Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds
aa r X i v : . [ m a t h . A P ] J un NONEXISTENCE RESULTS FOR ELLIPTIC DIFFERENTIAL INEQUALITIESWITH A POTENTIAL ON RIEMANNIAN MANIFOLDS
P. Mastrolia , D. D. Monticelli and F. Punzo , Abstract.
In this paper we are concerned with a class of elliptic differential inequalities with a potentialboth on R m and on Riemannian manifolds. In particular, we investigate the effect of the geometry ofthe underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegativesolutions. Introduction
One of the most important and well-studied class of elliptic differential inequalities in Global Analysis,due to its ubiquitous presence in many applications, is(1.1) ∆ u + V ( x ) u σ ≤ , both on R m and on general Riemannian manifolds ( M, g ), where ∆ denotes the Laplace-Beltrami operatorassociated to the metric and σ >
1. In particular, in many instances it is also required that the solution u of the problem is positive.The aim of this paper is to investigate in depth the influence of the geometry of the underlyingcomplete, noncompact Riemannian manifold ( M, g ) of dimension m and of the potential V on theexistence of positive solutions to the class of elliptic differential inequalities(1.2) 1 a ( x ) div (cid:16) a ( x ) |∇ u | p − ∇ u (cid:17) + V ( x ) u σ ≤ , thus highlighting the interplay between analysis and geometry in this class of problems, which includes(1.1). Here and in the rest of the paper we assume that a : M → R satisfies(1.3) a > , a ∈ Lip loc ( M ) ,V > M and V ∈ L ( M ), and the constants p and σ satisfy p > σ > p −
1. In our results,the geometry of M appears through conditions on the growth of suitably weighted volumes of geodesicballs, involving both the potential V and the function a . We explicitly note that some of the results wefind are new also in the specific case of the model equation (1.1).This class of problems has a very long history, particularly in the Euclidean setting, starting with theseminal works of Gidas [4] and Gidas-Spruck [5]. In those papers the authors show, among other results,that any nonnegative solution of equation (1.1) is in fact identically null if and only if σ ≤ mm − , in case V ≡ m ≥ R m . Note that analogousresults have also been obtained for degenerate elliptic equations and inequalities (see, e.g., [3], [17]),and for the parabolic companion problems (see, e.g., [16], [18], [19], [20]). Using nonlinear capacityarguments, which exploit suitably chosen test functions, Mitidieri-Pohozaev prove nonexistence of weak Date : February 5, 2018.2010
Mathematics Subject Classification. Universit`a degli Studi di Milano, Italy. Email: [email protected]. Partially supported by FSE, Regione Lombardia. Universit`a degli Studi di Milano, Italy. Email: [email protected]. Universit`a degli Studi di Milano, Italy. Email: [email protected] three authors are supported by GNAMPA project “Analisi globale ed operatori degeneri” and are members of theGruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale diAlta Matematica (INdAM). or distributional solutions of wide classes of differential inequalities in R m , which include also many ofthe examples that we consider here. In particular, they show that equation (1.1) on R m does not admitany nontrivial nonnegative solution, provided that(1.4) lim inf R → + ∞ R − σσ − Z B √ R \ B R V − σ − dx < ∞ . For the case of equation (1.2) on R m , the authors show that no nonnegative, nontrivial solution existsin case V ≡ m > p and p − < σ ≤ m ( p − m − p . This can again be read as a condition relating thevolume growth of Euclidean balls, which depends on the dimension m of the space, and the exponent ofthe nonlinearity in the equation.The results in the case of a complete Riemannian manifold have a more recent history, in particular wecite the inspiring papers of Grygor’yan-Kondratiev [8] and Grygor’yan-Sun [9], whose approach originatesfrom the work of Kurta [12]. Using a capacity argument which only exploits the gradient of the distancefunction from a fixed reference point, in particular the authors showed in [8] that equation (1.2), in case p = 2, admits a unique nonnegative weak solution provided that there exist positive constants C , C such that for every R > ε > Z B R aV − β + ε dµ ≤ CR α + C ε (log R ) k , where dµ is the canonical Riemannian measure on M , B R is the geodesic ball centered at a point x ∈ M and α = 2 σσ − , β = 1 σ − , ≤ k < β. Let r ( x ) denote the geodesic distance of a point x ∈ M from a fixed origin o ∈ M . Note that condition(1.5) is satisfied if, for instance, for every R > V ( x ) ≤ C (1 + r ( x )) C and Z B R V − β dµ ≤ CR α (log R ) k for some positive constants C , C . The sharpness of the exponent α in this type of results is evident fromthe Euclidean case (1.1) with V ≡ m ≥
3, where α = m and the corresponding critical growth is σ = mm − . The sharpness of the exponent β is definitely a more delicate question and has recently beensettled on a general Riemannian manifold, in case a ≡ V ≡
1, in [9]. In that paper, using carefullychosen families of test functions, the authors showed that equation (1.1) with V ≡ k = β .In this work we intend to further focus our attention on these classes of differential inequalities, withthe objective of adding some new results to the already very interesting overall picture. Our resultsconcerning equation (1.1), in their simplest form, are contained in the two following theorems. Theorem 1.1.
Let ( M, g ) be a complete Riemannian manifold, σ > , V ∈ L loc ( M ) with V > a.e. on M . Define (1.6) α = 2 σσ − , β = 1 σ − and assume that there exist C , C > such that for every R > sufficiently large one has (1.7) Z B R \ B R/ V − β dµ ≤ CR α (log R ) β and (1.8) 1 C (1 + r ( x )) − C ≤ V ( x ) ≤ C (1 + r ( x )) C a.e. on M. Then the only nonnegative weak solution of (1.1) is u ≡ . Theorem 1.2.
With the same notation of Theorem 1.1, assume that there exist C ≥ , k ≥ , θ > , τ > max { σ − σ ( k + 1) , } such that for every sufficiently large R > Z B R \ B R/ V − β dµ ≤ CR α (log R ) k and (1.10) V ( x ) ≤ C (1 + r ( x )) C e − θ (log r ( x )) τ a.e. on M. Then the only nonnegative weak solution of (1.1) is u ≡ . A few remarks are now in order. For equation (1.1) on R m , condition (1.8) is not required in theresults of Mitidieri-Pohozaev. On the other hand, the potential V ( x ) = (cid:0) log(2 + | x | ) (cid:1) − and the choiceof the exponent σ = mm − with m ≥ M, g ), where only the case of a constant potential is considered, to thecase of a nonconstant V .On the other hand, while (as we have already noted) the exponent α = σσ − in the power of R inconditions (1.7) and (1.9) is indeed sharp, Theorem 1.2 also shows that the sharpness of the exponent ofthe term log R for this type of results is a notion which is also related to the behavior of the potential V at infinity. In particular, if V decays at infinity faster than any power of r ( x ), as in condition (1.10), thenthe critical threshold for the power of the logarithmic term in the volume growth condition (1.9) for thenonexistence of nonnegative, nontrivial solutions of (1.1) correspondingly increases to στσ − − > β = σ − .We explicitly note that the type of phenomenon described in Theorem 1.2 has not been pointed out beforein literature, to the best of our knowledge.The aforementioned theorems are a consequence of more general results concerning nonnegative weaksolutions of inequality (1.2), showing that similar phenomena also occur for this larger class of problems,which includes the case of inequalities involving the p -Laplace operator. We start with a definitiondescribing the weighted volume growth conditions on geodesic balls of ( M, g ), that will be used inobtaining the nonexistence results for nonnegative solutions of (1.2). We recall that with dµ we denotethe canonical Riemannian measure on M , while we define(1.11) dµ = a dµ the weighted measure on M with density a . Definition 1.3.
Let p > , σ > p − , V > a.e. on M and V ∈ L loc ( M ) . Define (1.12) α = pσσ − p + 1 , β = p − σ − p + 1 . We introduce the following three weighted volume growth conditions: i) We say that condition (HP1) holds if there exist C > , k ∈ [0 , β ) such that, for every R > sufficiently large and every ε > sufficiently small, (1.13) Z B R \ B R/ V − β + ε dµ ≤ CR α + C ε (log R ) k . ii) We say that condition (HP2) holds if there exists C > such that, for every R > sufficientlylarge and every ε > sufficiently small, (1.14) Z B R \ B R/ V − β + ε dµ ≤ CR α + C ε (log R ) β and Z B R \ B R/ V − β − ε dµ ≤ CR α + C ε (log R ) β . iii) We say that condition (HP3) holds if there exist C ≥ , k ≥ , θ > , τ > max { σ − p +1 σ ( k + 1) , } such that, for every sufficiently large R > and every ε > sufficiently small, (1.15) Z B R \ B R/ V − β + ε dµ ≤ CR α + C ε (log R ) k e − εθ (log R ) τ . We explicitly note that, when p = 2, the definitions of α , β given in (1.6) and (1.12) agree. Remark . The following are sufficient conditions that imply the above weighted volume growthconditions for geodesic balls in M .i) Suppose that there exist C > k > V ( x ) ≤ C (1 + r ( x )) C and(1.17) Z B R \ B R/ V − β dµ ≤ CR α (log R ) k for every R > C > C (1 + r ( x )) − C ≤ V ( x ) ≤ C (1 + r ( x )) C and(1.19) Z B R \ B R/ V − β dµ ≤ CR α (log R ) β for every R > C ≥ k ≥ θ > τ > max { σ − p +1 σ ( k + 1) , } such that(1.20) V ( x ) ≤ C ( r ( x )) C e − θ (log r ( x )) τ and(1.21) Z B R \ B R/ V − β dµ ≤ CR α (log R ) k for every R >
Theorem 1.5.
Let p > , σ > p − , V ∈ L loc ( M ) with V > a.e. on M and a ∈ Lip loc ( M ) with a > on M . If u ∈ W ,p loc ( M ) ∩ L σ loc ( M, V dµ ) is a nonnegative weak solution of (1.2) , then u ≡ on M provided that one of the conditions (HP1), (HP2) or (HP3) holds (see Definition 1.3). In the particular case p = 2, from Theorem 1.5 we can also derive nonexistence criteria for nonnegativeweak solutions of the semilinear inequality(1.22) 1 a ( x ) div ( a ( x ) ∇ u ) + b ( x ) u + V ( x ) u σ ≤ M. We refer the reader to Section 4 for a precise description of the results concerning inequality (1.22).The rest of the paper is organized as follows. In Section 2 we state and prove some preliminarytechnical results, that we put to use in Section 3, where we give the proof of Theorem 1.5. In Section4 we describe in more detail nonexistence results for nontrivial nonnegative weak solutions of (1.22).Finally in Section 5 we collect some counterexamples to Theorem 1.5 for the case p = 2, showing thatthe weighted volume growth conditions that we assume on geodesic balls are in many cases sharp. Acknowledgements.
The authors wish to thank Prof. Marco Rigoli for interesting discussions, andin particular, for suggesting Section 4. Preliminary results
For any relatively compact domain Ω ⊂ M and p > W ,p (Ω) is the completion of the space ofLipschitz functions w : Ω → R with respect to the norm k w k W ,p (Ω) = (cid:18)Z Ω |∇ w | p dµ + Z Ω | w | p dµ (cid:19) p . For any function u : M → R we say that u ∈ W ,p loc ( M ) if for every relatively compact domain Ω ⊂⊂ M one has u | Ω ∈ W ,p (Ω). Definition 2.1.
Let p > , σ > p − , V > a.e. on M and V ∈ L loc ( M ) . We say that u is a weaksolution of equation (1.2) if u ∈ W ,p loc ( M ) ∩ L σ loc ( M, V dµ ) and for every ϕ ∈ W ,p ( M ) ∩ L ∞ ( M ) , with ϕ ≥ a.e. on M and compact support, one has (2.1) − Z M a ( x ) |∇ u | p − (cid:28) ∇ u, ∇ (cid:18) ϕa ( x ) (cid:19)(cid:29) dµ + Z M V ( x ) u σ ϕ dµ ≤ on M. Remark . We note that, by (1.3), u ∈ W ,p loc ( M ) ∩ L σ loc ( M, V dµ ) is a weak solution of (1.2) if andonly if it is a weak solution ofdiv (cid:16) a ( x ) |∇ u | p − ∇ u (cid:17) + a ( x ) V ( x ) u σ ≤ M, i.e. if and only if for every ψ ∈ W ,p ( M ) ∩ L ∞ ( M ), with ψ ≥ M and compact support, one has(2.2) − Z M |∇ u | p − h∇ u, ∇ ψ i dµ + Z M V ( x ) u σ ψ dµ ≤ M, where dµ is the measure on M with density a , as defined in (1.11).Indeed, given any nonnegative ψ ∈ W ,p ( M ) ∩ L ∞ ( M ) with compact support, one can choose ϕ = aψ as a test function in (2.1) in order to obtain (2.2). Similarly, , given any nonnegative ϕ ∈ W ,p ( M ) ∩ L ∞ ( M ) with compact support, one can insert ψ = ϕa in (2.2) and find (2.1).The following two lemmas will be crucial ingredients in the proof the Theorem 1.5. Lemma 2.3.
Let s ≥ pσσ − p +1 be fixed. Then there exists a constant C > such that for every t ∈ (0 , min { , p − } ) , every nonnegative weak solution u of equation (1.2) and every function ϕ ∈ Lip( M ) with compact support and ≤ ϕ ≤ one has (2.3) tp Z M ϕ s u − t − |∇ u | p χ Ω dµ + 1 p Z M V u σ − t ϕ s dµ ≤ Ct − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ, where Ω = { x ∈ M : u ( x ) > } , χ Ω is the characteristic function of Ω and dµ is the measure on M withdensity a , as defined in (1.11) .Proof. Let η > u η = u + η , then u η ∈ W ,p loc ( M ) ∩ L σ loc ( M, V dµ ). Define ψ = ϕ s u − tη , then ψ is anadmissible test function for equation (2.2), with ∇ ψ = sϕ s − u − tη ∇ ϕ − tϕ s u − t − η ∇ u a.e. on M. Indeed, supp ψ = supp ϕ , 0 ≤ ψ ≤ η − t so that ψ ∈ L ∞ ( M ) and ψ ∈ W ,p ( M ) with Z M |∇ ψ | p dµ ≤ p − (cid:20) s p Z M ϕ ( s − p u − ptη |∇ ϕ | p dµ + t p Z M ϕ sp u − ( t +1) pη |∇ u | p dµ (cid:21) ≤ p − (cid:20) s p η − pt Z M |∇ ϕ | p dµ + t p η − ( t +1) p Z supp ϕ |∇ u | p dµ (cid:21) < + ∞ . Equation (2.2) then gives(2.4) t Z M ϕ s u − t − η |∇ u | p dµ + Z M V u σ u − tη ϕ s dµ ≤ s Z M ϕ s − u − tη |∇ u | p − h∇ u, ∇ ϕ i dµ. Now we estimate the right-hand side of (2.4) using Young’s inequality, obtaining s Z M ϕ s − u − tη |∇ u | p − h∇ u, ∇ ϕ i dµ ≤ s Z M ϕ s − u − tη |∇ u | p − |∇ ϕ | dµ ≤ Z M (cid:18) t p − p ϕ s p − p u − ( t +1) p − p η |∇ u | p − (cid:19)(cid:18) st − p − p ϕ sp − u − t +1 p η |∇ ϕ | (cid:19) dµ ≤ p − p Z M tϕ s u − t − η |∇ u | p dµ + 1 p Z M s p t − ( p − ϕ s − p u p − ( t +1) η |∇ ϕ | p dµ. From (2.4) we have(2.5) tp Z M ϕ s u − t − η |∇ u | p dµ + Z M V u σ u − tη ϕ s dµ ≤ p Z M s p t − ( p − ϕ s − p u p − ( t +1) η |∇ ϕ | p dµ. We exploit again Young’s inequality on the right-hand side of (2.5), with q = σ − tp − t − , q ′ = qq − σ − tσ − p + 1 , δ = p − p obtaining1 p Z M s p t − ( p − ϕ s − p u p − ( t +1) η |∇ ϕ | p dµ = Z M (cid:16) δ q u p − ( t +1) η V q ϕ sq (cid:17)(cid:18) δ − q s p pt p − ϕ sq ′ − p V − q |∇ ϕ | p (cid:19) dµ ≤ δq Z M u σ − tη V ϕ s dµ + 1 q ′ p q ′ δ − q ′ q (cid:18) s p t p − (cid:19) q ′ Z M ϕ s − pq ′ V − q ′ q |∇ ϕ | pq ′ dµ ≤ δ Z M u σ − tη V ϕ s dµ + δ − p − σ − p +1 (cid:18) s p t p − (cid:19) σσ − p +1 Z M V − q ′ q |∇ ϕ | pq ′ dµ = p − p Z M u σ − tη V ϕ s dµ + Ct − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ. Substituting in (2.5) we have I = tp Z M ϕ s u − t − η |∇ u | p dµ + Z M V u σ u − tη ϕ s dµ − p − p Z M V u σ − tη ϕ s dµ (2.6) ≤ Ct − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ. Since ∇ u = 0 a.e. on the set M \ Ω, see [6, Lemma 7.7], we have I = Z M (cid:20) tp ϕ s u − t − η |∇ u | p + V u σ u − tη ϕ s − p − p V u σ − tη ϕ s (cid:21) χ Ω dµ − Z M \ Ω p − p V η σ − t ϕ s dµ. Now note that h tp ϕ s u − t − η |∇ u | p + V u σ u − tη ϕ s − p − p V u σ − tη ϕ s i χ Ω converges a.e. in M to the function (cid:20) tp ϕ s u − t − |∇ u | p + 1 p V u σ − t ϕ s (cid:21) χ Ω as η → + . By an application of Fatou’s lemma and using (2.6) we obtain Z M tp ϕ s u − t − |∇ u | p χ Ω dµ + Z M p V u σ − t ϕ s dµ = Z M (cid:20) tp ϕ s u − t − |∇ u | p + 1 p V u σ − t ϕ s (cid:21) χ Ω dµ ≤ lim inf η → + Z M (cid:20) tp ϕ s u − t − η |∇ u | p + V u σ u − tη ϕ s − p − p V u σ − tη ϕ s (cid:21) χ Ω dµ ≤ Ct − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ + lim inf η → + Z M \ Ω p − p V η σ − t ϕ s dµ = Ct − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ, that is inequality (2.3). (cid:3) Lemma 2.4.
Let s ≥ pσσ − p +1 be fixed. Then there exists a constant C > such that for every nonnegativeweak solution u of equation (1.2) , every function ϕ ∈ Lip( M ) with compact support and ≤ ϕ ≤ andevery t ∈ (0 , min { , p − , σ − p +12( p − } ) one has Z M ϕ s u σ V dµ ≤ Ct − p − p − ( p − σp ( σ − p +1) Z M \ K V − ( t +1)( p − σ − ( t +1)( p − |∇ ϕ | pσσ − ( t +1)( p − dµ ! σ − ( t +1)( p − pσ (2.7) (cid:18)Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ (cid:19) p − p Z M \ K ϕ s u σ V dµ ! ( t +1)( p − pσ , with K = { x ∈ M : ϕ ( x ) = 1 } and dµ is the measure on M with density a , as defined in (1.11) .Proof of Lemma 2.4. Under our assumptions ψ = ϕ s is a feasible test function in equation (2.2). Thuswe obtain(2.8) Z M ϕ s u σ V dµ ≤ Z M sϕ s − |∇ u | p − h∇ u, ∇ ϕ i dµ ≤ Z M sϕ s − |∇ u | p − |∇ ϕ | dµ. Now let Ω = { x ∈ M : u ( x ) > } and let χ Ω be the characteristic function of Ω. Since ∇ u = 0 a.e. onthe set M \ Ω, through an application of H¨older’s inequality we obtain Z M sϕ s − |∇ u | p − |∇ ϕ | dµ = Z M sϕ s − |∇ u | p − χ Ω |∇ ϕ | dµ (2.9) = s Z M (cid:16) ϕ p − p s |∇ u | p − u − p − p ( t +1) χ Ω (cid:17)(cid:16) ϕ sp − u p − p ( t +1) |∇ ϕ | (cid:17) dµ ≤ s (cid:18)Z M ϕ s |∇ u | p u − t − χ Ω dµ (cid:19) p − p (cid:18)Z M ϕ s − p u ( p − t +1) |∇ ϕ | p dµ (cid:19) p . Moreover from equation (2.3) we deduce(2.10) Z M ϕ s |∇ u | p u − t − χ Ω dµ ≤ Ct − − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ, with C > s . Thus from (2.8), (2.9) and (2.10) we obtain(2.11) Z M ϕ s u σ V dµ ≤ C (cid:18) t − − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ (cid:19) p − p (cid:18)Z M ϕ s − p u ( p − t +1) |∇ ϕ | p dµ (cid:19) p . Now we use again H¨older’s inequality with exponents q = σ ( t + 1)( p − , q ′ = qq − σσ − ( t + 1)( p − Z M ϕ s − p u ( p − t +1) |∇ ϕ | p dµ = Z M \ K (cid:16) ϕ sq u ( p − t +1) V q (cid:17)(cid:16) ϕ sq ′ − p V − q |∇ ϕ | p (cid:17) dµ ≤ Z M \ K ϕ s u σ V dµ ! ( t +1)( p − σ Z M \ K ϕ s − pσσ − ( t +1)( p − V − ( t +1)( p − σ − ( t +1)( p − |∇ ϕ | pσσ − ( t +1)( p − dµ ! σ − ( t +1)( p − σ . Substituting into (2.11) we get Z M ϕ s u σ V dµ ≤ Ct − p − p − ( p − σp ( σ − p +1) (cid:18)Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ (cid:19) p − p Z M \ K ϕ s u σ V dµ ! ( t +1)( p − pσ Z M \ K ϕ s − pσσ − ( t +1)( p − V − ( t +1)( p − σ − ( t +1)( p − |∇ ϕ | pσσ − ( t +1)( p − dµ ! σ − ( t +1)( p − pσ . Now inequality (2.7) immediately follows from the previous relation, by our assumptions on s, t and since0 ≤ ϕ ≤ (cid:3) From Lemma 2.4 we immediately deduce
Corollary 2.5.
Under the same assumptions of Lemma 2.4 there exists a constant
C > , independentof u , ϕ and t , such that (cid:18)Z M ϕ s u σ V dµ (cid:19) − ( t +1)( p − pσ (2.12) ≤ Ct − p − p − ( p − σp ( σ − p +1) (cid:18)Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ (cid:19) p − p (cid:18)Z M V − ( t +1)( p − σ − ( t +1)( p − |∇ ϕ | pσσ − ( t +1)( p − dµ (cid:19) σ − ( t +1)( p − pσ . Proof.
Inequality (2.12) easily follows form (2.7), since s ≥ pσσ − ( t +1)( p − and 0 ≤ ϕ ≤ M . (cid:3) Proof of Theorem 1.5
We divide the proof of Theorem 1.5 in three cases, depending on which of the conditions (HP1), (HP2)or (HP3) is assumed to hold (see Definition 1.3).
Proof of Theorem 1.5 . ( a ) Assume that condition (HP1) holds (see (1.13)). Let r ( x ) be the distance of x ∈ M from a fixed origin o , for any fixed R > t = R and denote by B R themetric ball centered at o with radius R . Fix any C ≥ C + p +2 pσ with C as in condition (1.13), define for x ∈ M (3.1) ϕ ( x ) = r ( x ) < R, (cid:16) r ( x ) R (cid:17) − C t for r ( x ) ≥ R and for n ∈ N (3.2) η n ( x ) = r ( x ) < nR, − r ( x ) nR for nR ≤ r ( x ) ≤ nR, r ( x ) ≥ nR. Let(3.3) ϕ n ( x ) = η n ( x ) ϕ ( x ) for x ∈ M, then ϕ n ∈ Lip( M ) with 0 ≤ ϕ n ≤
1, we have ∇ ϕ n = η n ∇ ϕ + ϕ ∇ η n a.e. in M and for every a ≥ |∇ ϕ n | a ≤ a − ( |∇ ϕ | a + ϕ a |∇ η n | a ) a.e. in M. Now we use ϕ n in formula (2.3) of Lemma 2.3 with any fixed s ≥ pσσ − p +1 and deduce that, for somepositive constant C and for every n ∈ N and every small enough t >
0, we have Z M V u σ − t ϕ sn dµ ≤ Ct − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ n | p ( σ − t ) σ − p +1 dµ (3.4) = Ct − ( p − σσ − p +1 Z M V − β + tσ − p +1 |∇ ϕ p ( σ − t ) σ − p +1 n dµ ≤ Ct − ( p − σσ − p +1 p ( σ − t ) σ − p +1 − "Z M V − β + tσ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ + Z B nR \ B nR V − β + tσ − p +1 ϕ p ( σ − t ) σ − p +1 |∇ η n | p ( σ − t ) σ − p +1 dµ ≤ Ct − ( p − σσ − p +1 [ I + I ] , where I := Z M \ B R V − β + tσ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ,I := Z B nR \ B nR ϕ p ( σ − t ) σ − p +1 |∇ η n | p ( σ − t ) σ − p +1 V − β + tσ − p +1 dµ. By (3.1), (3.2) and assumption (HP1) with ε = tσ − p +1 , see equation (1.13), for every n ∈ N and everysmall enough t > I ≤ sup B nR \ B nR ϕ ! p ( σ − t ) σ − p +1 (cid:18) nR (cid:19) p ( σ − t ) σ − p +1 Z B nR \ B nR V − β + tσ − p +1 dµ (3.5) ≤ C (cid:18) nRR (cid:19) − p ( σ − t ) σ − p +1 C t (cid:18) nR (cid:19) p ( σ − t ) σ − p +1 (2 nR ) α + C tσ − p +1 [log(2 nR )] k ≤ Cn α + C tσ − p +1 − p ( σ − t ) σ − p +1 ( C t +1) R α + C tσ − p +1 − p ( σ − t ) σ − p +1 [log(2 nR )] k . By our choice of C , for every small enough t > α + C tσ − p + 1 − p ( σ − t ) σ − p + 1 ( C t + 1) = t ( C − pσC + pC t + p ) σ − p + 1 ≤ − tσ − p + 1 < . Moreover, since t = R , we have R α + C tσ − p +1 − p ( σ − t ) σ − p +1 = R C pσ − p +1 t = e C pσ − p +1 t log R = e C pσ − p +1 In view of (3.5) and (3.6) for
R > t = R small enough, we obtain(3.7) I ≤ Cn − tσ − p +1 [log(2 nR )] k . In order to estimate I we recall that if f : [0 , ∞ ) → [0 , ∞ ) is a nonnegative decreasing function and(1.13) holds, then for any small enough ε > R > Z M \ B R f ( r ( x ))( V ( x )) − β + ε dµ ≤ C Z + ∞ R f ( r ) r α + C ε − (log r ) k dr for some positive constant C , see [9, formula (2.19)]. Moreover, there holds(3.9) |∇ ϕ | ≤ C tR C t r − C t − . Thus, using (3.19)-(3.9), I ≤ Z M \ B R V − β + tσ − p +1 ( R C t C tr − C t − ) p ( σ − t ) σ − p +1 dµ ≤ C Z ∞ R R p ( σ − t ) σ − p +1 C t (1 + C ) pσσ − p +1 t p ( σ − t ) σ − p +1 r − p ( σ − t ) σ − p +1 ( C t +1)+ α + C tσ − p +1 − (log r ) k dr. Now note that R p ( σ − t ) σ − p +1 C t = e p ( σ − t ) σ − p +1 C < e pσC σ − p +1 and that by our choice of C we have a := − p ( σ − t ) σ − p + 1 ( C t + 1) + α + C tσ − p + 1 = tσ − p + 1 ( pC t − pσC + p + C ) ≤ − tσ − p + 1 < . Then, by the above inequalities and performing the change of variables ξ := | a | log r , we get I ≤ Ct p ( σ − t ) σ − p +1 Z ∞ r − p ( σ − t ) σ − p +1 ( C t +1)+ α + C tσ − p +1 (log r ) k drr (3.10) ≤ C | a | − ( k +1) t p ( σ − t ) σ − p +1 Z ∞ e − ξ ξ k dξ ≤ C (cid:18) tσ − p + 1 (cid:19) − k − t p ( σ − t ) σ − p +1 Z ∞ e − ξ ξ k dξ ≤ Ct p ( σ − t ) σ − p +1 − k − . By (3.4), (3.7) and (3.10) Z B R V u σ − t dµ ≤ Z M V u σ − t ϕ sn dµ (3.11) ≤ Ct − ( p − σσ − p +1 [ n − tσ − p +1 (log(2 nR )) k + t p ( σ − t ) σ − p +1 − k − ] . Since
R > t = R < n → ∞ in (3.11)we obtain(3.12) Z B R V u σ − t dµ ≤ Ct p ( σ − t ) σ − p +1 − k − − ( p − σσ − p +1 . Observe that, for each small enough t > p ( σ − t ) σ − p + 1 − k − − ( p − σσ − p + 1 = p − σ − p + 1 − k − ptσ − p + 1 = β − k − ptσ − p + 1 ≥ δ ∗ > . Then, for any fixed sufficiently small t >
0, we have Z M V u σ − t χ B e /t dµ = Z B R V u σ − t dµ ≤ Ct δ ∗ . By Fatou’s Lemma, taking the lim inf as t → + in the previous inequality we obtain Z M V u σ dµ ≤ , which implies u ≡ M .( b ) Assume that condition (HP2) holds (see (1.14)). Let the functions ϕ , η n and ϕ n be defined on M asin formulas (3.1), (3.2) and (3.3), with R > t = R , C ≥ max n C + p +2 pσ , C σ − p +1 o and C as in condition (1.14). We now apply formula (2.12), using the family of functions ϕ n ∈ Lip ( M ) andany fixed s ≥ pσσ − p +1 , and thus we have (cid:18)Z M ϕ sn u σ V dµ (cid:19) − ( t +1)( p − pσ ≤ Ct − p − p − ( p − σp ( σ − p +1) (cid:18)Z M V − p − t − σ − p +1 |∇ ϕ n | p ( σ − t ) σ − p +1 dµ (cid:19) p − p (cid:18)Z M V − ( t +1)( p − σ − ( t +1)( p − |∇ ϕ n | pσσ − ( t +1)( p − dµ (cid:19) σ − ( t +1)( p − pσ . We now need need to estimate(3.13) Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ and Z M V − ( t +1)( p − σ − ( t +1)( p − |∇ ϕ | pσσ − ( t +1)( p − dµ. Arguing as in the previous proof of the theorem under the validity of condition (HP1), with the onlydifference that the condition k < β there is replaced here by k = β , using (1.14) we can deduce that(3.14) Z M V − p − t − σ − p +1 |∇ ϕ n | p ( σ − t ) σ − p +1 dµ ≤ C h n − tσ − p +1 (log (2 nR )) β + t p ( σ − t ) σ − p +1 − β − i . In order to estimate the second integral in (3.13) we start by defining Λ = ( p − σt ( σ − p +1)[ σ − ( t +1)( p − , and wenote that(3.15) ( p − σ ( σ − p + 1) t < Λ < p − σ ( σ − p + 1) t < ε ∗ for every small enough t >
0, and that( t + 1)( p − σ − ( t + 1)( p −
1) = β + Λ and pσσ − ( t + 1)( p −
1) = α + Λ p, with α, β as in Definition 1.3. By our definition of the functions ϕ n , for every n ∈ N and every smallenough t > Z M V − β − Λ |∇ ϕ n | α +Λ p dµ ≤ C (cid:20)Z M V − β − Λ η nα +Λ p |∇ ϕ | α +Λ p dµ + Z M V − β − Λ ϕ α +Λ p |∇ η n | α +Λ p dµ (cid:21) (3.16) ≤ C "Z M \ B R V − β − Λ |∇ ϕ | α +Λ p dµ + Z B nR \ B nR V − β − Λ ϕ α +Λ p |∇ η n | α +Λ p dµ := C ( I + I ) . Now we use condition (1.14) with ε = Λ, and we obtain I = Z B nR \ B nR V − β − Λ ϕ α +Λ p |∇ η n | α +Λ p dµ ≤ sup B nR \ B nR ϕ ! α +Λ p (cid:18) nR (cid:19) α +Λ p Z B nR \ B nR V − β − Λ dµ ! ≤ Cn − ( α +Λ p ) C t (cid:18) nR (cid:19) α +Λ p (2 nR ) α + C Λ (log (2 nR )) β ≤ Cn − ( α +Λ p ) C t − p Λ+ C Λ R − p Λ+ C Λ (log (2 nR )) β . By our definition of C , Λ and by relation (3.15) we easily find − C ( α + Λ p ) t − Λ p + Λ C < − pσtC σ − ( t + 1)( p −
1) + σt ( p − C [ σ − ( t + 1)( p − σ − p + 1)(3.17) ≤ − σtC [ σ − ( t + 1)( p − σ − p + 1) < − σtC ( σ − p + 1) < , for any small enough t >
0. Moreover by (3.15), since t = R , we have R − p Λ+ C Λ ≤ R C Λ ≤ R p − σC t ( σ − p − = e p − σC σ − p − . Thus, for any sufficiently large
R > I ≤ Cn − σtC σ − p +1)2 (log (2 nR )) β . In order to estimate I we note that if f : [0 , ∞ ) → [0 , ∞ ) is a nonnegative decreasing function and(1.14) holds, then for any small enough ε > R > Z M \ B R f ( r ( x ))( V ( x )) − β − ε dµ ≤ C Z + ∞ R f ( r ) r α + C ε − (log r ) β dr for some positive constant C , see (3.19) and [9, formula (2.19)]. Thus, noting that |∇ ϕ | ≤ C tR C t r − C t − a.e. on M and using (3.15), for every small enough t > I ≤ Z M \ B R V − β − Λ (cid:0) C tR C t r − C t − (cid:1) α +Λ p dµ ≤ C Z ∞ R R ( α +Λ p ) C t (1 + C ) α +Λ p t α +Λ p r − ( α +Λ p )( C t +1)+ α + C Λ − (log r ) β dr Now, since t = R , by relation (3.15)we have R ( α +Λ p ) C t = e ( α +Λ p ) C ≤ e ( α + ε ∗ p ) C ;moreover, as we noted already in (3.17), for t > b = − ( α + Λ p )( C t + 1) + α + C Λ < − C tσ ( σ − p + 1) < . With the change of variables ξ = | b | log r , using the previous relations we find I ≤ Ct α +Λ p Z ∞ r b (log r ) β drr = Ct α +Λ p | b | − β − (cid:18)Z ∞ e − ξ ξ β dξ (cid:19) (3.20) ≤ C ( σ − p + 1) C σ ! β +1 t α +Λ p − β − = Ct α +Λ p − β − . From equations (3.16), (3.18) and (3.20) it follows that Z M V − β − Λ |∇ ϕ n | α +Λ p dµ ≤ C (cid:20) t α +Λ p − β − + n − σtC σ − p +1)2 (log (2 nR )) β (cid:21) . (3.21)From (2.12), using (3.14) and (3.21) then we have (cid:18)Z B R u σ V dµ (cid:19) − ( t +1)( p − pσ ≤ (cid:18)Z M ϕ sn u σ V dµ (cid:19) − ( t +1)( p − pσ ≤ Ct − p − p − ( p − σp ( σ − p +1) (cid:18)Z M V − p − t − σ − p +1 |∇ ϕ n | p ( σ − t ) σ − p +1 dµ (cid:19) p − p × (cid:18)Z M V − β − Λ |∇ ϕ n | α +Λ p dµ (cid:19) α +Λ p ≤ Ct − p − p − ( p − σp ( σ − p +1) h n − tσ − p +1 (log (2 nR )) β + t p ( σ − t ) σ − p +1 − β − i p − p × (cid:20) t α +Λ p − β − + n − σtC σ − p +1)2 (log (2 nR )) β (cid:21) α +Λ p By taking the lim inf as n → + ∞ we get (cid:18)Z B R u σ V dµ (cid:19) − ( t +1)( p − pσ ≤ Ct − p − p − ( p − σp ( σ − p +1) + ( p − σ − t ) σ − p +1 − ( β +1)( p − p +1 − β +1 α +Λ p for every sufficiently small t >
0, with t = R . But − p − p − ( p − σp ( σ − p + 1) + ( p − σ − t ) σ − p + 1 − ( β + 1)( p − p + 1 − β + 1 α + Λ p = − ( p − p ( σ − p + 1) t, hence for every small enough t > Z B e /t u σ V dµ ! − ( t +1)( p − pσ ≤ Ct − ( p − p ( σ − p +1) t ≤ C, that is Z B e /t u σ V dµ ≤ C uniformly in t , for t > t → + we deduce(3.22) Z M u σ V dµ < + ∞ , and thus u ∈ L σ ( M, V dµ ). Now we exploit inequality (2.7) with the cutoff function ϕ n , and using again(3.14) and (3.21) we obtain Z M ϕ sn u σ V dµ ≤ Ct − p − p − ( p − σp ( σ − p +1) h n − tσ − p +1 (log (2 nR )) β + t p ( σ − t ) σ − p +1 − β − i p − p Z M \ B R ϕ sn u σ V dµ ! ( t +1)( p − pσ × (cid:20) t α +Λ p − β − + n − σtC σ − p +1)2 (log (2 nR )) β (cid:21) α +Λ p . Since ϕ n ≡ B R and 0 < ϕ n ≤ M , for all n ∈ N Z B R u σ V dµ ≤ Z M ϕ sn u σ V dµ, Z M \ B R ϕ sn u σ V dµ ≤ Z M \ B R u σ V dµ.
Using previous inequalities and taking the lim inf as n → + ∞ we get Z B R u σ V dµ ≤ Ct − p − p − ( p − σp ( σ − p +1) + ( p − σ − t ) σ − p +1 − ( β +1)( p − p +1 − β +1 α +Λ p Z M \ B R u σ V dµ ! ( t +1)( p − pσ = Ct − ( p − p ( σ − p +1) t Z M \ B R u σ V dµ ! ( t +1)( p − pσ ≤ C Z M \ B R u σ V dµ ! ( t +1)( p − pσ uniformly for t > t = R . Since u ∈ L σ ( M, V dµ ), Z M \ B R u σ V dµ → R → + ∞ . Moreover ( t +1)( p − pσ → p − pσ > R → + ∞ . It follows that Z M u σ V dµ = lim R → + ∞ Z B R u σ V dµ = 0 , which implies u ≡ M .( c ) Assume that condition (HP3) holds (see (1.15)). Consider the functions ϕ , η n and ϕ n defined in (3.1),(3.2) and (3.3), with R > t = R , C ≥ C + p +2 pσ and C as in condition (1.15). Arguingas in the previous proof of the theorem under the assumption of the validity of (HP1), by formula (2.3)with any fixed s ≥ pσσ − p +1 , we see that Z M V u σ − t ϕ sn dµ ≤ Ct − ( p − σσ − p +1 Z M V − p − t − σ − p +1 |∇ ϕ n | p ( σ − t ) σ − p +1 dµ (3.23) ≤ Ct − ( p − σσ − p +1 "Z M V − p − t − σ − p +1 |∇ ϕ | p ( σ − t ) σ − p +1 dµ + Z B nR \ B nR V − p − t − σ − p +1 ϕ p ( σ − t ) σ − p +1 |∇ η n | p ( σ − t ) σ − p +1 dµ := Ct − ( p − σσ − p +1 [ I + I ] , for some positive constant C and for every n ∈ N and every small enough t >
0. Now, recalling thedefinitions of ϕ and η n , by condition (1.15) with ε = tσ − p +1 , for every small enough t > I ≤ sup B nR \ B nR ϕ ! p ( σ − t ) σ − p +1 (cid:18) nR (cid:19) p ( σ − t ) σ − p +1 Z B nR \ B nR V − β + tσ − p +1 dµ ≤ Cn − p ( σ − t ) σ − p +1 C t (cid:18) nR (cid:19) p ( σ − t ) σ − p +1 (2 nR ) α + C tσ − p +1 (log(2 nR )) k e − θtσ − p +1 (log(2 nR )) τ = Cn tσ − p +1 ( C − pσC + pC t + p ) R C pσ − p +1 t (log(2 nR )) k e − θtσ − p +1 (log(2 nR )) τ . Note that, since t = R , we have R C pσ − p +1 t = e C pσ − p +1 t log R = e C pσ − p +1 and that by our choice of C , if t > C − pσC + pC t + p ) < −
1. Thuswe conclude that(3.24) I ≤ Cn − tσ − p +1 (log(2 nR )) k . In order to estimate I we note that if f : [0 , ∞ ) → [0 , ∞ ) is a nonnegative decreasing function and(1.15) holds, then for any small enough ε > R > Z M \ B R f ( r ( x ))( V ( x )) − β + ε dµ ≤ C Z + ∞ R f ( r ) r α + C ε − (log r ) k e − εθ (log r ) τ dr for some positive fixed constant C . Indeed, by the monotonicity of the involved functions, using condition(1.15) we obtain in a similar way as [9, formula (2.19)] Z M \ B R f ( r ( x ))( V ( x )) − β + ε dµ = + ∞ X i =0 Z B i +1 R \ B iR f ( r ( x ))( V ( x )) − β + ε dµ ≤ + ∞ X i =0 f (2 i R ) Z B i +1 R \ B iR V − β + ε dµ ≤ C + ∞ X i =0 f (2 i R ) e − εθ (log(2 i +1 R )) τ (2 i +1 R ) α + C ε (log(2 i +1 R )) k ≤ C + ∞ X i =0 f (2 i R ) e − εθ (log(2 i +1 R )) τ (2 i − R ) α + C ε (log(2 i − R )) k ≤ C + ∞ X i =0 Z i R i − R f ( r ) e − εθ (log r ) τ r α + C ε − (log r ) k dr = C Z + ∞ R f ( r ) e − εθ (log r ) τ r α + C ε − (log r ) k dr. Now, since for a.e. x ∈ M we have |∇ ϕ ( x ) | ≤ C tR C t ( r ( x )) − C t − , using (3.25) with ε = tσ − p +1 , we obtain that for every small enough t > t = R I ≤ Z M \ B R V − β + tσ − p +1 (cid:0) C tR C t ( r ( x )) − C t − (cid:1) p ( σ − t ) σ − p +1 dµ ≤ C Z + ∞ R R p ( σ − t ) C tσ − p +1 C p ( σ − t ) σ − p +1 t p ( σ − t ) σ − p +1 r − p ( σ − t )( C t +1) σ − p +1 + α + C tσ − p +1 − (log r ) k e − tθσ − p +1 (log r ) τ dr. Note that C p ( σ − t ) σ − p +1 ≤ (1 + C ) pσσ − p +1 and that R p ( σ − t ) C tσ − p +1 = R p ( σ − t ) C σ − p +1 1log R = e p ( σ − t ) C σ − p +1 ≤ e pσC σ − p +1 . Thus, with the change of variable r = e ξ , we deduce I ≤ Ct p ( σ − t ) σ − p +1 Z + ∞ r tσ − p +1 ( C − pσC + pC t + p ) (log r ) k e − tθσ − p +1 (log r ) τ r − dr = Ct p ( σ − t ) σ − p +1 Z + ∞ e tσ − p +1 ( C − pσC + pC t + p ) ξ ξ k e − tθσ − p +1 ξ τ dξ. Now recall that by our choice of C , for t > C − pσC + pC t + p ) <
0. Hence,setting ρ = (cid:16) tθσ − p +1 (cid:17) τ ξ , we have I ≤ Ct p ( σ − t ) σ − p +1 Z + ∞ ξ k e − tθσ − p +1 ξ τ dξ = Ct p ( σ − t ) σ − p +1 (cid:18) tθσ − p + 1 (cid:19) − k +1 τ Z + ∞ ρ k e − ρ τ dρ ≤ Ct p ( σ − t ) σ − p +1 − k +1 τ . (3.26)From (3.23), (3.24) and (3.26) we conclude that for every n ∈ N and every small enough t = R > Z B R V u σ − t dµ ≤ Z M V u σ − t ϕ sn dµ ≤ Ct − ( p − σσ − p +1 h t p ( σ − t ) σ − p +1 − k +1 τ + n − tσ − p +1 (log(2 nR )) k i for some fixed positive constant C . Passing to the limit as n → + ∞ in the previous relation yields(3.27) Z B R V u σ − t dµ ≤ Ct − ( p − σσ − p +1 + p ( σ − t ) σ − p +1 − k +1 τ . Now note that by our assumptions on τ, k we have − ( p − σσ − p + 1 + p ( σ − t ) σ − p + 1 − k + 1 τ = σσ − p + 1 − k + 1 τ − ptσ − p + 1 ≥ (cid:18) σσ − p + 1 − k + 1 τ (cid:19) := δ ∗ > t = R >
0. Thus (3.27) yields(3.28) Z B e /t V u σ − t dµ ≤ Ct δ ∗ for every small enough t >
0. Passing to the lim inf as t tends to 0 + in (3.28), we conclude by anapplication of Fatou’s Lemma that Z M V u σ dµ = 0 , so that u ≡ M . (cid:3) A problem with lower order terms
In this subsection we consider the semilinear equation(4.1) 1 a ( x ) div ( a ( x ) ∇ u ) + b ( x ) u + V ( x ) u σ ≤ M. We start with the following lemma.
Lemma 4.1.
Let u ∈ W , loc ( M ) ∩ L σ loc ( M, V dµ ) be a nonnegative weak solution of (4.1) , with a satisfying (1.3) , σ > , V > a.e. on M , V ∈ L loc ( M ) and b ∈ L mm +2 loc ( M ) . Assume there exists a weak solution z > , z ∈ Lip loc ( M ) of (4.2) 1 a ( x ) div ( a ( x ) ∇ z ) + b ( x ) z ≥ on M. Then w := uz ∈ W , loc ( M ) ∩ L σ loc ( M, V dµ ) is a nonnegative weak solution of (4.3) 1 a ( x ) z ( x ) div (cid:0) a ( x ) z ( x ) ∇ w (cid:1) + V ( x ) z σ − ( x ) w σ ≤ on M. Proof.
By our assumptions, for every ϕ ∈ W , ( M ) ∩ L ∞ ( M ) with compact support and ϕ ≥ M we have − Z M h∇ u, ∇ ϕ i dµ + Z M buϕ dµ + Z M V u σ ϕ dµ ≤ , (4.4) − Z M h∇ z, ∇ ϕ i dµ + Z M bzϕ dµ ≥ . (4.5) We explicitly note that, by our assumptions, all the integrals in (4.4) and (4.5) are finite. Moreover, bya density argument, we easily see that inequality (4.5) also holds for every ϕ ∈ W , ( M ) with compactsupport and ϕ ≥ M , not necessarily bounded.Now we fix ψ ∈ W , ( M ) ∩ L ∞ ( M ) with compact support and ψ ≥ M , and use ϕ = zψ ∈ W , ( M ) ∩ L ∞ ( M ) as a test function in (4.4) and ϕ = uψ ∈ W , ( M ) as a test function in (4.5).Subtracting the resulting inequalities one finds(4.6) − Z M h∇ u, ∇ ψ i z dµ + Z M h∇ z, ∇ ψ i u dµ + Z M V u σ ψz dµ ≤ . Since w = uz ∈ W , ( M ) ∩ L σ loc ( M, V dµ ) with ∇ w = 1 z ∇ u − uz ∇ z a.e. on M, inequality (4.6) becomes − Z M h∇ w, ∇ ψ i az dµ + Z M (cid:0) V z σ − w σ ψ (cid:1) az dµ ≤ . Then, see also Remark 2.2, w is a nonnegative weak solution of (4.3). (cid:3) Combining Lemma 4.1 with Theorem 1.5, one can easily obtain the following nonexistence results fornontrivial nonnegative weak solutions of equation (4.1).
Proposition 4.2.
Assume there exists a weak solution z > , z ∈ Lip loc ( M ) of equation (4.2) and let a satisfy (1.3) , σ > , V > a.e. on M , V ∈ L loc ( M ) and b ∈ L mm +2 loc ( M ) . Then any nonnegativeweak solution u ∈ W , loc ( M ) ∩ L σ loc ( M, V dµ ) of (4.1) is identically null, provided one of the followingconditions holds: i) there exist C > , k ∈ [0 , σ − ) such that, for every large enough R > and every ε > sufficiently small, Z B R V − σ − + ε az dµ ≤ CR σσ − + C ε (log R ) k , or ii) there exists C > such that, for every large enough R > and every ε > sufficiently small, Z B R V − σ − + ε az dµ ≤ CR σσ − + C ε (log R ) σ − and Z B R V − σ − − ε az dµ ≤ CR σσ − + C ε (log R ) σ − , or iii) there exist C ≥ , k ≥ , θ > , τ > max { σ − σ ( k + 1) , } such that, for every large enough R > and every ε > sufficiently small, Z B R \ B R V − σ − + ε az dµ ≤ CR σσ − + C ε (log R ) k e − εθ (log R ) τ . We now proceed to describe a general setting where one can indeed produce the desired auxiliaryfunction z , in the particular case when a ≡ M .Let us fix a point o ∈ M and denote by Cut( o ) the cut locus of o . For any x ∈ M \ (cid:2) Cut( o ) ∪ { o } (cid:3) , onecan define the polar coordinates with respect to o , see e.g. [7]. Namely, for any point x ∈ M \ (cid:2) Cut( o ) ∪{ o } (cid:3) there correspond a polar radius r ( x ) := dist ( x, o ) and a polar angle θ ∈ S m − such that the shortestgeodesics from o to x starts at o with the direction θ in the tangent space T o M . Since we can identify T o M with R m , θ can be regarded as a point of S m − . The Riemannian metric in M \ (cid:2) Cut( o ) ∪ { o } (cid:3) in the polar coordinates reads ds = dr + A ij ( r, θ ) dθ i dθ j , where ( θ , . . . , θ m − ) are coordinates in S m − and ( A ij ) is a positive definite matrix. It is not difficultto see that the Laplace-Beltrami operator in polar coordinates has the form(4.7) ∆ = ∂ ∂r + F ( r, θ ) ∂∂r + ∆ S r , where F ( r, θ ) := ∂∂r (cid:0) log p A ( r, θ ) (cid:1) , A ( r, θ ) := det( A ij ( r, θ )), ∆ S r is the Laplace-Beltrami operator onthe submanifold S r := ∂B ( o, r ) \ Cut( o ) . M is a manifold with a pole , if it has a point o ∈ M with Cut( o ) = ∅ . The point o is called pole andthe polar coordinates ( r, θ ) are defined in M \ { o } .A manifold with a pole is a spherically symmetric manifold or a model , if the Riemannian metric isgiven by(4.8) ds = dr + ψ ( r ) dθ , where dθ is the standard metric in S m − , and(4.9) ψ ∈ A := n f ∈ C ∞ ((0 , ∞ )) ∩ C ([0 , ∞ )) : f ′ (0) = 1 , f (0) = 0 , f > , ∞ ) o . In this case, we write M ≡ M ψ ; furthermore, we have p A ( r, θ ) = ψ m − ( r ), so the boundary area of thegeodesic sphere ∂S R is computed by S ( R ) = ω m ψ m − ( R ) ,ω m being the area of the unit sphere in R m . Also, the volume of the ball B R ( o ) is given by µ ( B R ( o )) = Z R S ( ξ ) dξ . Moreover we have ∆ = ∂ ∂r + ( n − ψ ′ ψ ∂∂r + 1 ψ ∆ S m − , or equivalently ∆ = ∂ ∂r + S ′ S ∂∂r + 1 ψ ∆ S m − , where ∆ S m − is the Laplace-Beltrami operator in S m − .Observe that for ψ ( r ) = r , M = R m , while for ψ ( r ) = sinh r , M is the m − dimensional hyperbolicspace H m .Let us recall some useful comparison results for sectional and Ricci curvatures, that will be used inthe sequel. For any x ∈ M \ (cid:2) Cut( o ) ∪ { o } (cid:3) , denote by Ric o ( x ) the Ricci curvature at x in the direction ∂∂r . Let ω denote any pair of tangent vectors from T x M having the form (cid:0) ∂∂r , X (cid:1) , where X is a unitvector orthogonal to ∂∂r . Denote by K ω ( x ) the sectional curvature at the point x ∈ M of the 2-sectiondetermined by ω . If M ≡ M ψ is a model manifold, then for any x = ( r, θ ) ∈ M \ { o } K ω ( x ) = − ψ ′′ ( r ) ψ ( r ) , and Ric o ( x ) = − ( m − ψ ′′ ( r ) ψ ( r ) . Observe moreover that (see [10], [11], [7, Section 15]), if M is a manifold with a pole o and(4.10) K ω ( x ) ≤ − ψ ′′ ( r ) ψ ( r ) for all x = ( r, θ ) ∈ M \ { o } , for some function ψ ∈ A , then(4.11) F ( r, θ ) ≥ ( m − ψ ′ ( r ) ψ ( r ) for all r > , θ ∈ S m − . On the other hand, if M is a manifold with a pole o and(4.12) Ric o ( x ) ≥ − ( m − ψ ′′ ( r ) ψ ( r ) for all x = ( r, θ ) ∈ M \ { o } , for some function ψ ∈ A , then(4.13) F ( r, θ ) ≤ ( m − ψ ′ ( r ) ψ ( r ) for all r > , θ ∈ S m − . We have the following
Lemma 4.3.
Let M be a manifold with a pole o and b ∈ L mm +2 loc ( M ) . Let b : R + → R be such that (4.14) b ( r, θ ) ≥ b ( r ) for all x = ( r, θ ) ∈ M \ { o } . Assume that ψ ∈ A , that ζ : R + → R is a positive weak solution in Lip loc ( R + ) of (4.15) (cid:0) ψ m − ζ ′ (cid:1) ′ + b ψ m − ζ ≥ in R + , and that either (A) ψ satisfies condition (4.10) and ζ is nondecreasing,or (B) ψ satisfies condition (4.12) and ζ is nonincreasing.Then z ( x ) := ζ ( r ( x )) ∈ Lip loc ( M ) is a positive weak solution of (4.2) , with a ≡ on M .Proof. In case condition ( A ) holds, the result is an easy consequence of (4.7), (4.11), the monotonicityof ζ and condition (4.14). Similarly, when condition ( B ) holds, the result follows immediately as before,using (4.13) in place of (4.11). (cid:3) We refer the interested reader to the stimulating paper of Bianchini, Mari, Rigoli [1] for resultsconcerning the existence of a positive solution of (4.15) and its precise asymptotic behavior as r tends to+ ∞ . These combined with Lemma 4.3 and Proposition 4.2 give somehow explicit nonexistence resultsfor equation (4.1). 5. Counterexamples
In this section, we will produce three counterexamples to the previous nonexistence results, all in theparticular case of equation (1.1). Here we follow a similar approach as one finds in [7] and [9]. In thesequel, α = σσ − and β = σ − as in Definition 1.3 with p = 2, while M will always denote a modelmanifold with a pole o and metric given by (4.7). Set B R ≡ B R ( o ) and r ≡ r ( x ) = dist ( x, o ) for any x ∈ M .Let spec ( − ∆) be the spectrum in L ( M ) of the operator − ∆. Note that (see [7, Section 10]) spec ( − ∆) ⊆ [0 , ∞ ) . Denote by ¯ λ ( M ) the bottom of spec ( − ∆), that is¯ λ ( M ) := inf spec ( − ∆) . By [2], for each fixed x ∈ M , there holds(5.1) ¯ λ ( M ) ≤ (cid:20) lim sup R → + ∞ log µ ( B R ( x )) R (cid:21) . For any ρ >
0, let λ ρ be the first Dirichlet eigenvalue of the Laplace operator in B ρ , that is the smallestnumber λ ρ for which the problem(5.2) ∆ u + λu = 0 in B ρ ,u = 0 on ∂B ρ . has a non-zero solution. Indeed, λ ρ coincides with the bottom of the spectrum of the operator − ∆ in L ( B ρ ) with domain C ∞ ( B ρ ). It is easily checked, see e.g. ****ref uberbook***, that λ ρ ≥
0; moreover, λ ρ ≥ λ ρ if ρ < ρ , and λ ρ → ¯ λ ( M ) as ρ → ∞ . In the sequel we shall make use of the following result (see [9]).
Proposition 5.1.
Let σ > , r > , A ∈ C (( r , ∞ )) with A > and R ∞ r drA ( r ) < ∞ . Let B ∈ C (( r , ∞ )) be such that Z ∞ r [ γ ( r )] σ | B ( r ) | dr < ∞ , where γ ( r ) := Z ∞ r dξA ( ξ ) for r ≥ r . Then the equation ( A ( r ) y ′ ) ′ + B ( r ) y σ = 0 for r > R , for R > r sufficiently large, admits a positive solution y ( r ) such that y ( r ) ∼ γ ( r ) as r → ∞ . Example . Let ψ ∈ A , see (4.9), with ψ ( r ) := r if 0 ≤ r < , [ r α − (log r ) β ] m − if r > β > β . Let 0 < δ < β − β and define V ( x ) ≡ V ( r ) := (log(2 + r )) δβ for all x ∈ M .
For any
R > S ( R ) = ω m R α − (log R ) β , µ ( B R ) ≃ CR α (log R ) β ;thus, thanks to (5.1), we have ¯ λ ( M ) = 0 . Moreover, there holds(5.3) Z B R V − β ( x ) dµ ≥ CR α (log R ) β − δ , with β − δ > β . Hence, in view of (5.3), neither condition (1.13) nor condition (1.14) is satisfied.Furthermore, observe that (1.18) holds true, while (1.19) fails. This is essentially due to the choice of ψ .Note that for any r > Z ∞ r dξS ( ξ ) < ∞ ;moreover, for r > γ ( r ) := Z ∞ r dξS ( ξ ) ≃ Cr α − (log r ) β . Hence for r > Z ∞ r [ γ ( r )] σ V ( r ) S ( r ) dr ≤ C Z ∞ r [log(2 + r )] δ/β r α − (log r ) β r σ ( α − (log r ) β σ dr (5.5) ≤ C Z ∞ r (log r ) β (1 − σ )+ δβ drr < ∞ . In view of (5.4)-(5.5), from Proposition 5.1 with A ( r ) = S ( r ) and B ( r ) = S ( r ) V ( r ), we have that thereexists a solution y = y ( r ) of(5.6) y ′′ ( r ) + S ′ ( r ) S ( r ) y ′ ( r ) + V ( r )[ y ( r )] σ = 0 , r > R , for some R > r . Furthermore, y ( r ) > r ∈ [ R , ∞ ) and y ( r ) ∼ γ ( r ) as r → ∞ .Now for any ρ >
0, let v ρ be the solution to the eigenvalue problem (5.2) with λ = λ ρ , which we canassume is normalized by setting v ρ ( o ) = 1. Thus we have that v ρ ≡ v ρ ( r ), that 0 < v ρ ( r ) ≤ v ρ ( r ) is decreasing for r ∈ [0 , ρ ]. For any ρ > R , define m := inf [ R ,ρ ) y ( r ) v ρ ( r ) and for any fixed ξ ∈ ( R , ρ )let e u ( x ) := mv ρ ( r ) in B ξ ,y ( r ) on ∂B cξ . Since ¯ λ ( M ) = 0, as in [9] one can prove that for some ξ ∈ ( R , ρ ) we have e u ∈ C ( M ), and thus e u ∈ W ( M ). Moreover ∆( mv ρ ) + λ ρ m σ − ( mv ρ ) σ = 0 in B ρ , ∆ y + V y σ = 0 in B cR . Define M ρ = max B ρ V and δ = min (cid:26) , m − λ σ − ρ M − σ − ρ (cid:27) ;then, since V > M , δ > B ρ we have∆( δmv ρ ) + V ( δmv ρ ) σ ≤ ∆( δmv ρ ) + M ρ ( δmv ρ ) σ ≤ ∆( δmv ρ ) + λ ρ ( δm ) σ − ( δmv ρ ) σ = 0 . On the other hand on B cR we have∆( δy ) + V ( δy ) σ ≤ δ ∆ y + δV y σ = 0 . Thus we see that the function u = δ e u is positive and satisfies∆ u + V u σ ≤ M. Example . Let ψ ∈ A with ψ ( r ) := r if 0 ≤ r < , [ r α − (log r ) β ] m − if r > . Let δ > V ( x ) ≡ V ( r ) := (log(2 + r )) − δβ for all x ∈ M .
For any
R > S ( R ) = ω m R α − (log R ) β , µ ( B R ) ≃ CR α (log R ) β , thus, thanks to (5.1), we conclude taht λ ( M ) = 0. Moreover, there holds(5.7) Z B R V − β ( x ) dµ ≥ CR α (log R ) β + δ . Observe that in view of (5.7), neither condition (1.13) nor condition (1.14) is satisfied. Moreover, notethat (1.18) holds, while (1.19) fails. This is essentially due to the choice of V .For any r > Z ∞ r dξS ( ξ ) < ∞ ;moreover, for r > γ ( r ) := Z ∞ r dξS ( ξ ) ≃ Cr α − (log r ) β . Hence for r > Z ∞ r [ γ ( r )] σ V ( r ) S ( r ) dr ≤ C Z ∞ r [log(2 + r )] − δ/β r α − (log r ) β r σ ( α − (log r ) βσ dr (5.9) ≤ C Z ∞ r r ) β ( σ − δβ drr < ∞ . In view of (5.8)-(5.9), from Proposition 5.1 with A ( r ) = S ( r ) and B ( r ) = S ( r ) V ( r ), we have that thereexists a solution y = y ( r ) of (5.6), for some R >
0. Furthermore, y ( r ) > r ∈ [ R , ∞ ) and y ( r ) ∼ γ ( r ) as r → ∞ . Since λ ( M ) = 0 and V > M , by the same arguments as in the previousExample 5.2, we can construct u ∈ C ( M ), with u = u ( r ) > M , which satisfies∆ u + V u σ ≤ M. Example . Let ψ ∈ A with ψ ( r ) := r if 0 ≤ r < ,e √ r if r > . For any sufficiently large
R > S ( R ) = ω m e ( m − √ r , µ ( B R ) ≃ Ce ( m − √ R [( m − √ R − , for some C >
0. Note that ¯ λ ( M ) = 0 by (5.1), sincelim sup R →∞ log( µ ( B R )) R = 0 . Let η = m − β = ( m − σ − , θ = σ + 1 σ − V ( x ) ≡ V ( r ) := e η √ r (1 + r ) − θβ for all x ∈ M. Then for ε >
R > Z B R V − β + ε ( x ) dµ ≥ C Z R e εη √ r (1 + r ) θ (1 − ε/β ) dr ≥ Ce εη √ R . On the other hand(5.11) Z B R V − β − ε ( x ) dµ ≤ C Z R e − εη √ r (1 + r ) θ (1+ ε/β ) dr ≤ CR θ +1+ θβ ε = CR α + θβ ε . Observe that, in view of (5.10), neither condition (1.13) nor the first inequality in condition (1.14) issatisfied. On the other hand, by (5.11) the second inequality in (1.14) holds. This is essentially due tothe choice of V . Note moreover that only the second inequality in (1.18) is not satisfied, while the firstinequality in (1.18) and (1.19) hold.Note that for any r > Z ∞ r dξS ( ξ ) < ∞ ;moreover, for r > γ ( r ) := Z ∞ r dξS ( ξ ) ≃ Ce − ( m − √ r √ r. Hence(5.13) Z ∞ r [ γ ( r )] σ V ( r ) S ( r ) dr ≤ C Z ∞ r e [ − ( m − σ − η ] √ r r σ − θβ < ∞ . In view of (5.12)-(5.13), from Proposition 5.1 with A ( r ) = S ( r ) and B ( r ) = S ( r ) V ( r ), we have thatthere exists a solution y = y ( r ) of (5.6), for some R >
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