Existence and non-existence results for a logistic-type equation on manifolds

We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold, which is mostly taken into account by means of conditions on the volume growth of geodesic balls.


Introduction
The aim of this paper is to study the problem of existence, non-existence and uniqueness of solutions of the equation on a complete, connected Riemannian manifold (M, , ). Here σ > 1, the coefficient b(x) is assumed to be non-negative while a(x) is not assumed to be of constant sign.
Equations of the form (0.1) arise in Riemannian geometry, as the equation for the change of the scalar curvature under a conformal change of the metric (see, e.g., [K]) and in mathematical biology, where they describe the steady state solutions of the logistic equation with diffusion (0.2) ∂ ∂t u = ∆u + a(x)u − b(x)u σ , (see, e.g., [AB], [AW], [DM1], [DM2]). In the latter context u represents the density of a population, and it is therefore assumed to be nonnegative, the non-linear term −b(x)u σ accounts for the fact that the population is self limiting, and the function a(x) represents the birth rate of the population, with no self limitation. In Euclidean setting, G.A. Afrouzi and K.J Brown, [AB], have studied the following special case of (0.1) where the positive parameter λ > 0 is the inverse of the diffusion rate, and g(x) is a changing sign coefficient which again represents the birth rate of the population. Their results describe the interplay between diffusion, and birth rate, and show that if diffusion is sufficiently small, solutions may exist even if a is predominantly negative, while if diffusion is large, then solutions exist only if the birth rate is sufficiently large. In fact, the mutual interactions between diffusion and birth rate is often taken into account by the principal eigenvalue λ * of the linear part of equation (0.3) (see Section 1 below for the relevant definitions). This is exemplified, e.g., in [AB] Theorem 2.2, where it is proved that if g is positive somewhere, so that λ * ≥ 0, and if λ > λ * , then equation (0.3) has a positive solution.
Further, (see Section 3 therein) under the additional assumption that g(x) is strictly negative in the complement of a ball, (0.3) has (exactly one) positive solution if λ > λ * , and no positive solution if λ ≤ λ * .
More recently, existence, non-existence, and uniqueness results have been obtained by Y.Du and L.Ma, [DM2], who study the equation Our results provide new insight on the interplay between diffusion and growth rate, that is, in our notation, between the relative size of the variable coefficients a(x) and b(x). Indeed, we show that if a(x) is sufficiently large in a suitable ball, while outside the ball the negative part of a(x) is not too big (so a possibly overall negative birth rate is compensated by a sufficiently large positive birth rate in the ball), then (0.1) has a positive solution (see Theorem 2.10 below), independently of the size of b(x). On the other hand, the content of Theorem 3.7 is that if a(x) is sufficiently small compared to b(x), and certain geometric conditions on the volume growth of the manifold hold, then (0.1) has no positive solution. We note that in the special case where b is constant, and the underlying manifold is Euclidean space, Theorem 3.7 generalizes and complements the non-existence result contained in [AB], Section 3.
Observe also that, according to Theorem 2.1 in [BRS2], if a(x) is sufficiently large that the bottom of the spectrum λ ∆+a(x) 1 (M ) of the Schrödinger operator ∆ + a(x) is negative, then one can guarantee the existence of a (minimal) solution of (0.1) irrespectively of the size of b(x). This is tightly related to the above mentioned relationship between the existence of steady state solutions and the principal eigenvalue λ * of the problem ∆u + λa(x)u = 0 on M . Indeed, as we shall explain in Section 1, λ * is precisely the largest value of λ for which λ ∆+λa(x) 1 (M ) ≥ 0.
Note however, that in our main existence result, Theorem 2.10, we avoid an assumption of this type and describe explicit conditions on the coefficients that guarantee existence, thus giving a new contribution to the subject.
It should also be stressed that having replaced Euclidean space with a Riemannian manifold, the behavior of the equation is now sensitive to the geometry of the underlying space, and therefore reflects not only the mutual relationship of the coefficients a(x) and b(x), but also their respective interaction with the geometry.
From the analytic point of view this introduces new difficulties. For instance, to prove our main non-existence result we need to determine an asymptotic a priori upper bound for the solution u, and the techniques that are usually employed in Euclidean setting are not available. We overcome the problem via a different approach of geometric flavor, which may be of independent interest (see Lemma 3.6).
From now on we denote by (M, , ) a connected, complete, non-compact Riemannian manifold of dimension m ≥ 2. We fix a reference point o in M, and denote by r(x) the Riemannian distance function from o, and by B R and ∂B R the geodesic ball and sphere, respectively, of radius R > 0 centered at o. Finally, we will denote by C, possibly with subscripts or superscripts, a positive constant which may vary from place to place and that may depend on any factor quantified (implicitly or explicitly) before its occurrence, but not on factors quantified afterwards. Given functions A and B, defined on a set Ω, we say that 1. On the principal eigenvalue λ * As mentioned in the Introduction, existence results for equation (0.3) typically depend on the assumption that the parameter λ be strictly greater than the principal eigenvalue of the linear part of the equation.
Recall that a constant λ 1 is said to be a principal eigenvalue for the linear equation ∆u + λa(x)u = 0 if for λ = λ 1 the equation has a positive solution.
On the other hand, in the literature on the non-compact Yamabe equation existence results often depend on the assumption that the sign of the bottom of the spectrum of the Schrödinger operator associated to the equation be negative (see, e.g. [BRS2]).
It is therefore natural to investigate the relationships between principal and spectral eigenvalues.
Let a(x) ∈ C ∞ (M ) and, given a fixed radius R, consider the eigenvalue problem If a(x o ) > 0 for some x o ∈ B R , then it is well known (see [MM], [HK]) that (1.1) has a positive principal eigenvalue λ 1 (B R ), which is variationally characterized by and a principal positive eigenfunction ϕ on B R satisfying We note in passing that, by the maximum principle, the condition that a(x) is positive somewhere in B R is also necessary for the existence of a positive principal eigenvalue.
It follows from (1.2) that λ 1 (R) is a non-increasing function of R, and we may set On the other hand, for µ ∈ R, let L µ be the operator L µ = ∆+ µa(x) and denote by λ 1 (L µ , R) the first Dirichlet eigenvalue of L µ on B R , so that and there exists a smooth positive eigenfunction ψ of L µ on B R satisfying Again λ 1 (L µ , R) is a non-increasing function of R and one may define which coincides with the bottom of the L 2 -spectrum of L µ in the case where the operator is essentially self-adjoint on C ∞ c (M ) (this happens, e.g., if the operator L µ is bounded from below on C ∞ c , see [BdCS], Proposition 2). By a result of W.F. Moss and J. Pieperbrink, [MP], and D. Fisher-Colbrie and R. Schoen, [FCS], we have that λ We are now ready to prove the following and choose a sequence R k such that R o < R k ր +∞. Denote by ϕ k the solution of (1.3) on B R k with principal eigenvalue λ 1 (R k ), normalized with ϕ k (x o ) = 1. Arguing as in the proof of [J], Theorem 1, one shows that {ϕ k } has a subsequence which converges locally uniformly on M to a C ∞ non-negative function ϕ satisfying ϕ(x o ) = 1 and ∆ϕ + λ * a(x)ϕ = 0 on M.
Furthermore, by the maximum principle (see [GT], p. 35), ϕ > 0 on M. It follows from (1.6) that λ On the other hand, let µ ≥ 0 be such that λ Lµ 1 (M ) ≥ 0. We claim that µ ≤ λ * so that the reverse inequality holds in the above formula, and the required conclusion follows. To this end, let u be a smooth positive function satisfying (1.6), and fix R > 0 sufficiently large that x o ∈ B R . Defining w = log u, it follows from (1.6) that (1.7) ∆w = −µa(x) − |∇w| 2 .
Given any v ∈ C ∞ 0 (B R ), v ≡ 0, we multiply both sides of (1.7) times v 2 , integrate by parts and use Young inequality to obtain Now the variational characterization of the principal eigenvalue shows that µ ≤ λ 1 (R) and the claim follows from the definition of λ * .
Then a non-negative number µ satisfies µ > λ * if and only if λ Lµ 1 (M ) < 0. Proof. Since in our assumptions λ * ≥ 0, we may assume that µ > 0. Assume by contradiction that λ Lµ 1 (M ) < 0 and µ ≤ λ * . By definition, there exists R sufficiently large that x o ∈ B R and λ 1 (L µ , R) < 0, so that, if ψ is the corresponding positive eigenfunction as in (1.5), we have In particular, the integral on the right hand side is positive, and since ψ ∈ H 1 0 (B R ), we have BR a(x)ψ 2 < µ, which gives the required contradiction. The reverse implication is an immediate consequence of Proposition 1.1.
We remark that statements similar to Proposition 1.1 and Corollary 1.2 hold (almost trivially) in the case of a bounded domain Ω with smooth boundary such that a(x o ) > 0 for some x o ∈ Ω We conclude this section by showing an application of the results obtained to the case of the Schrödinger operator ∆ + λa(x) on R m . We assume that the positive part a + (x) of a(x) does not vanish identically, so that the results described above hold, and that it satisfies the estimate a + (x) ≤ k |x| 2 for some positive constant k. According to [BRS1] 4t 2 , then the equation ∆ϕ + A(|x|)ϕ = 0 has a positive solution ϕ on R m . Thus, if λk ≤ (m − 2) 2 /4, then ϕ satisfies ∆ϕ + λa(x)ϕ ≤ 0, which according to the above mentioned result of Fisher-Colbrie and Schoen, [FCS], gives λ ∆+λa(x) 1 (M ) ≥ 0. We conclude that λ * is strictly positive, and, in fact, We will come back to this in Section 3 below.

Existence and uniqueness results
The established relationship between λ * and λ Lµ 1 (M ) allows us to apply to the present situation many of the results obtained in [BRS2]. In particular, we quote the following theorem which states the existence of minimal positive solutions of equation (0.1).
As mentioned in the introduction, if we assume that the function a(x) is positive somewhere on M then the condition λ L 1 (M ) < 0 amounts to the fact µ = 1 is larger than the principal eigenvalue λ * of the problem ∆u + λa(x)u = 0 on M , and Theorem 2.1 compares with the existence results in [AB], Theorem 2.2 and [DM2], Theorem 1 (for the latter, see also the remark after Theorem 2.3).
The proof of Theorem 2.1 uses the method of super-and sub-solutions, and the main task is the construction of a sub-solution, which is where the assumption on the sign of λ L 1 (M ) plays a crucial role. In the main result of this Section, Theorem 2.10 below, we describe conditions not expressed in terms of the sign of λ L 1 (M ), for which one can guarantee existence of a solution. Our first result, Theorem 2.4 below, states that, if one has a global sub-solution of (2.1) and the set where the non-negative coefficient b is suitably small, it is always possible to prove the existence of a maximal solution.
The idea of the proof consists in applying the method of sub-and super-solutions to a sequence of boundary value problems on domains which exhaust the manifold.
Since a global sub-solution of (2.1) is given, one first needs to find local supersolution. If b(x) is strictly positive a sufficiently large constant will do. Even if this is not the case, a super-solution can be found provided the set where b(x) vanishes is small (see Theorem 2.3 below).
To apply the approximation method it is also crucial that the approximating sequence is monotonic, and this follows from the next comparison result.
Proposition 2.2. Let D ⊂ M be an open set with smooth boundary ∂D, and assume that a(x), b(x) are functions in C(D) ∩ C 0,α loc (D), 0 < α < 1, and that b(x) is non-negative and does not vanish identically on any connected component of D.
Proof. Since a(x) has indefinite sign, the standard comparison principle does not apply. To circumvent this problem, set v − = δu for some δ ∈ (0, 1]. Since b(x) ≥ 0 and 1 − σ < 0 we have By the monotone iteration scheme there exists a C 2 solution w of (2.2) with w = u on ∂D, and v − ≤ w ≤ v + = v on D. In order to conclude it is enough to show that w = u, and to this end we apply an argument used in the proof of Lemma 2.2 in [BRS2], which we reproduce here for the sake of completeness and the convenience of the reader. Let Z be the vector field defined on D by the formula Since u and w are solutions of (2.2), a direct calculation yields Integrating over D, applying the divergence theorem, and using the fact that u ≡ w on ∂D yield It follows that ∇w − w u ∇u = 0 so that u = Bw on any connected component D 1 of D, for some constant B > 0. Inserting this into the inequality Since w > 0 and b ≥ 0, b ≡ 0 on D 1 this forces B = 1. Thus u = w on D 1 , and therefore u = w on D, as required.
We remark that Proposition 2.2 holds if the coefficients a(x) and b(x) are only assumed to be continuous, and if the functions u and v are in C 1 (D) ∩ C 0 (D), provided we interpret (2.2) and (2.3) in weak sense. Under these weaker assumptions, the vector field Z will be only continuous, in general, but the proof may be carried out using a suitable version of the divergence theorem (see, e.g., [RS], pp. 477-478).
Before stating Theorem 2.4, we also need to make precise the sense in which the set where b(x) vanishes is small.
Let a(x) ∈ C 0 (M ) and let L = ∆ + a(x). If Ω is a non-empty open set, the first Dirichlet eigenvalue λ L 1 (Ω) is variationally defined as in Section 1 by means of the formula and, if Ω is bounded and both Ω and a are sufficiently regular, the infimum is attained and there exists a unique normalized eigenfunction v defined on Ω satisfying We extend the definition to an arbitrary bounded subset S of M , by setting where the supremum is taken over all open bounded sets with smooth boundary Ω such that S ⊂ Ω. Note that, by definition, if S = ∅ then λ L 1 (S) = +∞. Finally, if S is an unbounded subset of M , we define where the infimum is taken over all bounded open sets with smooth boundary. Note that if {D n } is a increasing sequence of open sets with smooth boundary which exhausts M, then, by domain monotonicity, λ L 1 (S) = lim n λ 1 (D n ∩ S). Since the first Dirichlet eigenvalue of the Laplacian of a ball B r grows like r −2 as r → 0, λ L 1 (B r ) > 0 provided r is sufficiently small, and one may think that the condition λ L 1 (S) > 0 expresses the fact that S is small in a spectral sense. This notion of smallness is appropriate for our purposes. Indeed, P. Li, L.-F. Tam and D. Yang, [LTY], have established the following relationship between the first eigenvalue of the set where b(x) vanishes, and the existence of a non trivial super-solution of equation (2.1).
We remark that using Theorem 2.3 we may improve Theorem 2.1 above replacing the assumption that b(x) > 0 on M with the assumption that b is non-negative, and its zero set S o is such that λ L 1 (S o ) > 0. Indeed, the strict positivity of b is only used to guarantee that on every bounded domain a suitably large constant is a super-solution of equation (2.1) (see [BRS2], p.184).
Note that, if we assume that S o is a bounded domain with smooth boundary such that a(x) is positive somewhere in S o , then the condition λ L 1 (S o ) > 0 amounts to the fact that 1 is strictly smaller that the principal eigenvalue of the problem ∆u+λa(x)u = 0 with Dirichlet boundary conditions. Recalling the remark after the statement of Theorem 2.1 we conclude that if λ * < 1 < λ 1 (L µ , S o ) then equation (2.1) has a unique positive minimal solution on M . This again compares with Theorem 1 in [DM2].
We are now ready to state is non-negative, and strictly positive off a compact set, and that, denoting SinceD k is compactly contained in D k+1 , inf D k v > 0 and u − is bounded onD k . Thus, given n ≥ maxD k u − , there exists C > 0 large enough that the function The monotone iteration scheme yields a solution u k,n of the boundary value problem We now show that the sequence {u k,n } is uniformly bounded with respect to n ∈ N on compact subsets of D k . Assume first that K is a compact subset of D k which does not intersect S 0 . Then we may find a positive constant b o and a finite number of disjoint open balls B Next we show that u k,n is uniformly bounded in a neighborhood of S 0 . By definition there exist open sets with smooth boundary Ω and Ω ′ such that Since φ is positive on Ω ′ , it is bounded away from zero onΩ and there exists a positive constant c such that cφ > C 2 on Ω.
Note that We claim that u k,n ≤ cφ on Ω. Indeed, assume that this is not so, and let A = {x ∈ Ω ′ : u k,n − cφ > 0}. Then A is non-empty andĀ ⊂ Ω, and we deduce that w = u k,n − cφ attains a positive maximum in A. On the other hand w satisfies and therefore, by the generalized maximum principle, w/φ is constant on A, and since it vanishes on ∂A we conclude that w/φ = 0 on A, that is, w = 0 on A, contradiction. Thus u k,n ≤ cφ ≤ C 2 onΩ and it follows easily that u k,n is uniformly bounded on compact subset of D k .
By interior elliptic estimates, a subsequence of u k,n converges in C 2 loc to a solution u ∞ k of We consider the sequence {u ∞ k }. Clearly, u ∞ k ≥ u − > 0, and an exhaustion argument and Proposition 2.2 show that Since {u ∞ k } is monotone non-increasing, it converges to a function u which solves (2.1) and satisfies u ≥ u − ≥ 0, u − ≡ 0 on M . If u 1 is another positive solution of (2.1) on M , then u 1 ≤ u ∞ k by Proposition 2.2, and therefore u 1 ≤ u, thus proving the maximality of u. Finally, u is strictly positive for otherwise the non-negative function u − would attain a zero minimum, thus violating the minimum principle ( [GT], p. 35).
It is worth pointing out the following consequence of Proposition 2.2.
has at most one positive C 2 solution.
Proof. Let u, v be positive C 2 solutions of (2.9). Choose any ǫ > 0 and observe that, since b is non-negative, the function w ǫ = (1 + ǫ)v is a super-solution of (2.9) (i) satisfying Fix R 0 > 0 sufficiently large, so that for every R > R 0 we have b(x) ≡ 0 on B R and w ǫ −u > 0 on ∂B R . The latter is possible because of the limit relations (2.9) (ii) and (2.10). It follows from Proposition 2.2 that w ǫ ≥ u on B R for every R ≥ R 0 . Thus, u(x) ≤ (1 + ǫ)v(x) on M , and since ǫ > 0 was arbitrary, u ≤ v on M. Interchanging u and v yields the reverse inequality, and equality follows.
We remark that the assumption L > 0 in the statement of the proposition cannot be weakened to L ≥ 0. Indeed, in [BRS2], pp. 214-215, it is shown that on mdimensional hyperbolic space H m , equation (2.9) with a(x) ≡ m(m − 2)/4, b(x) ≡ 1 and σ = (m + 2)/(m − 2) has a family of positive distinct radial solutions which tend to zero at infinity at the same rate.
It may also be worth noting that the assumptions on u and v in the uniqueness result obtained above may be weakened provided some conditions on the coefficients a and b and on the volume growth of the manifold are imposed.
Theorem 2.6. Let a(x), b(x) ∈ C 0 (M ) and assume that, for some C > 0 and 0 ≤ µ < 2, Assume that u and v are C 2 nonnegative solutions of Proof. Note first of all that, by the maximum principle (see [GT], p. 35), v is strictly positive, and therefore, by the liminf condition, it is bounded away from 0 on M . Also, u is bounded above on M. We may assume that u is not identically zero, for else there is nothing to prove. Thus, β = sup M u v is finite and strictly positive. The conclusion of the theorem amounts to saying that β ≤ 1. Assume by contradiction that β > 1, and let φ = u − βv.
Clearly φ ≤ 0. We claim that sup M φ = 0. Indeed, there exists a sequence x n such that u(xn) v(xn) → β > 0, and since u(x n ) is bounded above, so must be v(x n ) (for else β = 0), and then By the mean value theorem we have is continuous, and nonnegative on M. Further, since u is bounded above, it follows h is bounded above by a constant H on the set {x : φ(x) > −1}. Also, since β > 1, σ > 1 and v is bounded away from zero, for some positive constant c. Inserting the above expressions in (2.13), noting that, since φ is non-positive, −a(x)φ ≥ a − (x)φ, and dividing through by b(x), we obtain On the other hand, since the volume growth condition (2.12) holds, and b(x) satisfies the lower estimate (2.11) (i), Theorem A in [PRS1] applies, and the weak maximum principle holds, namely, ∆φ ≤ 0 thus yielding the required contradiction.
As an immediate corollary we have Corollary 2.7. Let a and b satisfy the conditions listed in Theorem 2.6, and let u and v be nonnegative solutions of If both u and v satisfy the condition and (2.12) holds, then u = v.
As observed above, condition (2.14) amounts to requiring that u and v are bounded and bounded away from zero on M. We also note that the family of functions mentioned in the remark that follows Proposition 2.5 also shows that uniqueness fails if we do not assume that the liminf of u and v are strictly positive.
It is worth mentioning the following geometric consequence.
Corollary 2.8. Let (M, , ) be a complete Riemannian manifold of dimension m ≥ 3 and scalar curvature s(x) satisfying for some constants C > 0 and 0 ≤ µ < 2. Assume that (2.12) holds. Then any conformal diffeomorphism of M into itself which preserves the scalar curvature and whose stretching factor u satisfies (2.14), is an isometry.
We remark that here we require that u is bounded above and away from zero, so that the conformal diffeomorphism φ is a quasi-isometry. By contrast, in [PRS1] Corollary 3.4, φ is not assumed to be a quasi-isometry, but the scalar curvature s(x) is assumed to be bounded below.
We now proceed with the main result of this section, Theorem 2.10, where we show that, under suitable assumptions on the coefficients, equation (2.1) has a globally defined positive sub-solution, and therefore, a maximal positive solution. The proof is based on the method of super and sub-solutions. This is achieved by constructing a sub-solution inside and outside a suitable ball in such a way that they can be glued together to yield a global sub-solution.
We begin with the following lemma, which will be the key ingredient in the construction of a sub-solution in the complement of a ball.
Note that by the conditions imposed on the function g, the metric originally defined on R m \ {0} extends to a smooth metric on the whole of R m . Let R 1 andR be such that 0 < R 1 < R < R + T <R, and let ψ be a smooth radial cut-off function such that 0 ≤ ψ ≤ 1, ψ ≡ 1 on B R1 , and ψ ≡ 0 on M \B R+T .
where N is constant and letL be the Schrödinger operator L = ∆ +ā(x). We claim that if N is sufficiently large then Indeed, let u be any smooth function satisfying u > 0 in BR and u = 0 on ∂BR. Then and since (1 − ψ)u 2 > 0 in BR \ B R+T , the right hand side may be made negative provided N is large enough. The claim now follows from the variational characterization of λL 1 (BR). Let φ be the radial, normalized eigenfunction belonging to λL 1 (BR). By definition Thus, if we denote byL = L +ā(x) − γB(r(x)), then If ξ is a positive radial eigenfunction belonging to λL 1 (BR), so that ∆ξ +ā(x)ξ = γB(r(x))ξ − λL 1 (BR)ξ ≥ γB(r(x))ξ on BR, On the other hand, since B(r) > 0, a sufficiently large positive constant v + is a super solution of the above problem, that is and by the monotone iteration scheme there exists a solution w to the problem Note that w is radial, since the monotone iteration scheme produces radial solutions in radial setting. Moreover, sinceā(x) > −A(r(x)) and infB R+T w > 0, if c is sufficiently large then the function w + = cw satisfies and since w − ≡ 0 is a sub-solution of the problem, applying once again the monotone iteration scheme produces a radial non-negative C 2 solution u of Since u satisfies ∆u − A(r(x)) + B(r(x))u σ−1 u = 0, by the strong maximum principle (see [GT], p. 35) we deduce that u > 0 in B R+T \B R . Since u is radial, we deduce that u(x) = α(r(x)) with α satisfying (2.15).
We now show that α ′ < 0 in [R, R + T ). Indeed, assume this is not so. Since It remains to show that estimate (2.16) holds. Integrating (2.15) between R + t 1 and R + t 2 with 0 ≤ t 1 < t 2 ≤ T , yields and since the integrand is positive, we deduce that whence, recalling that α ′ < 0 and g is non-decreasing, and therefore Moreover, for every T o ∈ (0, T ] there exists t ∈ (0, T o ) such that Writing (2.17) with t 1 = 0 t 2 = T o , and using the above inequality we obtain as required.
We are now ready to prove the main result of this section.
Theorem 2.10. Let (M, , ) be a complete manifold, and assume that the differential inequality holds pointwise in the complement of the cut locus Cut o of o, for some function , and assume that b(x) satisfies the conditions in Theorem 2.4. Suppose that we can choose R > 0 and T o > 0 in such a way that Then the equation has a maximal positive solution on M .
Remarks Observe that, by the Laplacian Comparison Theorem, see, e.g. [GW], or [BRS2], Appendix, the validity of an inequality of the form (2.18) can be deduced from appropriate lower bounds for the radial Ricci curvature. To compare with the existence theorems in Euclidean setting that can be found in the literature, we note that, on R m , (2.18) holds with g(r) = r, while if we assume an "almost Euclidean behavior", namely, that the radial Ricci curvature satisfies an estimate then (2.18) holds with g(r) = r B ′ where B ′ = [1 + √ 1 + 4B 2 ]/2. We note in passing that the assumption on the Ricci curvature does not imply that the manifold is quasi-isometric to Euclidean space. If we assume instead that Ricc ≥ −(m − 1)B so that, loosely speaking, the reference model is hyperbolic space of constant negative sectional curvature, then(2.18) holds with the choice g(r) = 1 √ B sinh( √ Br). We also note that condition (2.19) is satisfied if a(x) is sufficiently large near the origin o and non-negative in a suitably large ball. Further, up to choosing a different reference point, it suffices to assume that a(x) has a positive spike somewhere. Of course (2.18) should then be written with respect to the new origin centered at the spike. Observe however that if we assume, e.g., that the Ricci curvature is bounded from below by a negative constant, then the validity of (2.18) is independent of the chosen origin.
Proof. According to Theorem 2.4 it suffices to show that (2.20) has a non-negative, non-identically zero global sub-solution. This will be obtained by joining suitable radial local sub-solutions. We set Applying the previous lemma with A − (r) and B(r) + ǫ instead of A(r) and B(r), respectively, we deduce that for every α o and ǫ > 0 there exists a solution α ∈ C 2 ([R, R + T o ]) of differential inequality Using the expression of the Laplacian of a radial function, and the inequalities (2.18), a(x) ≥ −A − (r(x)) and b(x) ≤ B(r(x)) and α ′ ≤ 0, it follows that the function v(x) = α(r(x)) is Lipschitz in B R+To \ B R and C 2 in the complement of the cut locus of o where it satisfies the pointwise inequality To construct a sub-solution in the ball B R we consider the problem and we look for a solution of the form Note that β ′ (r) = −2ηRα o < 0, so, using (2.22), the condition β ′ (R) ≤ α ′ (R) follows from On the other hand, a direct computation that uses A(r) ≥ min BR a(x) and B(r) ≤ max BR b(x) shows that and the right hand side is non-negative provided Using (2.19) we may choose α o small enough that, for every η ≤ (min BR a)/[2(1+τ )] (which is the largest possible value of η allowed by (2.27)) the right hand side of (2.27) is greater than or equal to the right hand side of (2.26), and therefore choose η in such a way that both (2.26) and (2.27) are satisfied. For such values of α o and η, the function β satisfies all the requirements. Proceeding as above one verifies that the function w(x) = β(α(r(x)) is Lipschitz in B R and C 2 in the complement of the cut locus where it satisfies the pointwise inequality and claim that u − ∈ C 0 (M ) ∩ H 1 loc (M ) is a weak global sub-solution of (2.20). This is easily seen if we assume that o is a pole of M , for then, given a positive test function ϕ ∈ C ∞ c (M ), applying Green's second identity, and using the fact that w and v are pointwise sub-solutions of (2.20) in B R and B R+To \ B R respectively, we obtain and the claim follows from the inequalities In the case where the cut locus of o is not empty, one can adapt an argument in [PRS2], Lemma 2.2, as follows: we consider an exhaustion Ω n of M \ Cut o by domains with smooth boundary, which are star shaped with respect to o, so that, denoting by ν the outward unit normal, we have ∇r, ν > 0 on ∂Ω n . Since the part of ∂B R contained in Ω n is smooth, we may also deform Ω n , if necessary, in such a way that , for every n, ∂Ω n is transversal to ∂B R and ∂B R+To .
Since ∇u − is locally bounded we have , apply the divergence theorem, use the fact that v and w are pointwise sub-solutions of (2.20) in the complement of the cut locus, to obtain To conclude, note that, by the transversality assumption, up to sets of lower dimension, we have ∂ Ω n ∩ B R = (Ω n ∩ ∂B R ) ∪ (∂Ω n ∩ B R ) and similarly when B R is replaced by B R+To \ B R , so that the boundary integrals become ϕα ′ ∇r, ν and the first integral is non-positive because β ′ (R) ≤ α ′ (R), while the last two are non-positive because α ′ , β ′ ≤ 0 and ∇r, ν ≥ 0 on ∂Ω n .
Remark 2.11. The above existence result is also relevant to the Yamabe problem, that is, the possibility of conformally deforming the assigned metric, with scalar curvature s(x), to a new one with prescribed scalar curvature K(x). Indeed assume that m ≥ 3, and denote by u 2/(m−2) the conformal factor, so that the deformed metric is given by , = u 2 (m−2) , . Then the scalar curvature of , is K(x) provided u is a, necessarily positive, solution of the Yamabe equation . By way of example, assume that Ricc ≥ −(m−1)B, so that s(x) ≥ −m(m−1)B, and, as noted above, (2.18) holds with g(r) = ( It is easy to see that if √ B > 4(m − 1)/[m(m − 2)] then the above condition is verified provided s(x) is sufficiently near −m(m − 1)B in the ball B R and R is large enough. We note that there are situations (for instance when the sectional curvature is suitably pinched) where Theorem 2.10 is applicable, while Theorem 2.1 is not.

Non existence results
The purpose of this section is to prove a non-existence result for non-negative C 2 solutions of the differential inequality We begin with the following general and suppose that there exists a positive C 2 solution of the differential inequality Then the differential inequality has no non-negative C 2 solutions on M satisfying for some p > 1 and β satisfying max{0, A} ≤ β ≤ H(K + 1) − 1.
If p = 2, assumption (3.7) implies (3.6) if a suitable bound on ϕ is available, e.g., if p < 2 and ϕ is bounded from above. A similar simplification occurs if ϕ is bounded above, respectively below, by a radial function, see, e.g., Theorem 3.10 below. We also remark that the inequality valid for f > 0, together with integration in polar coordinates, shows that condition (3.6) is implied by ϕ 2−p H u ∈ L 2(β+1) (M ). Finally, we note that the proof of the theorem could be achieved by adapting the argument in the proof of Theorem 1.3 in [PRS3]. However, the present assumptions allow us to give an alternative argument that we describe here for the sake of completeness.

A straightforward computation that uses (3.3) and (3.4) yields
vdiv ϕ 2α ∇v ≥ α(K − α + 1)(u 2 + ǫ) β+1 |∇ϕ| 2 ϕ 2 + (β + 1)( We choose α = H −1 (β + 1), so that our assumptions on β, H and K yield 0 < α ≤ K + 1. Therefore α(K − α + 1) ≥ 0, and using the assumptions b(x) ≥ 0 and β + 1 ≥ 0, we deduce that vdiv ϕ 2α ∇v ≥ ǫ(β + 1)a(x)(u 2 + ǫ) β Let r(t) ∈ C 1 (R) and s(t) ∈ C 0 (R) satisfy the conditions and let Z be the vector field defined by Z = vr(v)ϕ 2α ∇v. For fixed t and δ > 0 let also ψ δ be the Lipschitz function defined by Using (3.8) (3.9) and the definition of ψ δ we compute div ψ δ Z = ψ δ div Z + ∇ψ δ , Z whence, integrating, and using the divergence theorem and the Cauchy-Schwarz inequality we obtain By Hölder inequality the integral on the right hand side is bounded above by Inserting into the above inequality, letting δ → 0+ and using the co-area formula (see Theorem 3.2.12 in [F]) we deduce that (3.10) In the above formula, the surface integral is computed with respect to (m − 1)dimensional Hausdorff measure on ∂B t , which coincides with the Riemannian measure induced on the regular part of ∂B t (the intersection of ∂B t with the complement of the cut locus of o, see [F], 3.2.46, or [Ch], Proposition 3.4). As ǫ → 0, v = v ǫ → v 0 = ϕ −α u β+1 , whence, using the dominated convergence theorem in (3.10), we get Defining so that by the co-area formula H is Lipschitz and and noting that the coefficient of the second integral on the left hand side of (3.11) is non-negative by the conditions imposed on β, we obtain Our aim is to show that under assumption (3.6) v 0 is constant. The proof follows the lines of that of Lemma 1.1 in [RS]. Assume by contradiction that v 0 is not constant. Then there exists R o such that h(t) > 0 for every t ≥ R o , and therefore the right hand side of (3.12) is positive for t ≥ R o . Dividing through by h(t), squaring and integrating the resulting differential inequality between R and r with We choose a sequence of functions Since condition (3.9) holds for every n, so does (3.13), whence, letting n → +∞ and using the Lebesgue and monotone convergence theorems we deduce that there exists C > 0 which depends only on p such that (3.14) Now, recalling that α = (β + 1)/H and the definition of v 0 , we have ϕ 2α v p 0 = ϕ (2−p)(β+1)/H u 2(β+1) and the required contradiction is reached by letting r → +∞ and using assumption (3.6).
Thus v 0 is constant, and we deduce that there exists a constant C ≥ 0 such that Since u is not identically zero by (3.5), C > 0 and u is strictly positive on M. We insert the expression of ϕ in terms of u in (3.3), divide by CHu H−2 and subtract the result from (3.4) to obtain Since the coefficient of |∇u| 2 is non-positive, by (3.2), we conclude that b(x)u σ+1 ≤ 0, which contradicts (3.5).
Remark 3.2. Observe that the above proof actually shows that if A < H(K +1)−1 then ∇u = 0, so that u, and therefore ϕ, are necessarily constant. It follows from (3.3) and (3.4) that 0 ≥ a(x)u ≥ b(x)u σ+1 ≥ 0, so that, if a does not vanish identically, then u ≡ 0, without any assumption on b. On the other hand, if A = H(K + 1) − 1, then the conclusion depends on the fact that b is positive somewhere.
Remark 3.3. We also note that if u is assumed to be strictly positive, then the conclusion of the Theorem holds, with a much easier proof, if we assume that max{−1, A} ≤ H(K + 1) − 1, and that β > −1, β ≥ A, β ≤ H(K + 1) − 1.
Remark 3.4. Let L H be the Schrödinger operator defined by L H = ∆ + Ha(x).
Then the validity of (3.3) is related to the sign of λ LH 1 (M ). Indeed, assume that ϕ is a positive C 2 solution of (3.3). Let ψ ∈ C ∞ c (M ) and apply the divergence theorem to the vector field ψ 2 ∇ log ϕ. Since, by (3.3) and Young inequality div ψ 2 |∇ϕ| 2 ϕ 2 , and from the variational characterization of the bottom of the spectrum we we conclude that if K ≥ 0 then λ LH 1 (M ) ≥ 0. On the other hand, if λ LH 1 (M ) ≥ 0, then, by an extension of the result of Moss Pieperbrink, and Fisher-Colbrie Schoen quoted in Section 1 (see [PRS3], Lemma 1.2), there exists a positive C 1 function v which satisfies ∆v + Ha(x)v = 0 weakly on M. Further, if a(x) is assumed to be C 0,α for some α ∈ (0, 1), then v is C 2 and it is a classical solution of the above equation. It is clear that v is respectively a weak or a classical, solution of (3.3) for every K ≤ 0.
Corollary 3.5. Let a(x), b(x) ∈ C 0 (M ) and assume that b(x) is non-negative and does not vanish identically. Suppose also that, for some H ≥ 1, λ LH 1 (M ) ≥ 0. Then there are no positive C 2 solutions of the differential inequality If a(x) is assumed to be C 0,α , the the corollary follows immediately from Remark 3.4 and from Theorem 3.1 with K = 0. In the general case, the function v satisfies inequality (3.3) with K = 0 only in weak sense, and the argument of Theorem 3.1 needs to be slightly modified to be carried out in this situation. Note also that, since the corollary deals with strictly positive solutions, we can drop the assumption that σ ≥ −1.
In order to apply Theorem 3.1 and obtain the non-existence result mentioned at the beginning of this section, one needs to verify that the (non-)integrability condition (3.6) holds. In principle, this may be obtained combining a-priori upper estimates for u with appropriate bounds for the volume growth of balls. Both estimates can be deduced imposing lower bounds on the radial Ricci curvature of the manifold.
In the following lemma we deduce an a-priori integral estimate for nonnegative solutions of (3.4), which will enable us to obtain (3.6) under the sole assumption of a volume growth condition.
Lemma 3.6. Let (M, , ) be a complete Riemannian manifold, and let a(x), b(x) ∈ C 0 (M ) with b(x) > 0 on M. Assume that u ≥ 0 is a C 2 solution of the differential inequality for A ≤ 1 and σ > 1. Then for every p ≥ 1, p > A + 2 there exist constants C 1 , C 2 > 0 which depend only on p, σ and R 0 > 0 such that, for every R ≥ R 0 , Proof. Observe first that we may assume that u ≡ 0, for otherwise there is nothing to prove. Thus, there exists R 0 > 0 such that u ≡ 0 on B R for every R ≥ R 0 . Next, for every R ≥ R 0 , let ψ = ψ R : M → [0, 1] be a smooth cut-off function such that for some C which depends only on p and σ. Note that this is possible since the exponent p−1 p+σ−2 is strictly less than 1. Having fixed ǫ > 0, we let W be the vector field defined by W = ψ 2 (u + ǫ) p−3 u∇u.
A computation that uses (3.17) yields We estimate the last term on the right hand side using Cauchy-Schwarz's inequality and Young's inequality 2ab We integrate the above inequality, apply the divergence theorem, rearrange, let ǫ → 0+ and use the dominated convergence theorem, in this order, to deduce that If p = 1 the conclusion follows immediately using (3.19). If p > 1,, we denote by I and II the two integrals on the right hand side, and use Hölder inequality with conjugate exponents p + σ − 2 p − 1 (> 1) and p + σ − 2 σ − 1 , and the assumption that b(x) > 0 to estimate Inserting into (3.20), noting that the integral on the left hand side is strictly positive by the choice of R, and simplifying, we obtain The required conclusion follows again using (3.19) and the elementary inequality (a + b) τ ≤ 2 τ (a τ + b τ ) valid for a, b, τ ≥ 0.
We are now ready for our main non-existence result.
Then the only non-negative C 2 solution u of the differential inequality Proof. If we set p = 2H + 2 − σ, the conditions imposed on the parameters imply that p satisfies the assumptions listed in the statement of Lemma 3.6. The lemma and condition (3.22) (i) show that there exist constants C i > 0 such that for r > 0 sufficiently large. We use condition (3.21) to estimate from below the integral on the left hand side. On the other hand, since σ < 2H + 1, we may again use condition (3.21) to estimate from above the first integral on the right hand side, and (3.22) (ii) to estimate from above the second integral, and deduce that, for r sufficiently large, Br u 2H ≤ C r (µ−2) 2H σ−1 vol B 2r + r 2 log r , whence, using the volume growth condition (3.26) we conclude that Br u 2H ≤ Cr 2 log r for r ≫ 1.
Remark 3.8. The argument used in the proof shows that the condition that a+ b is bounded above may be removed provided we replace (3.26) with σ−1 log r as r → +∞.
Note that since the integral on the left hand side is a non-decreasing function of r this also imposes the further restriction µ ≤ (σ − 1)/H, with corresponding restrictions being imposed on the range of the other parameters.
Remark 3.9. In the case where the ambient manifold is Euclidean space, we can compare our Theorems 3.1 and 3.7 with the results in [AB], Section 3. We consider the equation (3.28) ∆u + λa(x)u − u 2 = 0 on R m , which, with a(x) = g(x) and a change of scaling, is easily seen to be equivalent to (0.3). We assume, as in [AB], that a(x) is positive somewhere, and that its positive part a + (x) satisfies the estimate a(x) ≤ k |x| 2 , for some positive constant k. According to the discussion at the end of Section 1, it follows that the principal eigenvalue λ * of the linear equation associated to (3.28) is strictly positive and satisfies On the other hand, if u is a non-negative solution of (3.28), Lemma 3.6 with A = 0, σ = 2 and p > 2 shows that and therefore (3.29) r In order to apply Corollary 3.5, the non-integrability condition must hold with p satisfying p = 2(β + 1) ≤ 2H. Summing up, if λ * λ ≥ min{1, m−2 4 } then Corollary 3.5 applies, and we conclude that every non-negative solution of (3.28) vanishes identically. To compare with Theorem 3.9 in [AB], we point out we are assuming a less stringent condition on a + and that we do not require that u tends to zero at infinity.
On the other hand, assume that a + (x) satisfies the more stringent condition assumed in [AB], A |x| 2+δ } for some positive constants A, k and δ. According to [AB], Theorem 3.5, every positive solution of (3.28) which tends to zero at infinity satisfies the estimate and, in fact, by Theorem 3.4 therein, every positive solution tends to zero at infinity, provided a(x) is strictly negative off a compact. Now, it is easy to see that if u satisfies (3.30), then Br u 2 ≤ Cr 2 , and, clearly, λ ∆+λa(x) 1 (R m ) ≥ 0 for every λ ≤ λ * . An application of Corollary 3.5 with β + 1 = H = 1 shows that u vanishes identically. We therefore recover the conclusion of Theorem 3.9 in [AB].
Using (3.32) we obtain the following version of Theorem 3.1.
An application of Theorem B in [RS] shows that v is constant. The conclusion now follows as in the proof of Theorem 3.1.
Note that, even in the case of Theorem 3.10, if A < −1, then the conclusion can be strengthened to assert that every non-negative solution of (3.10) vanishes identically, unless a(x) = b(x) ≡ 0.
In applying Theorem 3.10 it is of course crucial to being able to find positive solutions of (3.31) satisfying the asymptotic lower bound (3.33). By contrast, in order to apply Theorem 3.1 one needs a positive solution of (3.3), whose existence, in typical applications like the one exemplified by Theorem 3.7 above, is guaranteed by means of assumptions on the spectrum of a suitable operator.
Observe now that if v is a solution of the Poisson equation ∆v = a(x) then the function ϕ = e −v is a positive solution of ∆ϕ + a(x)ϕ = |∇ϕ| 2 ϕ , and furthermore, an upper bound for v yields a lower bound for ϕ. The Poisson equation on complete Riemannian manifolds has been extensively studied using heat kernel techniques to obtain bounds on the Green kernel. To illustrate an application of Theorem 3.10, we consider the elementary case where the positive part of a(x) is integrable. Then we have the following lemma (see, e.g., the proof of Theorem 3.2 in [NST]) Lemma 3.11. Let (M, , ) be a complete, non-parabolic manifold, and let ρ ∈ C 0,α (M ) ∩ L 1 (M ) (0 ≤ α < 1) be a non-negative function. Then, there exists a solution v ∈ C 2 of the Poisson equation Then there are no non-negative, non-identically zero C 2 (M ) solutions of the differential inequality (3.41) u∆u + a(x)u 2 ≥ b(x)u σ+1 − A|∇u| 2 on M.
It follows from (3.42) above with q + σ − 2 = 2, that every non-negative solution of (3.44) ∆u + a(x)u − b(x)u σ = 0 satisfies (3.45) Br u 2 ≤ Cr 2 log r, and the same estimate is clearly satisfied by the difference of two solutions. An application of Theorem 4.1 in [BRS2] shows that (3.44) has at most one positive solution. We remark in this respect that if we replace u 2 in (3.45) with u p with p > 2, then the conclusion of Theorem 4.1 in [BRS2] fails, as the example described on pages 214-215 therein shows.