The Hénon-Lane-Emden system: a sharp nonexistence result
aa r X i v : . [ m a t h . A P ] A p r The H´enon-Lane-Emden system:a sharp nonexistence result
Andrea Carioli ∗ and Roberta Musina † Abstract
We deal with very weak positive supersolutions to the H´enon-Lane-Emden sys-tem on neighborhoods of the origin. In our main theorem we prove a sharpnonexistence result.
Keywords: weighted Lane-Emden system, critical hyperbola, distributionalsolutions.
The system of elliptic equations − ∆ u = | x | a v p − − ∆ v = | x | b u q − (1.1)has been largely studied since Mitidieri’s paper [19] appeared, in 1990. We focusour attention on the related problem − ∆ u ≥ λ | x | a v p − − ∆ v ≥ λ | x | b u q − ( P a,b ) ∗ SISSA, via Bonomea, 265 – 34136 Trieste, Italy. Email: [email protected]. Partially supportedby INDAM-GNAMPA. † Dipartimento di Matematica e Informatica, Universit`a di Udine, via delle Scienze, 206– 33100 Udine, Italy. Email: [email protected]. Partially supported by Miur-PRIN201274FYK7 004.
1n punctured domains Ω \ { } , where p, q >
1, Ω ⊂ R n is a neighborhood of theorigin and n ≥
3. We are interested in nonnegative, distributional (or very weak )solutions to ( P a,b ), accordingly with the next definition. Definition 1.1
A nontrivial and nonnegative distributional solution to ( P a,b ) on Ω \ { } is a pair u, v of nonnegative functions satisfying u, v ∈ L (Ω \ { } ) , u q − , v p − ∈ L (Ω \ { } ) ,for which there exist λ , λ > such that the inequalities in ( P a,b ) hold in the senseof distributions on Ω \ { } . Problems (1.1) and ( P a,b ) change their nature depending on the sign of the quantity( p − q − −
1. One has to distinguish between the following cases:(AC) 1 p + 1 q < p + 1 q = 1 [Homogeneous case](C) 1 p + 1 q > λ , λ have to be regarded as (possiblynonlinear) eigenvalues and can not be a priori prescribed. If (AC) or (C) applies,then one can always assume that λ = λ = 1.In the present paper we prove a sharp nonexistence result in the spirit of thepaper [6] by Brezis and Cabr´e. More precisely, for fixed p, q > E p,q of parameters a, b ∈ R , for which there exist positive distributional solutions to( P a,b ) in neighborhoods of the origin. The set E p,q is defined as follows. • Anticoercive case.
If (AC) holds, then E p,q := (cid:26) ( a, b ) (cid:12)(cid:12) a, b > − n, ap + bp ′ + 2 > , aq ′ + bq + 2 > (cid:27) . • Homogeneous case.
We put E p,p ′ := (cid:26) ( a, b ) (cid:12)(cid:12) a, b > − n, ap + bp ′ + 2 ≥ (cid:27) . Coercive case.
If (C) holds, then E p,q := ( a, b ) (cid:12)(cid:12) a, b > − n, a + np + b + nq ( p − > n − a + np ( q −
1) + b + nq > n − . We are in position to state our main result.
Theorem 1.2
Let Ω ⊂ R n be a bounded domain containing the origin, and let p, q > . Then ( P a,b ) has a nontrivial and nonnegative distributional solution on Ω \ { } if and only if ( a, b ) ∈ E p,q . Trivially, any distributional solution u ≥ − ∆ u ≥ λ | x | a u p − (1.2)gives rise to the solution u, v = u to the corresponding system − ∆ u ≥ λ | x | a v p − − ∆ v ≥ λ | x | a u p − , (1.3)for λ = λ = λ . The converse might not be true, in general. Recall that by [6,Theorem 0.1], the inequality (1.2) has no nontrivial and nonnegative distributionalsolutions on Ω \ { } if λ > p = 3 and a ≤ − p > Corollary 1.3
Let Ω ⊂ R n be a bounded domain containing the origin, and let p > . Then (1.3) has a nontrivial and nonnegative distributional solution on Ω \{ } if and only if one of the following conditions is satisfied: i ) p > and a > − ii ) p = 2 and a ≥ − iii ) 1 < p < , a > − n and p < p a := 2( n −
1) + an − . p a coincides with Serrin’s critical exponent when a = 0.Theorem 1.2, combined with the action of the Kelvin transform K : L ( R n \ { } ) → L ( R n \ { } ) , ( K w )( x ) = | x | − n w (cid:16) x | x | (cid:17) , (1.4)immediately leads to the following sharp Liouville-type result (power-type solutionsare computed in the appendix). Theorem 1.4
The system of inequalities ( P a,b ) has a positive distributional solution u, v on R n \ { } if and only if the system (1.1) has a positive power-type solution,that is, having the form u ( x ) = c | x | α , v ( x ) = c | x | β . Theorems 1.2 and 1.4 are related to some known results. Serrin and Zou [29]constructed positive radial solutions of class C ( R n ) under the assumptions a = b =0, n/p + n/q ≤ n −
2. One can adapt the shooting method in [29] to find a boundedsolution to (1.1) in a ball Ω about the origin if a, b > − a, b ) ∈ E p,q . Therestriction a, b > − C (Ω) ∩ C (Ω \ { } ), seefor instance [1].Under the anticoercivity assumption (AC), Bidaut-Veron and Giacomini [2] in-vestigated an equivalent Hamiltonian system to prove the existence of a radial solu-tion u, v ∈ C ( R n ) ∩ C ( R n \ { } ) on R n if and only if a, b > − a + np + b + nq ≤ n − . To prove the existence part in Theorem 1.2 we write E p,q = E + p,q ∪ E − p,q , where E + p,q is the set of pairs ( a, b ) such that a, b > − n , a + np + b + nq > n − , (1.5)and E − p,q = E p,q \ E + p,q . If ( a, b ) ∈ E − p,q then the system ( P a,b ) admits power typesolutions, see the explicit computations in the Appendix. For ( a, b ) ∈ E + p,q we take4 large ball B about the origin containing Ω and we study the system − ∆ u = λ | x | a v p − − ∆ v = λ | x | b u q − u, v > Bu = v = 0 on ∂B (1.6)The existence of a solution to ( P a,b ) readily follows from the next theorem, thatmight have an independent interest. Theorem 1.5
If (1.5) holds, then there exist λ , λ > such that (1.6) has at leasta radial solution u, v satisfying u, v ∈ W , ∩ W , ( B ) , Z B | x | b | u | q dx < ∞ , Z B | x | a | v | p dx < ∞ . (1.7)Theorem 1.5 will be proved in Section 2, via variational arguments. A sim-ple computation shows that u, v can never be a power-type solution. Notice thatTheorem 1.5 provides existence also for certain exponents a, b ≯ − u, v to (1.1) on the punctured domain Ω \ { } . Her argument plainlycovers problem ( P a,b ) and can be used to prove Theorem 1.4 under the additionalassumption u, v ∈ C ( R n \ { } ).D’Ambrosio and Mitidieri [13, Theorem 3.5] used representation formulae toprove the nonexistence result in Theorem 1.4 for locally integrable distributionalsolutions on R n . Notice however that [13] include a much larger class of systems.The nonexistence part of Theorem 1.2 will be proved in Section 3.The available literature for (1.1) and related problems is very extensive. The in-terested reader can find exhaustive surveys in [2, 13, 16], besides remarkable results.A number of papers (see for instance [26, 28, 30, 31]) deal with the H´enon-Lane-Emden Conjecture , that originated from the nonexistence results in [19], [21]. Wecite also [3, 4, 5, 10, 11, 14, 15, 17, 20, 23, 24, 25, 27], and the references therein.5
Proof of Theorem 1.5: existence
The homogeneous case (H) is covered by [9, Theorem 1.3]. Thus, we assume that q = p ′ . Since the system (1.6) is not homogeneous, we can fix λ = λ = 1. To getexistence, we follow the outline of the proof in [9]. For brevity, we will skip somedetails.Our approach is based on the formal equivalence, already noticed for instance in[12], between the system (1.6 and the fourth order Navier problem − ∆ (cid:16) | x | − a ( p ′ − ( − ∆ u ) p ′ − (cid:17) = | x | b u q − u, − ∆ u > Bu = ∆ u = 0 on ∂B . (2.1)We use variational methods to show that (2.1) admits a radial weak solution u ina suitably defined energy space. To conclude the proof, one only has to check thatthe pair u, v := | x | − a ( p ′ − ( − ∆ u ) p ′ − solves (1.6).The first step consists in defining W ,p ′ N, rad ( B ; | x | − a ( p ′ − dx )as the completion of of the space of radial functions u ∈ C ( B ), such that u = 0 on ∂ Ω , ∆ u ≡ , with respect to the norm k u k = (cid:18)Z B | x | − a ( p ′ − | ∆ u | p ′ dx (cid:19) /p ′ . We claim that the infimum m := inf u ∈ W ,p ′ N, rad ( B ; | x | − a ( p ′− dx ) u =0 Z B | x | − a ( p ′ − | ∆ u | p ′ dx (cid:18)Z B | x | b | u | q dx (cid:19) p ′ /q is positive and achieved by some function u . For the sake of clarity, we distinguishthe coercive case from the anticoercive one.6 oercive case. If q < p ′ we take any exponent b , such that b > − n and( i ) ap + b p ′ + 2 > , ( ii ) b + np ′ < b + nq . Thanks to ( i ), we have that W ,p ′ N, rad ( B ; | x | − a ( p ′ − dx ) is compactly embedded into L p ′ ( B ; | x | b dx ) by [9, Lemma 2.8]. On the other hand, the space L p ′ ( B ; | x | b dx ) iscontinuously embedded into L q ( B ; | x | b dx ) by ( ii ) and H¨older inequality. The claimfollows by standard arguments. Anticoercive case.
Fix exponents a , b satisfying a + np + b + nq = n − , − n < a ≤ a, − n < b < b, that is possible as ( a, b ) ∈ E p,q and (1.5) holds. By [22, Theorem 4.10], we have thatthere exists a constant c > Z R n | x | − a ( p ′ − | ∆ ϕ | p ′ dx ≥ c (cid:18)Z R n | x | b | ϕ | q dx (cid:19) p ′ /q for any radially symmetric ϕ ∈ C ∞ c ( R n ). Since we are dealing with a boundeddomain, it is easy to infer that W ,p ′ N, rad ( B ; | x | − a ( p ′ − dx ) is compactly embedded into L q ( B ; | x | b dx ), and this proves the claim.Next, let u be an extremal for the infimum m . Use the arguments in [9, Lemma3.2] to show that u ∈ W , ∩ W , ( B ) , v := | x | − a ( p ′ − | ∆ u | p ′ − ( − ∆ u ) ∈ W , ∩ W , ( B ) , and that, up to a Lagrange multiplier, the pair u, v is a weak solution to the system − ∆ u = | x | a | v | p − v , − ∆ v = | x | b | u | q − u in the ball B . To check that u, v are positive on B use the (standard) argument in[9, Lemma 3.4]. The proof of the existence part is complete. (cid:3) Proof of Theorem 1.2: nonexistence
In this proof we denote by c any inessential positive constant.Let u, v be a nontrivial and nonnegative distributional solution to ( P a,b ) in Ω \{ } .We claim that the following facts hold: i ) u, v ∈ L (Ω), | x | b u q − , | x | a v p − ∈ L (Ω); ii ) u, v solve ( P a,b ) in the sense of distributions on Ω; iii ) u, v are superharmonic and positive on Ω; iv ) a, b > − n .The first two conclusions are immediate consequences of [7, Lemma 1]. Since u, v ∈ L (Ω) solve − ∆ u ≥ − ∆ v ≥
0, then u, v are superharmonic by well known andclassical facts. In particular u and v can be assumed to be lower semicontinuousand positive on Ω, and so iii ) holds. Finally, since v is lower semicontinuous andpositive, we can find δ > | x | a v p − ≥ δ | x | a in a closed ball B ⊂ Ω aboutthe origin. Now, from | x | a v p − ∈ L ( B ) we infer that the weight | x | a is locallyintegrable on B , that is, a > − n . The conclusion b > − n can be proved in a similarway.Next, up to dilations we can assume that Ω contains the closure of the unit ball B about the origin. Let G x ( · ) be the Green function for B and let h x ( · ) be itsregular part, that is, G x ( y ) = c n (cid:20) | y − x | n − − h x ( y ) (cid:21) , h x ( y ) = | x | n − | y | x | − x | n − . We claim that u ( x ) ≥ λ Z B G xB ( y ) | y | a v ( y ) p − dy,v ( x ) ≥ λ Z B G xB ( y ) | y | b u ( y ) q − dy (3.1)almost everywhere on B . Let us prove the first of the inequalities in (3.1), the secondone being similar. For any integer k ≥
1, we put f k = min { λ | x | a v p − , k } u k to the problem − ∆ u k = f k , u k ∈ H ( B ) . Green’s representation formula yields u k ( x ) = Z B G xB ( y ) f k dy, and the maximum principle for superharmonic functions implies that u − u k ≥ B . Thus Fatou’s Lemma gives u ( x ) ≥ lim inf k →∞ u k ( x ) ≥ Z B G xB ( y ) f ( y ) dy, for almost every x ∈ B , as claimed.We will use (3.1) to estimate the quantities U R = Z B R | x | b | u | q − dx , V R = Z B R | x | a | v | p − dx for any R > U R , V R are finite). For | x | < / y ∈ B we have the uniform lower bound G x ( y ) ≥ c n (cid:20) | x | + | y | ) n − − M (cid:21) , M = max | x |≤ , | y | < h x ( y ) . Therefore, if R is small enough and R ∈ (0 , R ), then G xB ( y ) ≥ cR − n for x, y ∈ B R .Using (3.1), we infer that u ( x ) ≥ cR − n Z B R | y | a v ( y ) p − dy , v ( x ) ≥ cR − n Z B R | y | b u ( y ) q − dy for almost every x ∈ B R , and hence U R ≥ cR (2 − n )( q − b + n V q − R , V R ≥ cR (2 − n )( p − a + n U p − R . With simple computations we arrive at U ( p − q − − R ≤ c R − ( q − p h a + np + b + np ( q − − ( n − i , (3.2) V ( p − q − − R ≤ c R − ( p − q h a + nq ( p − + b + nq − ( n − i . (3.3)9ow we distinguish three cases, depending whether (C), (H) or (AC) is satisfied. Coercive case (C).
We have that θ := 1 − ( p − q − >
0, and thus (3.3) gives R ( p − qθ h a + nq ( p − + b + nq − ( n − i ≤ c V R . But clearly V R → R →
0. Thus a + nq ( p −
1) + b + nq − ( n − > . A similar argument and (3.2) lead to the conclusion that ( a, b ) ∈ E p,q . Homogeneous case (H).
We have that q = p ′ , and therefore (3.3) gives1 ≤ c R − p h ap + bp ′ +2 i . Hence ap + bp ′ + 2 ≥
0, that is, ( a, b ) ∈ E p,p ′ . Anticoercive case (AC). As u, v are positive and superharmonic, they are uni-formly bounded from below on any ball B R ⊂ Ω about the origin. Hence, for R ∈ (0 , R ] we get V R = Z B R | x | a v p − dx ≥ cR n + a , U R = Z B R | x | b u q − dx ≥ cR n + b , that compared with (3.2), (3.3) give c ≤ R − ( q − p h ap + bp ′ +2 i , c ≤ R − ( p − q h aq ′ + bq +2 i , as ( p − q − − >
0. We immediately infer that aq ′ + bq + 2 ≥ , ap + bp ′ + 2 ≥ . (3.4)It remains to prove that strict inequalities hold in (3.4). We argue by contradic-tion. Assume for instance that aq ′ + bq + 2 = 0 . (3.5)10hen the second inequality in (3.4) and ( p − q − − > a ≤ − ≤ b .In addition, (3.3) becomes V R ≤ c R n + a . (3.6)We first consider the case − n < a < − , b = − q − a ( q − > − . Since v is bounded from below on a small ball B √ R , then − ∆ u ≥ c | x | a on B √ R .Thus, by the maximum principle, u ( x ) ≥ c (cid:0) | x | a +2 − (2 √ R ) a +2 (cid:1) on B √ R .In particular, u ( x ) ≥ c | x | a +2 on B √ R , so that − ∆ v ≥ c | x | − on B √ R ,as b + ( a + 2)( q −
1) = −
2. Again by the maximum principle, v ( x ) ≥ c log( √ R/ | x | )on B √ R , and in particular v ( x ) ≥ c (cid:12)(cid:12) log | x | (cid:12)(cid:12) on B R .We infer that V R ≥ c Z B R | x | a (cid:12)(cid:12) log | x | (cid:12)(cid:12) p − dx = O (cid:16) R n + a | log R | p − (cid:17) , in contradiction with (3.6). To exclude the case a = −
2, notice that in this case b = − − ∆ v ≥ c | x | − on Ω. Conclude as before. (cid:3) Acknowledgements
The authors are pleased to thank Enzo Mitidieri and Lorenzo D’Ambrosio for usefulremarks on how to improve the presentation of this paper.
Appendix: power-type solutions
In [1], Bidaut-Veron computed the positive “power-type” solutions u, v to − ∆ u = λ | x | a v p − − ∆ v = λ | x | b u q − (A.1)11n R n \ { } . Actually we are just interested in finding the set Q p,q of parameters a, b , for which ( P a,b ) admits power-type solutions.Let us start with a few remarks about the Kelvin transform defined in (1.4). Asimple computation shows that K maps distributional solutions u, v ∈ L (Ω \ { } )of ( P a,b ) into distributional solutions K u, K v ∈ L ( ˆΩ) to ( P κ ( a,b ) ), where ˆΩ is thereflection of Ω with respect to the unitary sphere and κ ( a, b ) = ( − a − n + p ( n − , − b − n + q ( n − , κ : R → R . Trivially, K maps power-type solutions into power-type solutions, that is, Q p,q isinvariant under the action of κ . Next we notice that κ is a central inversion withrespect to its fixed point F : κ ( a, b ) = 2 F − ( a, b ) , F = (cid:18) p n − − n, q n − − n (cid:19) . If the pair u ( x ) = | x | α , v ( x ) = | x | β solves (A.1) with respect to some λ , λ > α, β have to satisfy ( q − α − β = − b − − α + ( p − β = − a − . (A.2)In the non-homogeneous cases (AC) and (C), we have that the system (A.2) admitsthe unique solution α = − ap + bp ′ + 2( p − q − − p , β = − aq ′ + bq + 2( p − q − − q . The corresponding pair u α , v β solves ( P a,b ) with λ , λ given, up to positive multi-pliers, by λ = − (cid:18) ap + bp ′ + 2 (cid:19) (cid:18) a + nq ( p −
1) + b + nq − ( n − (cid:19) λ = − (cid:18) aq ′ + bq + 2 (cid:19) (cid:18) a + np + b + np ( q − − ( n − (cid:19) . In conclusion, nontrivial and positive power-type solutions to (A.1) exist if andonly if the couple of exponents ( a, b ) belongs to the open parallelogram Q p,q whose12ertices are X = ( − n, q ( n − − n ) , X ′ = κ ( X ) = ( p ( n − − n, − n ) ,V = ( − , − , V ′ = κ ( V ) . More explicitly, if (AC) holds we have that Q p,q = ( a, b ) ∈ R : min (cid:26) ap + bp ′ + 2 , aq ′ + bq + 2 (cid:27) > (cid:26) a + np + b + np ( q − , a + nq ( p −
1) + b + nq (cid:27) < n − , while in the coercive case (C) we find Q p,q = ( a, b ) ∈ R : max (cid:26) ap + bp ′ + 2 , aq ′ + bq + 2 (cid:27) < (cid:26) a + np + b + np ( q − , a + nq ( p −
1) + b + nq (cid:27) > n − . Points in the boundary of Q p,q correspond to trivial solutions to (A.1) in the senseof Bidaut-Veron [1], that is, at least one of the components is harmonic on R n \ { } .The coordinates of the vertices X, X ′ satisfy a + np + b + nq = n − . ( CL)
The remaining vertices V and V ′ lie on opposite sides of line in the a, b plane givenby ( CL ). More precisely, V is below ( CL ) in the anticoercive case (AC), while V isabove ( CL ) if (C) holds.In the homogenous case (H), the line ( CL ) becomes ap + bp ′ + 2 = 0 ( CL H )and with simple calculations we find that Q p,q collapses into Q p,p ′ = (cid:26) ( a, b ) ∈ R : a, b > − n , ap + bp ′ + 2 = 0 (cid:27) , X, X ′ .In the next pictures we represent the set Q p,q in the ( a, b ) plane. b = − na = − n Q p,q X X ′ (a) Non-homogeneous cases (AC) and (H) a = − n Q p,p ′ X X ′ (b) Homogeneous case (H), q = p ′ Notice that, in any case, Q p,q = E p,q ∩ κ ( E p,q ).In the next pictures we summarize our existence/nonexistence results. We haveexistence of weak solutions on bounded neighborhoods of the origin in the lightgray zone. Power-type solutions correspond to the darker area. The Brezis-Cabr´enonexistence result for the inequality (1.2) is related to the vertex V = ( − , −
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