B R ( x ) | u ∆ u || z | β dz + CR − Z B R ( x ) \ B R ( x ) u | z | β dz. (2.3) Furthermore, for large enough m , we obtain that for any ψ ∈ C ( R N ) with ≤ ψ ≤ Z R N (cid:20) | ∆ u | | x | β + | x | α | u | p +1 (cid:21) ψ m dx ≤ C Z R N | x | − α + βp + βp − Q ( ψ m ) p +1 p − dx + C Z R N | x | − α +( β +2)( p +1) p − R ( ψ m ) p +1 p − dx, and Z B R ( x ) (cid:20) | ∆ u | | z | β + | z | α | u | p +1 (cid:21) ψ m dz ≤ CR N − − β − α +2 βp − . (2.4) Here Q ( ψ m ) = |∇ ψ | + ψ − m ) h |∇ (∆ ψ m ) · ∇ ψ m | + | ∆ ψ m | + (cid:12)(cid:12) ∆ |∇ ψ m | (cid:12)(cid:12) i , R ( ψ m ) = ψ − m ) h | ∆ ψ m | | x · ∇ ψ m | + |∇ ψ m | + (cid:12)(cid:12) x · ∇ ( |∇ ψ m | ) (cid:12)(cid:12) i . proof. From the definition of a stable solution u , it implies that if we take arbitrarily ζ ∈ C ( R N ), then we obtain Z R N | x | α | u | p − uζdx = Z R N ∆ u | x | β ∆ ζdx, (2.5)and p Z R N | x | α | u | p − ζ dx ≤ Z R N | ∆ ζ | | x | β dx. (2.6)Now, in (2.5), we choose ζ = uψ with ψ ∈ C ( R N ), and find Z R N | x | α | u | p +1 ψ dx = Z R N ∆ u | x | β ∆( uψ ) dx. (2.7)We insert the test function ζ = uψ into (2.6) and get p Z R N | x | α | u | p +1 ψ dx ≤ Z R N [∆( uψ )] | x | β dx. Putting the above inequality and (2.7) back into (2.1) yields( p − Z R N | x | α | u | p +1 ψ dx ≤ Z R N | x | β h ∇ u · ∇ ψ ) − u ∆ u |∇ ψ | i dx + Z R N u | x | β h |∇ (∆ ψ ) · ∇ ψ | + | ∆ ψ | + 2 β | x | − ∆ ψ ( x · ∇ ψ ) i dx. Z R N | x | α | u | p +1 ψ dx ≤ C Z R N | u ∆ u || x | β |∇ ψ | dx + C Z R N u | x | β h |∇ (∆ ψ ) · ∇ ψ | + | ∆ ψ | + (cid:12)(cid:12) ∆ |∇ ψ | (cid:12)(cid:12) i dx + C Z R N u | x | β +2 h | ∆ ψ || x · ∇ ψ | + |∇ ψ | + (cid:12)(cid:12) x · ∇ ( |∇ ψ | ) (cid:12)(cid:12) i dx. (2.8)Since ∆( uψ ) = ∆ uψ + 2 ∇ u · ∇ ψ + u ∆ ψ , it implies from (2.7), (2.8) and Lemma 2.2,that Z R N | ∆ u | | x | β ψ dx ≤ C Z R N | u ∆ u || x | β |∇ ψ | dx + C Z R N u | x | β h |∇ (∆ ψ ) · ∇ ψ | + | ∆ ψ | + (cid:12)(cid:12) ∆ |∇ ψ | (cid:12)(cid:12) i dx + C Z R N u | x | β +2 h | ∆ ψ || x · ∇ ψ | + |∇ ψ | + (cid:12)(cid:12) x · ∇ ( |∇ ψ | ) (cid:12)(cid:12) i dx. (2.9)Replace ψ by ψ m in (2.8) and (2.9) with m > Z R N (cid:20) | ∆ u | | x | β + | x | α | u | p +1 (cid:21) ψ m dx ≤ C Z R N | u ∆ u || x | β ψ m − |∇ ψ | dx + C Z R N u | x | β h |∇ (∆ ψ m ) · ∇ ψ m | + | ∆ ψ m | + (cid:12)(cid:12) ∆ |∇ ψ m | (cid:12)(cid:12) i dx + C Z R N u | x | β +2 h | ∆ ψ m | | x · ∇ ψ m | + |∇ ψ m | + (cid:12)(cid:12) x · ∇ ( |∇ ψ m | ) (cid:12)(cid:12) i dx. Utilizing Young’s inequality, we obtain Z R N | u ∆ u || x | β ψ m − |∇ ψ | dx ≤ C Z R N | ∆ u | | x | β ψ m dx + C Z R N u | x | β ψ m − |∇ ψ | dx. Thus, it implies Z R N (cid:20) | ∆ u | | x | β + | x | α | u | p +1 (cid:21) ψ m dx ≤ C Z R N u | x | β ψ m − Q ( ψ m ) dx + C Z R N u | x | β +2 ψ m − R ( ψ m ) dx, where Q ( ψ m ) = |∇ ψ | + ψ − m ) h |∇ (∆ ψ m ) ·∇ ψ m | + | ∆ ψ m | + (cid:12)(cid:12) ∆ |∇ ψ m | (cid:12)(cid:12) i and R ( ψ m ) = ψ − m ) h | ∆ ψ m | | x · ∇ ψ m | + |∇ ψ m | + (cid:12)(cid:12) x · ∇ ( |∇ ψ m | ) (cid:12)(cid:12) i . Taking ( m − p + 1) ≥ m , weuse H¨older’s inequality to the both terms in the right hand side of the above inequalityand get Z R N u | x | β ψ m − Q ( ψ m ) dx = Z R N | x | αp +1 u ψ m − | x | − αp +1 − β Q ( ψ m ) dx ≤ (cid:18)Z R N | x | α | u | p +1 ψ m dx (cid:19) p +1 (cid:18)Z R N | x | − α + βp + βp − Q ( ψ m ) p +1 p − dx (cid:19) p − p +1 , Z R N u | x | β +2 ψ m − R ( ψ m ) dx = Z R N | x | αp +1 u ψ m − | x | − αp +1 − β − R ( ψ m ) dx ≤ (cid:18)Z R N | x | α | u | p +1 ψ m dx (cid:19) p +1 (cid:18)Z R N | x | − α +( β +2)( p +1) p − R ( ψ m ) p +1 p − dx (cid:19) p − p +1 . Therefore, we find Z R N (cid:20) | ∆ u | | x | β + | x | α | u | p +1 (cid:21) ψ m dx ≤ C Z R N | x | − α + βp + βp − Q ( ψ m ) p +1 p − dx + C Z R N | x | − α +( β +2)( p +1) p − R ( ψ m ) p +1 p − dx. Let us choose ψ ∈ C ( B R ( x )) a cut-off function verifying 0 ≤ ψ ≤ ψ ≡ B R ( x ), and |∇ k ψ | ≤ CR k for k ≤ 3. Substituting ψ into (2.8), (2.9) and the aboveinequality, we have Z B R ( x ) (cid:18) | ∆ u | | z | β + | z | α | u | p +1 (cid:19) dz ≤ CR − Z B R ( x ) \ B R ( x ) | u ∆ u || z | β dz + CR − Z B R ( x ) \ B R ( x ) u | z | β dz. and Z B R ( x ) (cid:20) | ∆ u | | z | β + | z | α | u | p +1 (cid:21) ψ m dz ≤ CR N − − β − α +2 βp − . (cid:3) Remark 2.1. If the domain R N is replaced by the subset Ω (boundedness or not) inLemma 2.1-Lemma 2.3, then the conclusions are also true. Lemma 2.4. (Pohozaev identity) Let u be a classical solution of (1.1), then we have N − − β Z Ω | ∆ u | | x | β dx − N + αp + 1 Z Ω | x | α | u | p +1 dx = 12 Z ∂ Ω | ∆ u | | x | β ( x · ν ) dS − p + 1 Z ∂ Ω | x | α | u | p +1 ( x · ν ) dS − Z ∂ Ω ∆ u | x | β ∇ ( x · ∇ u ) · νdS + Z ∂ Ω ∇ (cid:18) ∆ u | x | β (cid:19) · ν ( x · ∇ u ) dS, (2.10) where ν denotes the outward unit normal vector field.Proof. Multiplying (1.1) by ( x · ∇ u ), we obtain∆( | x | − β ∆ u )( x · ∇ u ) = | x | α | u | p − u ( x · ∇ u ) , in Ω \{ } . Hence, for every small ε > 0, we have Z Ω \ B ε ∆( | x | − β ∆ u )( x · ∇ u ) dx = Z Ω \ B ε | x | α | u | p − u ( x · ∇ u ) dx. (2.11)10pply the divergence theorem and integration by parts to calculate the right hand sideand the left hand side of (2.11) respectively, and get1 p + 1 Z Ω \ B ε | x | α (cid:0) x · ∇ (cid:0) | u | p +1 (cid:1)(cid:1) dx = − N + αp + 1 Z Ω \ B ε | x | α | u | p +1 dx + 1 p + 1 Z ∂ Ω | x | α | u | p +1 ( x · ν ) dS − p + 1 Z ∂B ε | x | α | u | p +1 ( x · ν ) dS, (2.12)and Z Ω \ B ε ∆( | x | − β ∆ u )( x · ∇ u ) dx = N X i,j =1 Z Ω \ B ε ( | x | − β ∆ u ) x i x i ( x j u x j ) dx = N X i,j =1 Z Ω \ B ε ( | x | − β ∆ u )( x j u x j ) x i x i dx − Z ∂ Ω | x | − β ∆ u ∇ ( x · ∇ u ) · νdS + Z ∂B ε | x | − β ∆ u ∇ ( x · ∇ u ) · νdS + Z ∂ Ω ∇ ( | x | − β ∆ u ) · ν ( x · ∇ u ) dS − Z ∂B ε ∇ ( | x | − β ∆ u ) · ν ( x · ∇ u ) dS. Again computing the first term in the right hand side of the above equality yields N X i,j =1 Z Ω \ B ε ( | x | − β ∆ u )( x j u x j ) x i x i dx = N X i,j =1 Z Ω \ B ε | x | − β ∆ u h δ ij u x i x j + x j u x i x i x j i dx = − N − − β Z Ω \ B ε | x | − β | ∆ u | dx + Z ∂ Ω | ∆ u | | x | − β x · νdS − Z ∂B ε | ∆ u | | x | − β x · νdS, and putting back into the above equality leads to Z Ω \ B ε ∆( | x | − β ∆ u )( x · ∇ u ) dx = − N − − β Z Ω \ B ε | ∆ u | | x | β dx + Z ∂ Ω | ∆ u | | x | − β x · νdS − Z ∂ Ω ∆ u | x | β ∇ ( x · ∇ u ) · νdS + Z ∂ Ω ∇ (cid:18) ∆ u | x | β (cid:19) · ν ( x · ∇ u ) dS − Z ∂B ε | ∆ u | | x | − β x · νdS + Z ∂B ε ∆ u | x | β ∇ ( x · ∇ u ) · νdS − Z ∂B ε ∇ (cid:18) ∆ u | x | β (cid:19) · ν ( x · ∇ u ) dS. (2.13)Since u ∈ C (Ω), α > − ≤ β ≤ N − 42 , the above integrations are well-defined.Now, we insert (2.12) and (2.13) into (2.11), take ε → Pohozaev identity to construct amonotonicity formula which is a crucial tool. More precisely, choose u ∈ W , loc (Ω) and11 x | α | u | p +1 ∈ L loc (Ω), fix x ∈ Ω, let 0 < r < R and B r ( x ) ⊂ B R ( x ) ⊂ Ω, and define M ( r ; x, u ) = r δ Z B r ( x ) | ∆ u | | z | β − p + 1 | z | α | u | p +1 + (1 + β ) λ N − − λ ) r λ +1 − N Z ∂B r ( x ) u ! + λ N − − λ ) ddr r λ +2 − N Z ∂B r ( x ) u ! + r ddr " r λ +1 − N Z ∂B r ( x ) (cid:18) λr − u + ∂u∂r (cid:19) + 1 + β − λ r λ +3 − N Z ∂B r ( x ) |∇ u | − (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) ! + 12 ddr " r λ +4 − N Z ∂B r ( x ) |∇ u | − (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) ! . (2.14)Here and in the following, we always set δ := 8 + 2 α + 2 βp − β − N and λ := 4 + α + βp − Theorem 2.1. Let p ≥ N + 4 + 2 α + βN − − β and let u ∈ W , loc (Ω) , | x | α | u | p +1 ∈ L loc (Ω) and | x | − β | ∆ u | ∈ L loc (Ω) be a weak solution of (1.1). Then M ( r ; x, u ) is nondecreasing in r ∈ (0 , R ) and satisfies the inequality ddr M ( r ; 0 , u ) ≥ C ( N, p, α, β ) r − N +2+2 λ Z ∂B r (cid:18) λr − u + ∂u∂r (cid:19) dS, where C ( N, p, α, β ) = ( N − β ) + 2 λ ( N − − β − λ ) − β > .Furthermore, if M ( r ; 0 , u ) ≡ const, for all r ∈ (0 , R ) , then u is homogeneous in B R \{ } , i.e., ∀ µ ∈ (0 , , x ∈ B R \{ } , u ( µx ) = µ − α + βp − u ( x ) . Proof. Define a function by F ( κ ) := κ δ Z B κ (cid:18) | ∆ u | | x | β − p + 1 | x | α | u | p +1 (cid:19) dx. Differentiating the function F ( κ ) in κ arrives at d F ( κ ) dκ = δκ δ − Z B κ (cid:18) | ∆ u | | x | β − p + 1 | x | α | u | p +1 (cid:19) dx + κ δ Z ∂B κ (cid:18) | ∆ u | | x | β − p + 1 | x | α | u | p +1 (cid:19) dS. (2.15)12ultiply the equation (1.1) by u and integrate by parts to get Z Ω | x | α | u | p +1 dx = − Z Ω ∇ (cid:18) ∆ u | x | β (cid:19) · ∇ udx + Z ∂ Ω ∂∂ν (cid:18) ∆ u | x | β (cid:19) udS = Z Ω | ∆ u | | x | β dx − Z ∂ Ω ∆ u | x | β ( ∇ u · ν ) dS + Z ∂ Ω ∂∂ν (cid:18) ∆ u | x | β (cid:19) udS, implies Z Ω | ∆ u | | x | β dx − Z Ω | x | α | u | p +1 dx = Z ∂ Ω ∆ u | x | β ( ∇ u · ν ) dS − Z ∂ Ω ∂∂ν (cid:18) ∆ u | x | β (cid:19) udS. (2.16)In addition, it is easily to see that δκ δ − Z B κ (cid:18) | ∆ u | | x | β − p + 1 | x | α | u | p +1 (cid:19) dx = λκ δ − Z B κ (cid:18) | ∆ u | | x | β − | x | α | u | p +1 (cid:19) dx − κ δ − Z B κ (cid:18) N − − β | ∆ u | | x | β − N + αp + 1 | x | α | u | p +1 (cid:19) dx. Therefore, combining the result with (2.10), (2.15) and (2.16), we obtain d F ( κ ) dκ = λκ δ − (cid:20)Z ∂B κ ∆ u | x | β ( ∇ u · ν ) dS − Z ∂B κ ∂∂ν (cid:18) ∆ u | x | β (cid:19) udS (cid:21) + κ δ − Z ∂B κ ∆ u | x | β ∇ ( x · ∇ u ) · νdS − κ δ − Z ∂B κ ∇ (cid:18) ∆ u | x | β (cid:19) · ν ( x · ∇ u ) dS. (2.17)Denote u κ ( x ) := κ α + βp − u ( κx ). Now, computing the first term in the right hand sideof (2.17) leads to λκ δ − Z ∂B κ | x | − β ∆ u ( ∇ u · ν ) dS = λκ Z ∂B R ∆ κ α + βp − u (cid:16) ∇ κ α + βp − u · ν (cid:17) dS · κ − N = λκ Z ∂B ∆ u κ ( ∇ u κ · ν ) dσ. (2.18)Similarly, we calculate the second term in the right hand side of (2.17) and get λκ δ − Z ∂B κ ∂∂ν (cid:18) ∆ u | x | β (cid:19) udS = λκ δ − Z ∂B κ | x | − β (cid:2) ( ∇ (∆ u ) · ν ) − β | x | − ∆ u ( x · ν ) (cid:3) udS = λκ Z ∂B κ h(cid:16) ∇ (cid:16) ∆ κ α + βp − u (cid:17) · ν (cid:17) − βκ − ∆ (cid:16) κ α + βp − u (cid:17) ( x · ν ) i κ α + βp − udS · κ − N = λκ Z ∂B h ( ∇ (∆ u κ ) · ν ) − β ∆ u κ i u κ dσ. (2.19)Similar to the above calculation, we find κ δ − Z ∂B κ | x | − β ∆ u ∇ ( x · ∇ u ) · νdS = 1 κ Z ∂B ∆ u κ ∇ ( x · ∇ u κ ) · νdσ, (2.20)13nd κ δ − Z ∂B κ ∇ ( | x | − β ∆ u ) · ν ( x · ∇ u ) dS = 1 κ Z ∂B h ( ∇ (∆ u κ ) · ν ) − β ∆ u κ i ( x · ∇ u κ ) dσ. (2.21)We use spherical coordinates r = | x | , θ = x | x | ∈ S N − and write u κ ( x ) = u κ ( r, θ ), thenwe insert (2.18)-(2.21) into (2.17) to obtain d F ( κ ) dκ = λκ Z ∂B ∆ u κ ( ∇ u κ · ν ) dσ − λκ Z ∂B h ( ∇ (∆ u κ ) · ν ) − β ∆ u κ i u κ dσ + 1 κ Z ∂B ∆ u κ ∇ ( x · ∇ u κ ) · νdσ − κ Z ∂B h ( ∇ (∆ u κ ) · ν ) − β ∆ u κ i ( x · ∇ u κ ) dσ = 1 κ Z ∂B λ (cid:18) ∂ u κ ∂r + ( N − ∂u κ ∂r + ∆ θ u κ (cid:19) ∂u κ ∂r − λ (cid:20) ∂ u κ ∂r + ( N − − β ) ∂ u κ ∂r − ( N − β ) ∂u κ ∂r − (2 + β )∆ θ u κ (cid:21) u κ + (cid:20) ∂ u κ ∂r + ( N − ∂u κ ∂r + ∆ θ u κ (cid:21) (cid:18) ∂ u κ ∂r + ∂u κ ∂r (cid:19) − (cid:20) ∂ u κ ∂r + ( N − − β ) ∂ u κ ∂r − ( N − β ) ∂u κ ∂r − (2 + β )∆ θ u κ (cid:21) ∂u κ ∂r = 1 κ Z ∂B − ∂ u κ ∂r ∂u κ ∂r − λ ∂ u κ ∂r u κ + (cid:18) ∂ u κ ∂r (cid:19) + ( λ + 1 + β ) ∂ u κ ∂r ∂u κ ∂r + ( N − λ + 2 + β ) (cid:18) ∂u κ ∂r (cid:19) − λ ( N − − β ) ∂ u κ ∂r u κ + λ ( N − β ) ∂u κ ∂r u κ + 1 κ Z ∂B ∆ θ u κ ∂ u κ ∂r + ( λ + 3 + β )∆ θ u κ ∂u κ ∂r + λ (2 + β )∆ θ u κ u κ := T + T , where ∆ θ represents the Laplace-Beltrami operator on ∂B and ∇ θ is the tangentialderivative on ∂B .Differentiating u κ in κ implies du κ dκ ( x ) = 1 κ (cid:20) λu κ ( x ) + r ∂u κ ∂r ( x ) (cid:21) = ⇒ r ∂u κ ∂r = κ du κ dκ − λu κ . (2.22)Differentiating the equation (2.22) in κ and r respectively yields r ∂∂r du κ dκ = κ d u κ dκ + (1 − λ ) du κ dκ and κ ∂∂r du κ dκ =(1 + λ ) ∂u κ ∂r + r ∂ u κ ∂r . Then, combining the above two equalities with (2.22), we obtain that, on ∂B ∂ u κ ∂r = r ∂ u κ ∂r = κ d u κ dκ − λκ du κ dκ + λ (1 + λ ) u κ . (2.23)14imilarly, we find r ∂ u∂r + 2 r ∂ u∂r = κ ∂∂r d u κ dκ − λκ ∂∂r du κ dκ + λ (1 + λ ) ∂u κ ∂r ,r ∂∂r d u κ dκ = κ d u κ dκ + (2 − λ ) d u κ dκ . Then, on ∂B , we have ∂ u∂r = κ d u κ dκ − λκ d u κ dκ + 3 λ (1 + λ ) κ du κ dκ − λ (1 + λ )(2 + λ ) u κ . (2.24)Substituting (2.22) and (2.23) into the expression of T arrives at T = Z ∂B κ ∆ θ u κ d u κ dκ − ( λ − − β )∆ θ u κ du κ dκ = Z ∂B − κ ∇ θ u κ ∇ θ d u κ dκ + ( λ − − β ) ∇ θ u κ ∇ θ du κ dκ = − d dκ (cid:20) κ Z ∂B |∇ θ u κ | (cid:21) + λ − − β ddκ Z ∂B |∇ θ u κ | + κ Z ∂B (cid:12)(cid:12)(cid:12)(cid:12) ∇ θ du κ dκ (cid:12)(cid:12)(cid:12)(cid:12) ≥ − d dκ (cid:20) κ Z ∂B |∇ θ u κ | (cid:21) + λ − − β ddκ Z ∂B |∇ θ u κ | . Let us note the two equalities − κ du κ dκ d u κ dκ = ddκ − κ ddκ (cid:18) du κ dκ (cid:19) ! + 3 κ du κ dκ d u κ dκ + κ (cid:18) d u κ dκ (cid:19) ,κu κ d u κ dκ = d dκ κ ( u κ ) ! − u κ du κ dκ − κ (cid:18) du κ dκ (cid:19) . Inserting (2.22)-(2.24) into the expression of T , and combining with the above twoequalities, we get T = Z ∂B − κ d u κ dκ du κ dκ + κ (cid:18) d u κ dκ (cid:19) + (1 + β ) κ d u κ dκ du κ dκ + h ( N − λ + β ) − λ (5 + λ + 2 β ) i κ (cid:18) du κ dκ (cid:19) + λ (2 + λ − N ) κu κ d u κ dκ + λ (3 + β )( λ + 2 − N ) u κ du κ dκ = Z ∂B ddκ − κ ddκ (cid:18) du κ dκ (cid:19) ! + 2 κ (cid:18) d u κ dκ (cid:19) + (4 + β ) κ d u κ dκ du κ dκ + h ( N − β ) + 2 λ ( N − − β − λ ) i κ (cid:18) du κ dκ (cid:19) + λ (2 + λ − N ) d dκ (cid:18) κ ( u κ ) (cid:19) + λ (2 + λ − N )(1 + β ) u κ du κ dκ Z ∂B ddκ − κ ddκ (cid:18) du κ dκ (cid:19) ! + λ (2 + λ − N )2 d dκ (cid:0) κ ( u κ ) (cid:1) + λ λ − N )(1 + β ) ddκ ( u κ ) . Since p ≥ N + 4 + 2 α + βN − − β , the deleted terms of T satisfies2 κ (cid:18) d u κ dκ (cid:19) +(4 + β ) κ d u κ dκ du κ dκ + h ( N − β ) + 2 λ ( N − − β − λ ) i κ (cid:18) du κ dκ (cid:19) = 2 κ (cid:20) κ d u κ dκ + (cid:18) β (cid:19) du κ dκ (cid:21) + C ( N, p, α, β ) κ (cid:18) du κ dκ (cid:19) ≥ , where C ( N, p, α, β ) =( N − β ) + 2 λ ( N − − β − λ ) − (cid:18) β (cid:19) =( N − β ) + 2 λ ( N − − β − λ ) − β > . Now, we rescale and write those κ derivatives in T and T as follows. Z ∂B ddκ ( u κ ) = ddκ (cid:18) κ λ +1 − N Z ∂B κ u (cid:19) , Z ∂B d dκ h κ ( u κ ) i = d dκ (cid:18) κ λ +2 − N Z ∂B κ u (cid:19) , Z ∂B ddκ " κ ddκ (cid:18) du κ dκ (cid:19) = ddκ " κ ddκ κ λ +1 − N Z ∂B κ (cid:18) λκ − u + ∂u∂r (cid:19) ! ,ddκ (cid:18)Z ∂B |∇ θ u κ | (cid:19) = ddκ " κ λ +3 − N Z ∂B κ |∇ u | − (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) ! ,d dκ (cid:18) κ Z ∂B |∇ θ u κ | (cid:19) = d dκ " κ λ +4 − N Z ∂B κ |∇ u | − (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) ! . Substituting these terms into d F ( κ ) dκ yields d F ( κ ) dκ ≥ λ (2 + λ − N )(1 + β )2 ddκ (cid:18) κ λ +1 − N Z ∂B κ u (cid:19) + λ (2 + λ − N )2 d dκ (cid:18) κ λ +2 − N Z ∂B κ u (cid:19) − ddκ " κ ddκ κ λ +1 − N Z ∂B κ (cid:18) λκ − u + ∂u∂r (cid:19) ! + λ − − β ddκ " κ λ +3 − N Z ∂B κ |∇ u | − (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) ! − d dκ " κ λ +4 − N Z ∂B κ |∇ u | − (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) ! . M ( r ; x, u ) is well defined andnon-decreasing in r ∈ (0 , R ).Next, we let M ( r ; 0 , u ) ≡ const, for all r ∈ (0 , R ), then, for any r , r ∈ (0 , R ) with r < r , we have0 = M ( r ; 0 , u ) − M ( r ; 0 , u ) = Z r r ddµ M ( µ ; 0 , u ) dµ ≥ C ( N, p, α, β ) Z B r \ B r | x | λ − N (cid:18) λµ − u + ∂u∂µ (cid:19) dx. Thus, we get λµ − u + ∂u∂µ = 0 , a.e. in B R \{ } . Integrating in r shows that u ( µx ) = µ − α + βp − u ( x ) , ∀ µ ∈ (0 , , x ∈ B R \{ } . (cid:3) Remark 2.2. From the proof of Theorem 2.1, we can find that if the linear combinationof Pohozaev identity and the identity (2.16) minus some terms of T and T , then it isequivalent to the derivative form of the monotonicity formula (2.14). First, we give the expression of N α,β ( p ). Now, we define four functions by f ( N ) := p α + βp − (cid:18) α + βpp − (cid:19) (cid:18) N − − α + βp − (cid:19) (cid:18) N − − α + βpp − (cid:19) , g ( N ) := p (cid:18) α + βpp − (cid:19) (cid:18) N − − α + βpp − (cid:19) + p α + βp − (cid:18) N − − α + βp − (cid:19) , F ( N ) := ( N + β ) ( N − − β ) , G ( N ) := ( N + β )( N − − β )2 . Differentiating the functions f ( N ) and F ( N ) in N , we obtain f ′ ( N ) = p α + βp − (cid:18) α + βpp − (cid:19) (cid:18) N − − β − α + 2 βp − (cid:19) , F ′ ( N ) = 14 ( N + β )( N − N − − β ) . 17 simple computation yields f (4 + β + 2 λ ) − F (4 + β + 2 λ ) = ( p − λ (2 + β + λ ) > , f (4 + β + (4 p + 1) λ ) − F (4 + β + (4 p + 1) λ )= 4 p λ (2 + β + λ )(2 + β + 4 pλ ) − (4 p + 1) λ h (4 + 2 β ) + (4 p + 1) λ i < , g (4 + β + 2 λ ) − G (4 + β + 2 λ ) = 2( p − λ (2 + β + λ ) > , f ′ (4 + β + 2 λ ) − F ′ (4 + β + 2 λ ) = ( p − λ (2 + β + λ )(2 + β + 2 λ ) > . Therefore, we take the least real root N ( p, α, β ) of the following algebra equation be-tween 4 + β + 2 λ with 4 + β + (4 p + 1) λ ( p − p + 6 p − p + 1) y − (8 p − p + 48 p − p + 8) y − ( p − p + 1) h (32 α + 104 β + 16 αβ + 18 β + 112) p + (16 αβ + 16 α − β + 16 β + 96 α + 160) p + 8 β + 2 β − i y + nh(cid:16) 48 + 44 β + 12 β + 8 αβ + 12 α + αβ + β (cid:17) p + (cid:16) 64 + 56 α + 28 αβ + 10 α + 40 β + 10 β + 4 αβ + 3 α β + β (cid:17) p + 28 α + 16 + 14 α + 2 α + 12 β + 12 αβ + 3 α β + 2 β + αβ i p − p ) − ( p − (32 β + 8 β ) o y + ( p − β ( β + 4) − h (8 + (2 + β ) α + (6 + β ) β ) p + (6 + α + β ) α + 2 β + 8 i × h (8 + 2 β ) p + (6 β + 6 α + αβ + β + 8) p + 2 α + α + αβ i p = 0 . Define N α,β ( p ) := N ( p, α, β ) . (3.1)Then, for any 4 + β + 8 + 2 α + 2 βp − < N < N α,β ( p ), we find f ( N ) > ( N + β ) ( N − − β ) . (3.2)Furthermore, combining the above inequality with the inequality a + b ≥ √ ab , for all a, b ≥ 0, we have g ( N ) > ( N + β )( N − − β )2 . (3.3)18n the other hand, we easily check that the equality f ( N ) − F ( N ) > α = β and 4 + α + 8 + 4 αp − < N < α + 8 + 4 αp − α = β = 0 and 4 + 8 p − < N < p + 1) p − r pp + 1 + s pp + 1 − r pp + 1 ! .Let us recall that if we takeΓ = 4 + α + βp − (cid:18) α + βpp − (cid:19) (cid:18) N − − α + βp − (cid:19) (cid:18) N − − α + βpp − (cid:19) , then u Γ ( r ) = Γ p − r − α + βp − is a singular solution of (1.1) in R N \{ } . By the well-known weighted Hardy-Rellichinequality ([15]) with the best constant Z R N | ∆ ψ | | x | β dx ≥ ( N + β ) ( N − − β ) Z R N ψ | x | β dx, ∀ ψ ∈ H loc ( R N ) , we conclude that the singular solution u Γ is stable in R N \{ } if and only if f ( N ) = p Γ ≤ ( N + β ) ( N − − β ) 16 = F ( N ) . Here − − p N − ≤ β ≤ N − 42 . Proof of Theorem 1.1. Since u ∈ W , ( B \ B ), | x | α | u | p +1 ∈ L loc ( R N \{ } ) and | x | − β | ∆ u | ∈ L loc ( R N \{ } ), we can assume that there exists a Ψ ∈ W , ( S N − ) ∩ L p +1 ( S N − ), such that in polar coordinates u ( r, θ ) = r − α + βp − Ψ( θ ) . Substituting into (1.1) to get∆ θ Ψ − Υ∆ θ Ψ + ΓΨ = | Ψ | p − Ψ , where Υ = λ ( N − − λ ) + (cid:18) α + βpp − (cid:19) (cid:18) N − − α + βpp − (cid:19) , Γ = λ ( N − − λ ) (cid:18) α + βpp − (cid:19) (cid:18) N − − α + βpp − (cid:19) . Z S N − | ∆ θ Ψ | + Υ |∇ θ Ψ | + ΓΨ = Z S N − | Ψ | p +1 . (3.4)Since u is a stable solution, we can take a test function r − N − − β Ψ( θ ) ξ ε ( r ) and obtain p Z R N | x | α | u | p − (cid:16) r − N − − β Ψ( θ ) ξ ε ( r ) (cid:17) dx ≤ Z R N (cid:12)(cid:12)(cid:12) ∆ (cid:16) r − N − − β Ψ( θ ) ξ ε ( r ) (cid:17)(cid:12)(cid:12)(cid:12) | x | β dx. (3.5)Here, for any ε > 0, we choose ξ ε ∈ C (cid:0)(cid:0) ε , ε (cid:1)(cid:1) such that ξ ε ≡ (cid:0) ε, ε (cid:1) and r | ξ ′ ε ( r ) | + r | ξ ′′ ε ( r ) | ≤ C, for all r > 0. Then one can easily deduce that Z ∞ r − ξ ε ( r ) dr ≥ Z ε ε r − dr = 2 | ln ε | , and Z ∞ h r | ξ ′ ε ( r ) | + r | ξ ′′ ε ( r ) | + | ξ ′ ε ( r ) ξ ε ( r ) | + r | ξ ε ( r ) ξ ′′ ε ( r ) | i dr ≤ C. Applying the coordinate transformation to the left hand side of (3.5), we get p Z + ∞ Z S N − r α | u | p − (cid:16) r − N − − β Ψ( θ ) ξ ε ( r ) (cid:17) r N − drdθ = p (cid:18)Z S N − | Ψ | p +1 dθ (cid:19) (cid:18)Z + ∞ r − ξ ε ( r ) dr (cid:19) . (3.6)A direct calculation finds∆ (cid:16) r − N − − β Ψ( θ ) ξ ε ( r ) (cid:17) = − ( N + β )( N − − β )4 r − N − β ξ ε ( r )Ψ( θ ) + r − N − β ξ ε ( r )∆ θ Ψ+ (3 + β ) r − N − − β ξ ′ ε ( r )Ψ( θ ) + r − N − − β ξ ′′ ε ( r )Ψ( θ ) , and inserting into the right hand side of (3.5) yields Z R N | x | − β (cid:12)(cid:12)(cid:12) ∆ (cid:16) r − N − − β Ψ( θ ) ξ ε ( r ) (cid:17)(cid:12)(cid:12)(cid:12) dx ≤ (cid:20)Z S N − (cid:18) | ∆ θ Ψ | + ( N + β )( N − − β )2 |∇ θ Ψ | + ( N + β ) ( N − − β ) 16 Ψ (cid:19) dθ (cid:21) × (cid:18)Z + ∞ r − ξ ε ( r ) dr (cid:19) + O (cid:26)Z + ∞ (cid:2) r | ξ ′ ε ( r ) | + r | ξ ′′ ε ( r ) | + | ξ ′ ε ( r ) | ξ ε ( r ) + rξ ε ( r ) | ξ ′′ ε ( r ) | (cid:3) dr (cid:27) × Z S N − (cid:2) Ψ( θ ) + |∇ θ Ψ( θ ) | (cid:3) dθ. (3.7)20ut (3.6) and (3.7) back into (3.5), take ε → 0, and pass to the limit to obtain p Z S N − | Ψ | p +1 dθ ≤ Z S N − (cid:20) | ∆ θ Ψ | + ( N + β )( N − − β )2 |∇ θ Ψ | + ( N + β ) ( N − − β ) 16 Ψ (cid:21) dθ. Now, combining the above inequality with (3.4), we have Z S N − ( p − | ∆ θ Ψ | + (cid:18) p Υ − ( N + β )( N − − β )2 (cid:19) |∇ θ Ψ | + (cid:18) p Γ − ( N + β ) ( N − − β ) (cid:19) Ψ ≤ . Since 4 + β + 8 + 2 α + 2 βp − < N < N α,β ( p ), it implies from the definition of N α,β ( p ),(3.2) and (3.3) that Ψ( θ ) ≡ . Therefore, we get u ≡ . (cid:3) Proof of Theorem 1.2. We divide the proof into three cases. Case I. ≤ N < β + 8 + 2 α + 2 βp − N < β + 8 + 2 α + 2 βp − R → + ∞ , Z B R ( x ) (cid:20) | ∆ u | | z | β + | z | α | u | p +1 (cid:21) dz ≤ CR N − − β − α +2 βp − → . Therefore, we get u ≡ . Case II. N = 4 + β + 8 + 2 α + 2 βp − Z R N (cid:20) | ∆ u | | z | β + | z | α | u | p +1 (cid:21) dz < + ∞ , implies lim R → + ∞ Z D (cid:20) | ∆ u | | z | β + | z | α | u | p +1 (cid:21) dz = 0 , D := B R ( x ) \ B R ( x ). Applying (2.3) and H¨older’s inequality yields Z B R ( x ) (cid:20) | ∆ u | | z | β + | z | α | u | p +1 (cid:21) dz ≤ CR − Z D | u ∆ u || z | β dz + CR − Z D u | z | β dz ≤ C C R − (cid:18)Z D | z | α | u | p +1 dz (cid:19) p +1 (cid:18)Z D | z | − α + β ( p +1) p − dz (cid:19) p − p +1) + CR − (cid:18)Z D | z | α | u | p +1 dz (cid:19) p +1 (cid:18)Z D | z | − α + β ( p +1) p − dz (cid:19) p − p +1 ≤ C C R h N − − β − α +2 βp − i p − p +1) (cid:18)Z D | z | α | u | p +1 dz (cid:19) p − p +1) + CR h N − − β − α +2 βp − i p − p +1 (cid:18)Z D | z | α | u | p +1 dz (cid:19) p − p +1 , where C = (cid:18)Z D | ∆ u | | z | β dz (cid:19) . From N = 4 + β + 8 + 2 α + 2 βp − R → + ∞ . Therefore, we obtain u ≡ . Case III. β + 8 + 2 α + 2 βp − < N < N α,β ( p ).First, we will obtain some properties of the function M . Lemma 4.1. lim r → + ∞ M ( r ; 0 , u ) < + ∞ .Proof. The proof mainly use the estimate (2.4) and the monotonicity of the function M ( r ; 0 , u ) in r .Applying (2.4) to estimate the first term in the right hand side of (2.14) yields r α +2 βp − +4+ β − N Z B r (cid:20) 12 (∆ u ) | x | β − p + 1 | x | α | u | p +1 (cid:21) dx ≤ Cr α +2 βp − +4+ β − N r N − − β − α +2 βp − ≤ C. Utilize H¨older’s inequality to estimate the second term in the right hand side of (2.14) r α +2 βp − +1 − N Z ∂B r u ≤ r Z rr µ α +2 βp − +1 − N Z ∂B µ u dS ! dµ ≤ r Z B r \ B r (cid:16) | x | α +2 βp − +1 − N − αp +1 (cid:17) p +1 p − ! p − p +1 (cid:18)Z B r | x | α | u | p +1 (cid:19) p +1 ≤ Cr h α +2 βp − − N − αp +1 + N p − p +1 i r p +1 h N − − β − α +2 βp − i ≤ C. ddr (cid:18) r λ +2 − N Z ∂B r u (cid:19) ≤ r Z rr Z ι + rι ddµ µ λ +2 − N Z ∂B µ u ! dµdι ≤ C. By the interpolation inequality and H¨older’s inequality, we get Z B r |∇ u | ≤ Cr Z B r | ∆ u | + Cr − Z B r u ≤ Cr Z B r | x | β | ∆ u | | x | β + Cr − (cid:18)Z B r | x | α | u | p +1 (cid:19) p +1 (cid:18)Z B r | x | − αp − dx (cid:19) p − p +1 ≤ Cr N − − α +2 βp − . (4.1)Then, it implies that r α +2 βp − +3 − N Z ∂B r |∇ u | dS ≤ r Z rr µ α +2 βp − +3 − N Z ∂B µ |∇ u | dS ! dµ ≤ C. Therefore, we get the boundedness of the fifth and sixth terms in the right hand side of(2.14). Utilizing H¨older’s inequality and (4.1), we find1 r Z rr Z ι + rι µ ddµ " µ λ +1 − N Z ∂B µ (cid:16) λµ − u + ∂u∂r (cid:17) dµdι = 12 r Z rr (cid:26) ( ι + r ) λ +4 − N Z ∂B ι + r h λ ( ι + r ) − u + ∂u∂r i − ι λ +4 − N Z ∂B r h λι − u + ∂u∂r i (cid:27) − r Z rr Z ι + rι µ λ +3 − N Z ∂B µ (cid:18) λµ − u + ∂u∂r (cid:19) ≤ Cr Z B r \ B r | x | λ +2 − N u + | x | (cid:18) ∂u∂r (cid:19) ! dx ≤ C. Consequently, we obtain the desired result. Lemma 4.2. For all κ > , define blowing down sequences u κ ( x ) := κ α + βp − u ( κx ) , then u κ strongly converges to u ∞ in W , loc ( R N ) ∩ L p +1 loc ( R N ) . Furthermore, u ∞ is ahomogeneous stable solution of (1.1). roof. Since u is a stable solution of (1.1), we can find p Z R N | x | α | u κ | p − ζ ( x ) dx = p Z R N | κx | α κ β | u ( κx ) | p − ζ ( x ) dx = pκ β − N Z R N | y | α | u ( y ) | p − ψ ( y ) dy taking ψ ( y ) := ζ ( x ) , x = yκ ≤ κ β − N Z R N | ∆ ψ ( y ) | | y | β dy = Z R N | ∆ ζ | | x | β dx. (4.2)Thus, u κ is a stable solution of (1.1). Furthermore, from (2.4), it implies that Z B r ( x ) h | y | − β (∆ u κ ) + | y | α | u κ | p +1 i dy = κ β + α +2 βp − − N Z B κr ( x ) h | z | − β | ∆( z ) | + | z | α | u ( z ) | p +1 i dz ≤ Cr N − − β − α +2 βp − , and applying H¨older’s inequality yields Z B r ( x ) | u κ | dz ≤ Z B r ( x ) | z | α | u κ | p +1 dz ! p +1 Z B r ( x ) | z | − αp − dz ! p − p +1 ≤ Cr N − λ . Clearly, we also obtain Z B r ( x ) | ∆ u κ | dz = Z B κr ( x ) κ λ +4 − N | z | β | ∆ u ( z ) | | z | β dz ≤ Cr N − − λ . By the application of the elliptic regularity theory, it implies that u κ are uniformlybounded in W , loc ( R N ). Again u ∈ C ( R N ) implies u κ ∈ L p +1 loc ( R N ). Then we cansuppose that u κ ⇀ u ∞ weakly in W , loc ( R N ) ∩ L p +1 loc ( R N ) (if necessary, we can extracta subsequence). Now, using the standard embeddings, we get u κ → u ∞ strongly in W , loc ( R N ). Therefore, applying the interpolation inequality between L q spaces with q ∈ (1 , p + 1), we get that, for any ball B r k u κ − u ∞ k L q ( B r ) ≤ k u κ − u ∞ k tL ( B r ) k u κ − u ∞ k − tL p +1 ( B r ) → , as κ → + ∞ , (4.3)where t ∈ (0 , 1) satisfying 1 q = t + 1 − tp + 1 . Next, combining with the definition of u κ and244.2), we conclude that, for any ζ ∈ C ( R N ) Z R N ∆ u ∞ | x | β ∆ ζ − | x | α | u ∞ | p − u ∞ ζ = lim κ →∞ Z R N ∆ u κ | x | β ∆ ζ − | x | α | u κ | p − u κ ζ, Z R N (∆ ζ ) | x | β − p | x | α | u ∞ | p − ζ = lim κ →∞ Z R N (∆ ζ ) | x | β − p | x | α | u κ | p − ζ ≥ , that is, u ∞ ∈ W , loc ( R N ) ∩ L p +1 loc ( R N ) is a stable solution of (1.1) in R N .From the boundedness and monotonicity of M ( r ; 0 , u ), it implies that for any 0 25 direct application of the interior L p -estimates getslim κ → + ∞ Z B X j ≤ |∇ j u κ | = 0 , implies Z ∞ X i =1 Z ∂B r X j ≤ |∇ j u κ i | dr ≤ ∞ X i =1 Z B r \ B r X j ≤ |∇ j u κ i | ≤ . Then, let us note that there exists a γ ∈ (1 , 2) such thatlim κ →∞ k u κ k W , ( ∂B γ ) = 0 . Now, combing the above results with the scaling invariance of M ( r ; 0 , u ), we obtainlim i →∞ M ( κ i γ ; 0 , u ) = lim i →∞ M ( γ ; 0 , u κ i ) = 0 . 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