Liouville-type theorems with finite Morse index for Δ_λ-Laplace operator
aa r X i v : . [ m a t h . A P ] J a n Liouville-type theorems with finite Morse index for ∆ λ -Laplace operator Belgacem Rahal a a Facult´e des Sciences, D´epartement de Math´ematiques, B.P 1171 Sfax 3000, Universit´e de Sfax, Tunisia.
Abstract
In this paper we study solutions, possibly unbounded and sign-changing, of the following problem − ∆ λ u = | x | a λ | u | p − u , in R n , n ≥ , p > , and a ≥ , where ∆ λ is a strongly degenerate elliptic operator, the functions λ = ( λ , ..., λ k ) : R n → R k satisfies some certain conditions, and | . | λ the homogeneous norm associated to the ∆ λ -Laplacian.We prove various Liouville-type theorems for smooth solutions under the assumption that theyare stable or stable outside a compact set of R n . First, we establish the standard integral estimatesvia stability property to derive the nonexistence results for stable solutions. Next, by mean of thePohozaev identity, we deduce the Liouville-type theorem for solutions stable outside a compactset. Keywords:
Liouville-type theorems, ∆ λ -Laplace operator, Stable solutions, Stability outside acompact set, Pohozaev identity. PACS:
Primary : 35J55, 35J65 ; Secondary : 35B65. Introduction and main results
The Liouville type theorem is the nonexistence of solutions in the entire space or in half-space.The classical Liouville type theorem stated that a bounded harmonic (or holomorphic) functiondefined in entire space must be constant. This theorem, known as Liouville theorem, was firstannounced in 1844 by Liouville [15] for the special case of a doubly-periodic function. Later inthe same year, Cauchy [3] published the first proof of the above stated theorem. This classicalresult has been extended to nonnegative solutions of the semilinear elliptic equation − ∆ u = | u | p − u in R n , p > , (1.1)in the whole space R n by Gidas and Spruck [10, 11] see also the paper of Chen and Li [4]. Theyproved that if 1 < p < n + n − , then the above equation only has the trivial solution u ≡ p ≥ p c ( n ) and n ≥
11, or p = n + n − and n ≥
3, where p c ( n ) is the so-called Joseph-Lundgren exponent. The study of stable solutionsin the H´enon type elliptic equation : − ∆ u = | x | a | u | p − u , in R n , p > a > − Email address: [email protected] (Belgacem Rahal )
Preprint submitted to arXiv 18 septembre 2018
In the past years, the Liouville property has been refined considerably and emerged as one ofthe most powerful tools in the study of initial and boundary value problems for nonlinear PDEs.It turns out that one can obtain from Liouville-type theorems a variety of results on qualitativeproperties of solutions such as universal, pointwise, a priori estimates of local solutions ; universaland singularity estimates ; decay estimates ; blow-up rate of solutions of nonstationary problems,etc., see [19, 21] and references therein.Liouville-type theorems for degenerate elliptic equations have been attracted the interest ofmany mathematicians. The classical Liouville theorem was generalized to p -harmonic functionson the whole space R n and on exterior domains by Serrin and Zou [22], see also [5] for related re-sults. The Liouville theorems for some linear degenerate elliptic operators such as X -elliptic opera-tors, Kohn-Laplacian (and more general sublaplacian on Carnot groups) and degenerate Ornstein-Uhlenbeck operators were proved in [14, 13].More recently, Yu [24] studied the equation − L α u = f ( u ) in R n × R n , where L α = ∆ x + (1 + α ) ∆ y , α > Q = n + (1 + α ) n is the homogeneous dimension ofthe space. Under some assumptions on the nonlinear term f , he showed that the above equationpossesses no positive solutions and the main technique used is the moving plane method in theintegral form.In this paper, we are concerned with the Liouville-type theorems for the following problem − ∆ λ u = | x | a λ | u | p − u , in R n : = R n × R n × ... × R n k , (1.2)where n ≥ a ≥ p > ∆ λ = λ ∆ x (1) + ... + λ k ∆ x ( k ) , | x | λ : = k X j = Y i , j λ i ( x ) ǫ j | x ( j ) | σ ,σ = + P ki = ( ǫ i − ≤ ǫ ≤ ... ≤ ǫ k , x = ( x (1) , ..., x ( k ) ) ∈ R n . Here the functions λ i : R n → R are continuous, strictly positive and of class C outside the coordinate hyperplanes, i.e. λ i > i = , ..., k in R n \ Q , where Q = { x = ( x , ..., x n ) ∈ R n : Q ni = x i = } , and ∆ x ( i ) denotes the classi-cal Laplacian in R n i , i = , ..., k . As in [12] we assume that λ i satisfy the following properties :( H ) λ ( x ) = λ i ( x ) = λ i ( x (1) , ..., x ( i − ), i = , ..., k .( H ) For every x ∈ R n , λ i ( x ) = λ i ( x ∗ ), i = , ..., k , where x ∗ = ( | x (1) | , ..., | x ( k ) | ) if x = ( x (1) , ..., x ( k ) ).( H ) There exists a group of dilations { δ t } t > δ t : R n → R n , δ t ( x ) = δ t ( x (1) , ..., x ( k ) ) = ( t ǫ x (1) , ..., t ǫ k x ( k ) ) , where 1 ≤ ǫ ≤ ǫ ≤ ... ≤ ǫ k , such that λ i is δ t -homogeneous of degree ǫ i −
1, i.e. λ i ( δ t ( x )) = t ǫ i − λ i ( x ) , ∀ x ∈ R n , t > , i = , ..., k . This implies that the operator ∆ λ is δ t -homogeneous of degree two, i.e. ∆ λ ( u ( δ t ( x ))) = t ( ∆ λ u )( δ t ( x )) , ∀ u ∈ C ∞ ( R n ) . We denote by Q the homogeneous dimension of R n with respect to the group of dilations { δ t } t > ,i.e. Q : = ǫ n + ǫ n + ... + ǫ k n k . The ∆ λ -Laplace operator was first introduced by Franchi and Lanconelli [8], and recently reconsi-dered in [12] under an additional assumption that the operator is homogeneous of degree two withrespect to a group dilation in R n . It was proved in [1], that the autonomous case, i.e. a =
0, (1.2)has no positive classical solution if 1 < p ≤ QQ − , with Q = ǫ + ǫ + ... + ǫ n , ( n i = i = , ..., n ).The ∆ λ -operator contains many degenerate elliptic operators. We now give some examplesof ∆ λ -Laplace operators (see also [12]). We use the following notation : we split R n as follows R n = R n × ... × R n k and write x = ( x (1) , ..., x ( k ) ) , x ( i ) = ( x ( i )1 , ..., x ( i ) n i ) ∈ R n i , | x ( i ) | = n i X j = | x ( i ) j | , i = , , ..., k . We denote the classical Laplace operator in ∈ R n i by ∆ x ( i ) = n i X j = ∂ x ( i ) j . Example 1.
Let α be a real positive constant and k =
2. We consider the Grushin-type operator ∆ λ = ∆ x + | x | α ∆ y , where λ = ( λ , λ ) with λ ( x ) = , λ ( x ) = | x (1) | α , x ∈ R n × R n . Our group of dilations is δ t ( x ) = δ t ( x (1) , x (2) ) = ( tx (1) , t α + x (2) ) , and the homogenous dimension with respect to ( δ t ) t > is Q = n + ( α + n . Example 2.
Given a multi-index α = ( α , ..., α k − ), α j ≥ j = , ..., k −
1, define ∆ α : = ∆ x (1) + | x (1) | α ∆ x (2) + ... + | x ( k − | α k − ∆ x ( k ) . Then ∆ α = ∆ λ with λ = ( λ , ..., λ k ) and λ i = | x ( i − | α i − , i = , ..., k . Here we agree to let | x (0) | α = λ satisfies ( H ) is given by δ t : R n → R n , δ t ( x ) = δ t ( x (1) , ..., x ( k ) ) = ( t ǫ x (1) , ..., t ǫ k x ( k ) ) , with ǫ = ǫ i = α i − ǫ i − + i = , ..., k . In particular, if α = ... = α k − =
1, the operator ∆ α and the dilation δ t becomes, respectively ∆ α = ∆ x (1) + | x (1) | ∆ x (2) + ... + | x ( k − | ∆ x ( k ) , and δ t ( x ) = ( tx (1) , t x (2) , ..., t k x ( k ) ) . Example 3.
Let α , β and γ be positive real constants. For the operator ∆ λ = ∆ x (1) + | x (1) | α ∆ x (2) + | x (1) | β | x (2) | γ ∆ x (3) , where λ = ( λ , λ , λ ) with λ ( x ) = , λ ( x ) = | x (1) | α , λ ( x ) = | x (1) | β | x (2) | γ , x ∈ R n × R n × R n , we find the group of dilations δ t ( x ) = δ t ( x (1) , x (2) , x (3) ) = ( tx (1) , t α + x (2) , t β + ( α + γ + x (3) ) . The aim of the present paper was to establish the Liouville-type theorems with finite Morseindex for the equation (1.2). In order to state our results we need the following :
Definition 1.1.
We say that a solution u of (1.2) belonging to C ( R n ) • is stable, if Q u ( ψ ) : = Z R n |∇ λ ψ | − p Z R n | x | a λ | u | p − ψ ≥ , ∀ ψ ∈ C c ( R n ) , where ∇ λ = ( λ ∇ x (1) , ..., λ k ∇ x ( k ) ) . • has Morse index equal to K ≥ if K is the maximal dimension of a subspace X K of C c ( R n ) suchthat Q u ( ψ ) < for any ψ ∈ X K \{ } . • is stable outside a compact set K ⊂ R n if Q u ( ψ ) ≥ for any ψ ∈ C c ( R n \K ) . Remark 1.1. a) Clearly, a solution stable if and only if its Morse index is equal to zero. b) It is well know that any finite Morse index solution u is stable outside a compact set
K ⊂ R n .Indeed, there exists m ≥ and X m : = Span { φ , ..., φ m } ⊂ C c ( R n ) such that Q u ( φ ) < for any φ ∈ X m \{ } . Hence, Q u ( ψ ) ≥ for every ψ ∈ C c ( R n \K ) , where K : = ∪ m j = supp ( φ j ) . In the following, we state Liouville-type results for solutions u ∈ C ( R n ) of (1.2). In whatfollows, we divide our study to stable solutions and solutions which are stable outside a compactset. Stable solutions
To state the following result we need to introduce some notation. We set Γ M ( p ) = p − + p p ( p −
1) and denote by Ω R = B (0 , R ǫ ) × B (0 , R ǫ ) × ... × B k (0 , R ǫ k ), where B i (0 , R ǫ i ) ⊂ R n i , i = , ..., k , the balls of center 0 and radius R ǫ i . Proposition 1.1.
Let u ∈ C ( R n ) be a stable solution of (1.2) . Then, for any γ ∈ (cid:2) , Γ M ( p )) , thereexists a positive constant C independent of R, such that Z Ω R (cid:18) | x | a λ | u | p + γ + |∇ λ ( | u | γ − u ) | (cid:19) dx ≤ CR Q − p + γ ) + ( γ + ap − , for all R > . (1.3)Proposition 1.1 provides an important estimate on the integrability of u and ∇ λ u . As we will see,our nonexistence results will follow by showing that the right-hand side of (1.3) vanishes underthe right assumptions on p when R → + ∞ . More precisely, as a corollary of Proposition 1.1, wecan state our first Liouville type theorem. Theorem 1.1.
Let u ∈ C ( R n ) be a stable solution of (1.2) with,p c ( Q , a ) = + ∞ if Q ≤ + a , ( Q − − a + a + Q ) + √ ( a + ( a + Q − Q − Q − a − if Q > + a . Then u ≡ . Solutions which are stable outside a compact set
In this subsection we prove some integral identities extending to the ∆ λ setting the classicalPohozaev identity for semilinear Poisson equation [18]. Pohozaev identity has been extended byseveral authors to general elliptic equations and systems, both in Riemannian and sub-Riemanniancontext, see, e.g., [2, 9, 20] and the references therein. To prove our identities we closely follow theoriginal procedure of Pohozaev, just replacing the vector field P = P ni = x i ∂ x i in [18], page 1410],by T = k X i = ǫ i x ( i ) ∇ x ( i ) , the generator of the group of dilation ( δ t ) t ≥ in ( H )(we say that T generates ( δ t ) t ≥ since a function u is δ t -homogeneous of degree m if and only if T u = mu ). Proposition 1.2.
Let u ∈ C ( R n ) be a solution of (1.2) and φ ∈ C c ( Ω R ) . If T ( | x | λ ) = | x | λ , then Z Ω R " Q − |∇ λ u | − Q + ap + | x | a λ | u | p + φ = Z Ω R " ∇ λ u ∇ λ φ T ( u ) + " − |∇ λ u | + | x | a λ p + | u | p + T ( φ ) . (1.4)Thanks to Proposition 1.2, we derive Theorem 1.2.
Let u ∈ C ( R n ) be a solution of (1.2) which is stable outside a compact set of R n ,with p s ( Q , a ) = + ∞ if Q ≤ , Q + + aQ − if Q > .If T ( | x | λ ) = | x | λ , then u ≡ . Example which satisfies T ( | x | λ ) = | x | λ The degenerate elliptic operators we consider are of the form ∆ λ = λ ∆ x (1) + ... + λ k ∆ x ( k ) . We denote by | x ( j ) | the euclidean norm of x ( j ) ∈ R n j and assume the functions λ i are of the form λ i ( x ) = k Y j = | x ( j ) | α ij , i = , ..., k , (2.1) such that1) α i j ≥ i = , ..., k , j = , ..., i − α i j = j ≥ i .3) P kl = ǫ l α jl = ǫ j − j = , ..., k with 1 = ǫ ≤ ǫ ≤ ... ≤ ǫ k .Clearly, λ i is δ t -homogeneous of degree ǫ i − { δ t } t > δ t : R n → R n , δ t ( x ) = δ t ( x (1) , ..., x ( k ) ) = ( t ǫ x (1) , ..., t ǫ k x ( k ) ) . Now, using the relation P kl = ǫ l α jl = ǫ j −
1, we get T ( | x | λ ) = | x | λ is satisfied.This paper is organized as follows. In section 3, we give the proof of Proposition 1.1 andTheorem 1.1. Section 4 is devoted to the proof of Proposition 1.2 and Theorem 1.2. The Liouville theorem for stable solutions : proof of Theorem 1.1
In this section we prove all the results concerning the classification of stable solutions, i.e.,Proposition 1.1 and Theorem 1.1. First, to prove Proposition 1.1, we need the following technicalLemma.Let R > Ω R = B (0 , R ǫ ) × B (0 , R ǫ ) × ... × B k (0 , R ǫ k ), where B i (0 , R ǫ i ) ⊂ R n i , i = , ..., k ,and consider k functions ψ , R ,..., ψ k , R such that ψ , R ( r (1) ) = ψ r (1) R ǫ ! , ..., ψ k , R ( r ( k ) ) = ψ k r ( k ) R ǫ k ! , with ψ , R , ... ψ k , R ∈ C ∞ c ([0 , + ∞ )) , ≤ ψ , R , ... ψ k , R ≤ ,ψ i ( t ) = , , , + ∞ ) , and for some constant C > ψ , R , ... ψ k , R satisfy (cid:12)(cid:12)(cid:12) ∇ x (1) ψ , R (cid:12)(cid:12)(cid:12) ≤ CR − ǫ , ..., (cid:12)(cid:12)(cid:12) ∇ x ( k ) ψ k , R (cid:12)(cid:12)(cid:12) ≤ CR − ǫ k , (cid:12)(cid:12)(cid:12) ∆ x (1) ψ , R (cid:12)(cid:12)(cid:12) ≤ CR − ǫ , ..., (cid:12)(cid:12)(cid:12) ∆ x ( k ) ψ k , R (cid:12)(cid:12)(cid:12) ≤ CR − ǫ k , where r ( i ) = | x ( i ) | , i = , ..., k . Lemma 3.1. (1)
There exists a constant C > independent of R such that a) | λ i ( x ) | ≤ CR ǫ i − , ∀ x ∈ Ω R , i = , ..., k . b) |∇ λ ψ R | + | ∆ λ ψ R | ≤ CR − , where ψ R = Q ki = ψ i , R . (2) The homogeneous norm, | . | λ , is δ t -homogeneous of degree one, i.e. | δ t ( x ) | λ = t | x | λ , ∀ x ∈ R n , t > . (3) There exists a constant C > independent of R such that | x | λ ≤ CR , ∀ x ∈ Ω R . Proof.
Proof of (1) a).
For any x = ( x (1) , ..., x ( k ) ) ∈ Ω R , we have x ( i ) ∈ B i (0 , R ǫ i ), i = , ..., k , this implies | x ( i ) | R ǫ i ≤ i = , ..., k . Therefore, if we write x = ( x (1) , ..., x ( k ) ) = R ǫ × x (1) R ǫ , ..., R ǫ k × x ( k ) R ǫ k ! , and let y = ( y (1) , ..., y ( k ) ) = (cid:16) x (1) R ǫ , ..., x ( k ) R ǫ k (cid:17) , then y ∈ Ω . Hence by assumption ( H ) made on functions λ i , we get λ i ( x ) = λ i ( R ǫ y (1) , ..., R ǫ k y ( k ) ) = R ǫ i − λ i ( y (1) , ..., y ( k ) ) = R ǫ i − λ i ( y ) . (3.1)Moreover, since λ i , i = , ..., k are continuous, then | λ i ( y ) | ≤ C , ∀ y ∈ Ω . (3.2)Therefore, from (3.1) and (3.2), we obtain | λ i ( x ) | ≤ CR ǫ i − , ∀ x ∈ Ω R , i = , ..., k . Proof of (1) b).
Using assumption ( H ) made on functions λ i , i = , ..., k , with r = ( r (1) , ..., r ( k ) ) = ( | x (1) | , ..., | x ( k ) | ), we have λ ( r ) = , λ i ( r ) = λ i ( r (1) , ..., r ( i − ) , ∀ i = , ..., k . If we denote by ψ R = Q ki = ψ i , R , we get ∇ λ ψ R = ( λ ( r ) ∇ x (1) ψ R , ..., λ k ( r ) ∇ x ( k ) ψ R ) = λ ( r ) ∇ x (1) ψ , R k Y i = ψ i , R , ..., λ k ( r ) ∇ x ( k ) ψ k , R k − Y i = ψ i , R , and ∆ λ ψ R = λ ( r ) ∆ x (1) ψ R + ... + λ k ( r ) ∆ x ( k ) ψ R = λ ( r ) ∆ x (1) ψ , R k Y i = ψ i , R + ... + λ k ( r ) ∆ x ( k ) ψ k , R k − Y i = ψ i , R . Since | λ i ( r ) | = | λ i ( x ) | ≤ CR ǫ i − , ∀ x ∈ Ω R , i = , ..., k , then there exists a constant C > R such that |∇ λ ψ R | ≤ CR − and | ∆ λ ψ R | ≤ CR − . Proof of (2).
Let x ∈ R n . The homogeneity of the functions λ i implies that | δ t ( x ) | λ : = k X j = Y i , j ( λ i ( δ t ( x ))) ǫ j | t ǫ j x ( j ) | + P ki = ǫ i − = k X j = Y i , j t ǫ j t ǫ i − ( λ i ( x )) ǫ j | x ( j ) | + P ki = ǫ i − = t + P ki = ( ǫ i − k X j = Y i , j ( λ i ( x )) ǫ j | x ( j ) | + P ki = ǫ i − = t | x | λ (3.3) Proof of (3).
For any x = ( x (1) , ..., x ( k ) ) ∈ Ω R , we have x ( i ) ∈ B i (0 , R ǫ i ), i = , ..., k , this implies | x ( i ) | R ǫ i ≤ i = , ..., k . Therefore, if we write x = ( x (1) , ..., x ( k ) ) = R ǫ × x (1) R ǫ , ..., R ǫ k × x ( k ) R ǫ k ! , and let y = ( y (1) , ..., y ( k ) ) = (cid:16) x (1) R ǫ , ..., x ( k ) R ǫ k (cid:17) , then y ∈ Ω (0).Using (3.3), we get | x | λ = | ( R ǫ y (1) , ..., R ǫ k y ( k ) ) | λ = R | ( y (1) , ..., y ( k ) ) | λ = R | y | λ . Since | λ i ( y ) | ≤ C , ∀ y ∈ Ω , i = , ..., k , then there exists a constant C > R suchthat | x | λ ≤ CR , ∀ x ∈ Ω R . This completes the proof of Lemma 3.1. (cid:3)
Proof of Proposition 1.1.
The proof follows the main lines of the demonstration of proposition 4in [7], with more modifications. We split the proof into four steps :
Step 1.
For any φ ∈ C c ( R n ) we have Z R n |∇ λ ( | u | γ − u ) | φ dx = ( γ + γ Z R n | x | a λ | u | p + γ φ dx + γ + γ Z R n | u | γ + ∆ λ ( φ ) dx . (3.4)Multiply equation (1.2) by | u | γ − u φ and integrate by parts to find γ Z R n |∇ λ u | | u | γ − φ dx + Z R n ∇ λ u ∇ λ ( φ ) | u | γ − u dx = Z R n | x | a λ | u | p + γ φ dx , therefore Z R n | x | a λ | u | p + γ φ dx = γ ( γ + Z R n |∇ λ ( | u | γ − u ) | φ dx + γ + Z R n ∇ λ ( | u | γ + ) ∇ λ ( φ ) dx = γ ( γ + Z R n |∇ λ ( | u | γ − u ) | φ dx − γ + Z R n | u | γ + ∆ λ ( φ ) dx . Identity (3.4) then follows by multiplying the latter identity by the factor ( γ + γ . Step 2.
For any φ ∈ C c ( R n ) we have p − ( γ + γ ! Z R n | x | a λ | u | p + γ φ dx ≤ Z R n | u | γ + " |∇ λ φ | + γ + γ − ! ∆ λ ( φ ) dx . (3.5)The function | u | γ − u φ belongs to C c ( R n ), and thus it can be used as a test function in the quadraticform Q u . Hence, the stability assumption on u gives p Z R n | x | a λ | u | p + γ φ dx ≤ Z R n |∇ λ ( | u | γ − u φ ) | dx . (3.6)A direct calculation shows that the right hand side of (3.6) equals to Z R n " | u | γ + |∇ λ φ | + |∇ λ ( | u | γ − u ) | φ + ∇ λ φ ∇ λ ( | u | γ + ) dx = Z R n | u | γ + " |∇ λ φ | − ∆ λ ( φ ) dx + Z R n |∇ λ ( | u | γ − u ) | φ dx . (3.7)From (3.6) and (3.7), we obtain that p Z R n | x | a λ | u | p + γ φ dx ≤ Z R n | u | γ + " |∇ λ φ | − ∆ λ ( φ ) dx + Z R n |∇ λ ( | u | γ − u ) | φ dx . (3.8)Putting this back into (3.4) gives p − ( γ + γ ! Z R n | x | a λ | u | p + γ φ dx ≤ Z R n | u | γ + " |∇ λ φ | + γ + γ − ! ∆ λ ( φ ) dx . Step 3.
For any γ ∈ (cid:2) , Γ M ( p )) and any integer m ≥ max { p + γ p − , } there exists a constant C ( p , m , γ ) > p , m and γ Z R n | x | a λ | u | p + γ ψ mR dx ≤ C ( p , m , γ ) Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − dx , (3.9) Z R n |∇ λ ( | u | γ − u ) | ψ mR dx ≤ C ( p , m , γ ) Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − dx , (3.10)where ψ R = Q ki = ψ i , R . Moreover, the constant C ( p , m , γ ) can be explicitly computed.From (3.5), we obtain that α Z R n | x | a λ | u | p + γ φ dx ≤ Z R n | u | γ + |∇ λ φ | + β Z R n | u | γ + ∆ λ φ dx . (3.11)where we have set α = (cid:16) p − ( γ + γ (cid:17) and β = − γ γ . Notice that α > β <
0, since p > γ ∈ (cid:2) , Γ M ( p )). Now, we set φ = ψ mR . The function φ belongs to C c ( R n ), since m ≥ m is an integer, henceit can be used in (3.11). A direct computation gives α Z R n | x | a λ | u | p + γ ψ mR dx ≤ Z R n | u | γ + ψ m − R (cid:16) m |∇ λ ψ R | + β m ( m − |∇ λ ψ R | + β m ψ R ∆ λ ψ R (cid:17) dx , (3.12)hence Z R n | x | a λ | u | p + γ ψ mR dx ≤ C Z R n | u | γ + ψ m − R (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) dx , (3.13)with C = m + β m ( m − α > − β m α ≥ Z R n | x | a λ | u | p + γ ψ mR ≤ C Z R n | u | γ + ψ m − R (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) dx = C Z R n | x | ( γ + ap + γ λ | u | γ + ψ m − R | x | − ( γ + ap + γ λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) dx ≤ γ + p + γ Z R n | x | a λ | u | p + γ ψ (2 m − p + γγ + R + ( p − C p + γ Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − . (3.14)At this point we notice that m ≥ max { p + γ p − , } implies (2 m − p + γ p − ≥ m and thus ψ (2 m − p + γγ + R ≤ ψ mR in R n , since 0 ≤ ψ R ≤ R n .Therefore, we obtain Z R n | x | a λ | u | p + γ ψ mR ≤ γ + p + γ Z R n | x | a λ | u | p + γ ψ mR + ( p − C p + γ Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − . The latter immediately implies Z R n | x | a λ | u | p + γ ψ mR dx ≤ C Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − dx , (3.15)which proves inequality (3.9) with C ( p , m , γ ) = C .To prove (3.10), we combine (3.4) and (3.5). This leads to Z R n |∇ λ ( | u | γ − u ) | φ dx ≤ A Z R n | u | γ + |∇ λ φ | dx + B Z R n | u | γ + φ ∆ λ φ dx , where A = ( γ + γα + ( γ + γ > B = β ( γ + γα + ( γ + γ ∈ R .Now, we insert the test function φ = ψ mR in the latter inequality to find, Z R n |∇ λ ( | u | γ − u ) | ψ mR dx ≤ Z R n | u | γ + ψ m − R (cid:16) Am |∇ λ ψ R | + Bm ( m − |∇ λ ψ R | + Bm ψ R ∆ λ ψ R (cid:17) dx , and hence Z R n |∇ λ ( | u | γ − u ) | ψ mR dx ≤ C Z R n | u | γ + ψ m − R (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) dx , (3.16)with C = max { Am + Bm ( m − , | B | m } > Z R n |∇ λ ( | u | γ − u ) | ψ mR ≤ C Z R n | x | a λ | u | p + γ ψ (2 m − p + γγ + R ! γ + p + γ Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − ! p − p + γ ≤ C Z R n | x | a λ | u | p + γ ψ mR ! γ + p + γ Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − ! p − p + γ . Finally, inserting (3.15) into the latter we obtain Z R n |∇ λ ( | u | γ − u ) | ψ mR dx ≤ C C + γ p − Z R n | x | − ( γ + ap − λ (cid:16) |∇ λ ψ R | + | ψ R || ∆ λ ψ R | (cid:17) p + γ p − dx , which gives the desired inequality (3.10). Step 4.
For any γ ∈ (cid:2) , Γ M ( p )), there exists a constant C > R such that Z Ω R (cid:18) | x | a λ | u | p + γ + |∇ λ ( | u | γ − u ) | (cid:19) dx ≤ CR Q − p + γ ) + ( γ + ap − , ∀ R > . (3.17)The proof of (3.17) follows immediately by adding inequality (3.9) to inequality (3.10) andusing Lemma 3.1. (cid:3) Proof of Theorem 1.1.
By Proposition 1.1, there exists a positive constant C independent of R such that Z Ω R | x | a λ | u | p + γ ≤ CR Q − p + γ ) + a ( γ + p − . (3.18)Then it su ffi ces to show that we can always choose a γ ∈ (cid:2) , Γ M ( p )), such that Q − p + γ ) + a ( γ + p − < R → + ∞ in (3.18), we deduce that Z R n | x | a λ | u | p + γ = , which implies that u ≡ R n .Next, we claim that, under the assumptions on the exponent p assumed in Theorem 1.1, we canalways choose γ ∈ [1 , Γ M ( p )) such that Q − p + γ ) + a ( γ + p − < . (3.19)As in [7], we consider separately the case Q ≤ + a and the case Q > + a . Case 1. Q ≤ + a and p >
1. In this case we have2( p + Γ M ( p )) + a ( Γ M ( p ) + > p − + p − + a (2 p + p − > (10 + a )( p − and therefore Q − p + Γ M ( p )) + a ( Γ M ( p ) + p − < Q − (10 + a ) ≤ . (3.20)The latter inequality and the continuity of the function x Q − p + x ) + a ( x + p − immediately implythe existence of γ ∈ [1 , Γ M ( p )) satisfying (3.19). Case 2. Q > + a and 1 < p < p c ( Q , a ). In this case we consider the real-valued func-tion x g ( x ) : = x +Γ M ( x )) + a ( Γ ( x ) + x − on (1 , + ∞ ). Since g is strictly decreasing function satisfyinglim x → + g ( x ) = + ∞ and lim x → + ∞ g ( x ) = + a , there exists a unique p > Q = g ( p ).We claim that p = p c ( Q , a ). Indeed, Q = g ( p ) ⇔ ( Q − p − − (4 + a ) p = (4 + a ) p p ( p − ⇔ ( Q − − a )( Q − p + ( − Q − + a + Q + a )) p + ( Q − = , which implies that( Q − − a )( Q − p + ( − Q − + a + Q + a )) p + ( Q − = , (3.21)and ( Q − p − − (4 + a ) p > (4 + a )( p − . (3.22)The roots of (3.21) p = ( Q − − a + a + Q ) + p ( a + ( a + Q − Q − Q − a − = p c ( Q , a ) , (3.23) p = ( Q − − a + a + Q ) − p ( a + ( a + Q − Q − Q − a − < p , (3.24)while (3.22) easily implies p > Q − − aQ − a − > p . This proves that p = p . Hence p c ( Q , a ) = ( Q − − a + a + Q ) + p ( a + ( a + Q − Q − Q − a − g ( p c ( Q , a )) = Q and g is a strictly decreasing function,it follows that ∀ < p < p c ( Q , a ) , Q < g ( p ) . (3.25)Now we can conclude as in the first case, i.e, the continuity of x Q − p + x ) + a ( x + p − immediatelyimplies the existence of γ ∈ [1 , Γ M ( p )) satisfying (3.19). (cid:3) The Liouville theorem for solutions which are stable outside a compact set of R n : proof ofTheorem 1.2 In this section, we prove Proposition 1.2 and Theorem 1.2.
Proof of Proposition 1.2.
Let u ∈ C ( R n ) be a solution of (1.2) and φ ∈ C c ( Ω R ). Multiplyingequation (1.2) by T ( u ) φ and integrating by parts in Ω R , we obtain − Z Ω R ∆ λ uT ( u ) φ dx = − Z Ω R ∆ λ u ǫ j x ( j ) ∇ x ( j ) u φ dx = Z Ω R λ i ∇ x ( i ) u ∇ x ( i ) (cid:16) ǫ j x ( j ) ∇ x ( j ) u φ (cid:17) dx = Z Ω R λ i ∇ x ( i ) u ǫ j δ i j ∇ x ( j ) u φ dx + Z Ω R λ i ∇ x ( i ) u ǫ j x ( j ) ∇ x ( i ) ( ∇ x ( j ) u ) φ dx + Z Ω R λ i ∇ x ( i ) u ǫ j x ( j ) ∇ x ( j ) u ∇ x ( i ) φ dx : = I + I + I , (4.1)Here and in the sequel, we use the Einstein summation convention : an index occurring twice in aproduct is to be summed from 1 up to the space dimension.Obviously I : = Z Ω R λ i ∇ x ( i ) u ǫ j δ i j ∇ x ( j ) u φ dx = Z Ω R λ i |∇ x ( i ) u | ǫ i φ dx . (4.2)Moreover, an integration by parts in I gives I : = Z Ω R λ i ∇ x ( i ) u ǫ j x ( j ) ∇ x ( i ) ( ∇ x ( j ) u ) φ dx = − Z Ω R ∇ x ( j ) ( λ i ) |∇ x ( i ) u | ǫ j x ( j ) φ dx − I − Z Ω R λ i |∇ x ( i ) u | ǫ j n j φ dx − Z Ω R λ i |∇ x ( i ) u | ǫ j x ( j ) ∇ x ( j ) φ dx = − Z Ω R λ i |∇ x ( i ) u | T ( λ i ) φ dx − I − Q Z Ω R |∇ λ u | φ dx − Z Ω R |∇ λ u | T ( φ ) dx . Since λ i is δ t -homogeneous of degree ǫ i −
1, then T ( λ i ) = ( ǫ i − λ i . Hence I = − Z Ω R ( ǫ i − λ i |∇ x ( i ) u | φ dx − I − Q Z Ω R |∇ λ u | φ dx − Z Ω R |∇ λ u | T ( φ ) dx = (2 − Q ) Z Ω R |∇ λ u | φ dx − I − I − Z Ω R |∇ λ u | T ( φ ) dx . Then I = − Q Z Ω R |∇ λ u | φ dx − I − Z Ω R |∇ λ u | T ( φ ) dx . (4.3) It is easily seen that I : = Z Ω R λ i ∇ x ( i ) u ǫ j x ( j ) ∇ x ( j ) u ∇ x ( i ) φ dx = Z Ω R ∇ λ u ∇ λ φ T ( u ) dx . (4.4)Hence, by (4.1), − Z Ω R ∆ λ uT ( u ) φ dx = − Q Z Ω R |∇ λ u | φ dx − Z Ω R |∇ λ u | T ( φ ) dx + Z Ω R ∇ λ u ∇ λ φ T ( u ) dx . (4.5)On the other hand, an integration by parts gives Z Ω R | x | a λ | u | p − uT ( u ) φ dx = p + Z Ω R | x | a λ ∇ x ( j ) ( | u | p + ) ǫ j x ( j ) φ dx = − Qp + Z Ω R | x | a λ | u | p + φ − ap + Z Ω R | x | a − λ | u | p + T ( | x | λ ) φ − p + Z Ω R | x | a λ | u | p + T ( φ ) dx . If T ( | x | λ ) = | x | λ , then Z Ω R | x | a λ | u | p − uT ( u ) φ dx = p + Z Ω R | x | a λ ∇ x ( j ) ( | u | p + ) ǫ j x ( j ) φ dx = − Q + ap + Z Ω R | x | a λ | u | p + φ − p + Z Ω R | x | a λ | u | p + T ( φ ) dx . (4.6)Clearly (1.4) follows directly from (4.5) and (4.6). (cid:3) Proof of Theorem 1.2.
Let u be a solution of (1.2) which is stable outside a compact set. Webegin defining some smooth compactly supported functions which will be used several times inthe sequel. More precisely, for R ∗ >
0, we choose a function ζ i , R ∈ C c ( R n i ), i = , ..., k , 0 ≤ ζ i , R ≤ R n i and ζ i , R ( x ( i ) ) = | x ( i ) | < R ∗ + | x ( i ) | > R ǫ i ,ζ i , R ( x ( i ) ) = R ∗ + < | x ( i ) | < R ǫ i , |∇ x ( i ) ζ i , R | + | ∆ x ( i ) ζ i , R | ≤ CR − ǫ i for { R ǫ i < | x ( i ) | < R ǫ i } . The rest of the proof splits into several steps.
Step 1.
Let p >
1. There exists R ∗ > γ ∈ (cid:2) , Γ M ( p )) and every R ǫ i > R ∗ + Z Σ ( R ) (cid:18) | x | a λ | u | p + γ + |∇ λ ( | u | γ − u ) | (cid:19) dx ≤ C R ∗ + CR Q − p + γ ) + ( γ + ap − , (4.7)where Σ ( R ) = Ω R \ B (0 , R ∗ + × ... × B k (0 , R ∗ + C R ∗ and C are positive constants dependingon p , γ , R ∗ but not on R . Since u is stable outside a compact set of R n , there exists R ∗ > Z Σ ( R ) (cid:18) | x | a λ | u | p + γ + |∇ λ ( | u | γ − u ) | (cid:19) dx ≤ C ( p , m , γ ) Z R n | x | − a ( γ + p − λ (cid:16) |∇ λ ζ R | + | ζ R || ∆ λ ζ R | (cid:17) p + γ p − dx ≤ C R ∗ + CR Q − p + γ ) + ( γ + ap − , where ζ R = Q ni = ζ i , R . Hence, the desired integral estimate (4.7) follows. Step 2. If Q = < p < + ∞ or Q ≥ < p < Q + + aQ − , then u ≡ γ = Step
1, we get | x | ap + λ u ∈ L p + ( R n ) and |∇ λ u | ∈ L ( R n ) for1 < p < p s ( Q , a ).Take φ = ψ R = Q ki = ψ i , R in (1.4) where ψ i , R defined as above. Since | x | ap + λ u ∈ L p + ( R n ) and |∇ λ u | ∈ L ( R n ), then Z Σ R |∇ λ u | dx → , as R → + ∞ and Z Σ R | x | a λ | u | p + dx → , as R → + ∞ , (4.8)where Σ R = Ω R \ Ω R .Recalling that λ i and λ i ∇ x ( i ) u are δ t -homogeneous of degree ǫ i − T generates ( δ t ) t ≥ , we have T ( λ i ) = ( ǫ i − λ i and T ( λ i ∇ x ( i ) u ) = λ i ∇ x ( i ) u . (4.9)Integrating by parts and using (4.9), we derive Z Ω R ∇ λ u ∇ λ ψ R T ( u ) = Z Ω R λ i ∇ x ( i ) u λ i ∇ x ( i ) ψ R ǫ j x ( j ) ∇ x ( j ) u = − Z Ω R T ( λ i ∇ x ( i ) u ) λ i ∇ x ( i ) ψ R u − Z Ω R λ i ∇ x ( i ) uT ( λ i ) ∇ x ( i ) ψ R u − Z Ω R λ i ∇ x ( i ) uT ( ∇ x ( i ) ψ R ) u − Q Z Ω R ∇ λ u ∇ λ ψ R u = − ( Q + Z Ω R ∇ λ u ∇ λ ψ R u − Z Ω R ( ǫ i − λ i ∇ x ( i ) u ∇ x ( i ) ψ R u − Z Ω R λ i ∇ x ( i ) uT ( ∇ x ( i ) ψ R ) u = Q + Z Ω R u ∆ λ ψ R + Z Ω R ǫ i − u λ i ∆ x ( i ) ψ R + Z Ω R u λ i ∇ x ( i ) [ T ( ∇ x ( i ) ψ R )] (4.10)By Lemma 3.1, (4.10) and using H¨older’s inequality, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω R ∇ λ u ∇ λ ψ R T ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CR − Z Σ R u = CR − Z Σ R | x | − ap + λ | x | ap + λ u ≤ CR ( Q − ap − ) p − p + − Z Σ R | x | a λ | u | p + ! p + . (4.11) Similarly, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω R " − |∇ λ u | + | x | a λ p + | u | p + T ( ψ R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z Σ R (cid:16) |∇ λ u | + | x | a λ | u | p + (cid:17) . (4.12)From (4.8), (4.11) and (4.12), we obtainlim R → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω R ∇ λ u ∇ λ ψ R T ( u ) + " − |∇ λ u | + | x | a λ p + | u | p + T ( ψ R ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . As a consequence, (1.4) becomes Q − Z R n |∇ λ u | dx − Q + ap + Z R n | x | a λ | u | p + dx = . (4.13)On the other hand, multiplying equation (1.2) by u ψ R and integrating by parts yields Z R n |∇ λ u | ψ R dx − Z R n | x | a λ | u | p + ψ R dx = Z R n u ∆ λ ψ R dx . Since 1 < p < p s ( Q , a ), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n u ∆ λ ψ R dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n | x | a λ | u | p + dx ! p + Z Σ R | x | − ap − λ | ∆ λ ψ R | p + p − dx ! p − p + ≤ CR Q p − p + − − ap + → R → + ∞ . Then Z R n |∇ λ u | dx = Z R n | x | a λ | u | p + dx . (4.14)To complete the proof we combine (4.13) and (4.14) to get Q − − Q + ap + ! Z R n | x | a λ | u | p + dx = , but Q − − Q + ap + ,
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