On the maximum and anti-maximum principles

Unspecified Book Proceedings Series

On the maximum and anti-maximum principles Yehuda Pinchover Abstract. In this paper we discuss some new results concerning the maximum and anti-maximum principles for second order, linear, elliptic partial differential equations defined on a noncompact, Riemannian manifold.

1. Introduction The perturbation theory of positive solutions is a useful tool in the qualitative theory of second order, linear, elliptic and parabolic equations and in spectral theory of Schr¨odinger operators. For instance, some new results concerning maximum principles, eigenfunction estimates and the uniqueness of the Cauchy problem were obtained via this approach. In this perturbation theory, the notions of small and semismall perturbations play a fundamental role (see Definition 2.4). The definitions of small and semismall perturbations were introduced by the author [9] and M. Murata [7], respectively. It turns out that these classes of perturbations are appropriate in the study of stability properties of positive solutions. For example, the behavior of the positive minimal Green function GΩ P (x, y) of an elliptic operator P defined in a domain Ω, the structure of the cone CP (Ω) of all positive solutions of the equation P u = 0 in Ω and the corresponding Martin boundary are stable under small or semismall perturbations. Roughly speaking, the decay properties of positive solutions of minimal growth at infinity (see Definition 2.2) are preserved under semismall perturbations, while small perturbations preserve also the behavior of all positive global solutions. Recently, in [10], among other results, the author used semismall perturbations to prove refined maximum and anti-maximum principles for solutions which ‘decay’ at infinity. The main aim of the present paper is to use small perturbations and to extend the refined maximum principle and the anti-maximum principle for solutions which do not grow too fast. In Section 3, we discuss extensions of the classical generalized maximum principle which holds in relatively compact subdomains to a refined (generalized) maximum principle which holds in any domain. Recall that if CP (Ω) 6= ∅, then the generalized maximum principle asserts that for any subdomain Ω1 ⊂⊂ Ω a function u 6= 0 which satisfies P u ≥ 0 in Ω1 and u ≥ 0 on ∂Ω1 is strictly positive in 1991 Mathematics Subject Classification. Primary 35B05, 35B50, 31C35; Secondary 35J15. This research was supported by the Fund for the Promotion of Research at the Technion. c °0000 (copyright holder)

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Ω1 . Roughly speaking, our maximum principles are of Phragm´en-Lindel¨of type and state that if P admits Green function GΩ P (x, y) in Ω, and if P u ≥ 0 in Ω, where u does not grow too fast near infinity in Ω, then u ≥ 0 (see Theorem 3.1 and Theorem 3.10). As a consequence, we prove existence and uniqueness theorems for solutions of the nonhomogeneous equation (see Theorem 3.3 and Theorem 3.12). A quite recent topic in the theory of second order elliptic operators is the socalled anti-maximum principle discovered by Ph. Cl´ement and L. A. Peletier [4] (see also [3, 10, 12, and the references therein]). This interesting phenomenon occurs above the generalized principal eigenvalue λ0 , where λ0 = λ0 (1, P, Ω) := sup{λ ∈ R : CP −λ (Ω) 6= ∅} (recall that the generalized maximum principle holds true below λ0 ). Let uλ be a solution of the equation (1.1)

(P − λ)u = f 0

in Ω.

The anti-maximum principle reads roughly that under some ‘smallness’ conditions on f and uλ in a neighborhood of infinity in Ω, there exists ² > 0 which may depend on f such that uλ < 0, for all λ ∈ (λ0 , λ0 + ²). In Theorem 4.2, we prove a refined anti-maximum principle for the equation P u = f , where f is not necessarily nonnegative and the functions f and u ‘decay’ at infinity. This theorem extends some known anti-maximum principles and give a simpler proof to the anti-maximum principle in [10]. In Theorem 5.2, we prove a weak anti-maximum principle for the equation P u = f for functions f and u which do not grow too fast at infinity. In a subsequent paper, we study the anti-maximum principle for indefinite-weight elliptic problems [11]. 2. Preliminaries In this section we collect some terminology and results which we need in this paper. Let P be a linear, second order, elliptic operator defined on a noncompact, connected, C 3 -smooth Riemannian manifold Ω of dimension d. Here P is an elliptic operator with real, H¨older continuous coefficients which in any coordinate system (U ; x1 , . . . , xd ) has the form (2.1)

P (x, ∂x ) = −

d X i,j=1

aij (x)∂i ∂j +

d X

bi (x)∂i + c(x),

i=1

where ∂i = ∂/∂xi . We assume that for every x ∈ Ω the real quadratic form (2.2)

d X

aij (x)ξi ξj , ξ = (ξ1 , . . . , ξd ) ∈ Rd

i,j=1

is positive definite. We denote the cone of all positive (classical) solutions of the elliptic equation P u = 0 in Ω by CP (Ω). In case that the coefficients of P are smooth enough, we denote by P ∗ the formal adjoint of P . Let {Ωk }∞ k=1 be an exhaustion of Ω, i.e. a sequence of smooth, relatively compact domains such that Ω1 6= ∅, Ωk ⊂ Ωk+1 and ∪∞ k=1 Ωk = Ω. Assume that k CP (Ω) 6= ∅. Then for every k ≥ 1 the Dirichlet Green function GΩ P (x, y) exists and Ωk ∞ is positive. By the generalized maximum principle, {GP (x, y)}k=1 is an increasing

ON THE MAXIMUM AND ANTI-MAXIMUM PRINCIPLES

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sequence which, by the Harnack inequality, converges uniformly in any compact subdomain of Ω either to GΩ P (x, y) , the positive minimal Green function of P in Ω and P is said to be a subcritical operator in Ω , or to infinity and in this case P is critical in Ω. The operator P is said to be supercritical in Ω if CP (Ω) = ∅ [6, 8]. It follows that P is critical (resp. subcritical) in Ω if and only if P ∗ is critical (resp. subcritical) in Ω. Furthermore, if P is critical in Ω, then CP (Ω) is a onedimensional cone and any positive supersolution of the equation P u = 0 in Ω is a solution. In this case φ ∈ CP (Ω) is called a ground state of P in Ω. We fix a reference point x0 ∈ Ω1 . From time to time, we consider the convex set KP (Ω) := {u ∈ CP (Ω) | u(x0 ) = 1} of all normalized positive solutions. Remark 2.1. We would like to point out that the criticality theory, and in particular the results of this paper, are also valid for the class of weak solutions of elliptic equations in divergence form and also for the class of strong solutions of strongly elliptic equations with locally bounded coefficients. Nevertheless, for the sake of clarity, we prefer to present our results only for the class of classical solutions. Subcriticality is a stable property in the following sense. If P is subcritical in Ω and V ∈ C0α (Ω) is a real function, then there exists ² > 0 such that P − µV is subcritical, for all |µ| < ² [6, 8]. On the other hand, if P is critical in Ω and V ∈ C α (Ω) is a nonzero, nonnegative function, then for any ² > 0 the operator P + ²V is subcritical and P − ²V is supercritical in Ω. We associate to Ω a fixed exhaustion {Ωn }∞ n=1 . For every k ≥ 1, we denote ∗ Ωk = Ω \ Ωk and for every k > k0 we denote by Ωk,k0 the ‘annulus’ Ωk \ Ωk0 . Let f, g ∈ C(Ω) be positive functions. We say that f is equivalent to g on Ω and use the notation f ≈ g, if there exists a positive constant C such that C −1 g(x) ≤ f (x) ≤ Cg(x) for all x ∈ Ω . We denote by f + (resp. f − ) the positive (resp. negative) part of a function f . So, f = f + − f − . By 1, we denote the constant function on Ω taking at any point x ∈ Ω the value 1. Let B be a Banach space and B ∗ its dual. If T is a (bounded) operator in B we denote by T ∗ its adjoint. The range and the kernel of T are denoted by R(T ) and N (T ). For every f ∈ B and g ∗ ∈ B ∗ we use the notation hg ∗ , f i := g ∗ (f ). We denote the spectrum of an operator T acting on B by σ(T ). Definition 2.2 (Agmon). Let P be an elliptic operator defined on Ω. A function u ∈ C(Ω∗n ) is said to be a positive solution of the operator P of minimal growth in a neighborhood of infinity in Ω if u satisfies the following two conditions: (i) The function u is a positive solution of the equation P u = 0 in Ω∗n ; (ii) If v is a continuous function on Ω∗k for some k > n which is a positive solution of the equation P u = 0 in Ω∗k , and u ≤ v on ∂Ωk , then u ≤ v on Ω∗k . Let V ∈ C α (Ω) be a real function and denote the generalized principal eigenvalue by λ0 = λ0 (V, P, Ω) := sup{λ ∈ R : CP −λV (Ω) 6= ∅}.

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Definition 2.3. Let Pi , i = 1, 2 be two subcritical operators in Ω . We say that Ω the Green functions GΩ P1 (x, y) and GP2 (x, y) are equivalent (resp. semi-equivalent) Ω Ω Ω if GP1 ≈ GP2 on Ω × Ω \ {(x, x) | x ∈ Ω} (resp. GΩ P1 (·, y0 ) ≈ GP2 (·, y0 ) on Ω \ {y0 } for some fixed y0 ∈ Ω). The following notions of small and semismall perturbations play a fundamental role throughout this paper and are closely related to the problem of (semi-) equivalence of Green functions. Moreover, in many cases sufficient conditions for the equivalence of two Green functions turn out to be sufficient conditions for small perturbations (see, [1, 6, 7, 8, 9, and the references therein]). Definition 2.4. Let P be a subcritical operator in Ω and fix x0 ∈ Ω1 . Let W ∈ C α (Ω) be a real function. (i) We say that W is a small perturbation of P in Ω if ( ) Z Ω GΩ P (x, z)|W (z)|GP (z, y) dz = 0 . (2.3) lim sup k→∞ x,y∈Ω∗ Ω∗ GΩ P (x, y) k k (ii) We say that W is a semismall perturbation of P in Ω if ( ) Z Ω GΩ P (x0 , z)|W (z)|GP (z, y) sup (2.4) lim dz = 0 . k→∞ y∈Ω∗ Ω∗ GΩ P (x0 , y) k k Remark 2.5. (i) A small perturbation is semismall [7]. We do not know of any example of a semismall perturbation which is not a small perturbation. It is known [7, 9] that if the operator P is subcritical and W is a (resp. semi-) small perturbation of (resp. P ∗ ) P in Ω, then the Green functions of P and P − tW are (resp. semi-) equivalent provided that P − tW is subcritical. Moreover, if W is a semismall perturbation of P ∗ in Ω, P − λW is critical in Ω, and φ0 is the ground ∗ state of P − λW , then φ0 ≈ GΩ P (·, x0 ) in Ω1 [7]. (ii) If W is a semismall perturbation of the operator P , and P +W is subcritical M in Ω, then the Martin spaces ΩM P and ΩP +W are homeomorphic and in particular, Ω CP (Ω) and CP +W (Ω) are affine homeomorphic [7]. On the other hand, if GΩ P1 ≈ GP2 , then it is only known that the cones CP1 (Ω) and CP2 (Ω) are homeomorphic [8]. (iii) Let P be a subcritical operator in Ω and assume that V > 0. Then V is a (resp. semi-) small perturbation of the operator P if and only if 1 is a (resp. semi-) small perturbation of the operator V −1 P . Therefore, when considering perturbations by a positive function, we may assume that the perturbation is by the function V = 1, and we leave to the reader to formulate the results for a general positive function V (see [11] for the anti-maximum principle for the indefinite-weight case). 3. Refined maximum principles The purpose of this section is to extend the classical generalized maximum principle to two versions of (generalized) maximum principles which hold in any domain. Roughly speaking, our maximum principles are of Phragm´en-Lindel¨of type and state that if P u ≥ 0 in Ω, P is subcritical in Ω and u satisfies some growth condition near infinity in Ω, then u ≥ 0 (see theorems 3.1 and 3.10). As a consequence, we prove existence and uniqueness theorems for suitable solutions of the nonhomogeneous equation (see theorems 3.3 and 3.12).

ON THE MAXIMUM AND ANTI-MAXIMUM PRINCIPLES

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Let P be a subcritical operator in Ω and let φ ∈ C(Ω) be a positive function such that φ is a solution of the equation P u = 0 in Ω∗1 which has a minimal growth in a neighborhood of infinity in Ω. We denote by B the real ordered Banach space B = {u ∈ C(Ω) | |u(x)| ≤ cφ(x) for some c > 0 and all x ∈ Ω} equipped with the norm kukB = inf{c > 0 | |u(x)| ≤ cφ(x) ∀ x ∈ Ω}. The ordering on B is the natural pointwise ordering of functions. For the purpose of spectral theory, we consider also the canonical complexification of B without changing our notation. In [10, Theorem 4.4] the following refined maximum principle for supersolutions in B is proved. Theorem 3.1. Let P be a subcritical operator in Ω and let φ ∈ C(Ω) be a positive function such that φ is a solution of the equation P u = 0 in Ω∗1 which has a minimal growth in a neighborhood of infinity in Ω. Suppose that v ∈ B satisfies the equation P v = f ≥ 0 in Ω, where f ∈ C α (Ω). Then v ≥ 0 in Ω. In the critical case we have (see [10, Proposition 4.5]) Proposition 3.2. Let P be a critical operator in Ω and let φ0 be a ground state of the operator P in Ω. Suppose that P u ≥ 0 in Ω and that for some C > O, u ≥ −Cφ0 in Ω∗1 . Then u = C1 φ0 , where C1 is a real constant. The refined maximum principle (Theorem 3.1) implies the following theorem concerning the existence, uniqueness and integral representation for solutions of the nonhomogeneous equation (see [10, Theorem 4.6]). Theorem 3.3. Let P be a subcritical operator in Ω and let φ be a positive solution of the equation P u = 0 in Ω∗1 which has a minimal growth in a neighborhood of infinity in Ω. (i) Let f ∈ C α (Ω), 0 < α ≤ 1, be a real function such that Z (3.1) GΩ P (x, y)|f (y)|dy ≤ Cφ(x) Ω

for all x ∈ Ω∗2 . Then there exists a unique solution u ∈ B of the equation P u = f in Ω. Moreover, Z u(x) = GΩ P (x, y)f (y)dy. Ω

(ii) Suppose that f ∈ C α (Ω), 0 < α ≤ 1, and f ≥ 0. Then f satisfies estimate (3.1) if and only if there exists a solution u ∈ B of the equation P u = f in Ω. In this case, u is the minimal nonnegative solution of the equation P v = f in Ω. Remark 3.4. Condition (3.1) holds true if f /φ is a semismall perturbation of P in Ω. In particular, it holds for all f ∈ B if 1 is a semismall perturbation. We present now a new version of the maximum principle which extends Theorem 3.1 and is valid for solutions which do not grow too fast. We denote by B = BK the real ordered Banach space B = BK := {f ∈ C(Ω) (3.2)

|

|f (x)| ≤ cu(x) for some fixed c > 0 and u ∈ KP (Ω), and for all x ∈ Ω}

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YEHUDA PINCHOVER

equipped with the norm kf kB :=

inf

u∈KP (Ω)

inf{c > 0 | |f (x)| ≤ cu(x) ∀ x ∈ Ω}.

The ordering on B is the natural pointwise ordering of functions. For the purpose of spectral theory, we consider also the canonical complexification of B without changing our notation. Clearly, B ⊂ B and there exists C > 0 such that kf kB ≤ Ckf kB , for all f ∈ B. Recall that φ ∈ C(Ω) is a fixed positive function such that φ is a solution of the equation P u = 0 in Ω∗1 which has a minimal growth in a neighborhood of infinity in Ω. We consider also the set K,φ B∞ := {f | f ∈ C(Ω∗N ) for some N ∈ N, and ∀² > 0 ∃ n ≥ N, C² ≥ 0, u² ∈ KP (Ω) s.t. |f (x)| ≤ ²u² (x) + C² φ(x) in Ω∗n } . K,φ Remark 3.5. (i) Note that B∞ ∩ C(Ω) is a closed subspace of B. We denote K this Banach subspace by B0 = B0 . Clearly, B ⊂ B0 . (ii) Consider the following closed subspace of B ( )

(3.3)

AK :=

f ∈ C(Ω)| lim

inf

sup {|f (x)|/u(x)} = 0

n→∞ u∈KP (Ω) x∈Ω∗

,

n

which contains functions that grow ‘slower’ than functions in K. Clearly, AK ⊂ B0 and if B ⊂ AK , then AK = B0 . There are examples where B 6⊂ AK [2]. Note that A. Ancona proved recently that if P is symmetric with respect to a Riemannian measure σ, then B ⊂ AK [2]. We first prove some simple lemmas which will imply the refined maximum principle in B0 . Lemma 3.6. Let P be a critical or subcritical operator in Ω. Suppose that K,φ v ∈ C(Ω∗2 ) ∩ B∞ is a solution of the equation P u = 0 in Ω∗2 such that v = 0 in ∂Ω2 . Then v = 0. Proof. Suppose that v(x1 ) > 0 for some x1 ∈ Ω∗2 and let ² > 0. By definition, there exist n ∈ N, C² ≥ 0 and u² ∈ KP (Ω) such that |v(x)| ≤ ²u² + C² φ in Ω∗n . By the generalized maximum principle |v(x)| ≤ ²u² (x) + C² φ(x) in Ω∗2 . Let C0,² = inf{C ≥ 0 | ²u² (x) + Cφ(x) − v(x) ≥ 0 in Ω∗2 } . It follows that ²u² (x) + C0,² φ(x) − v(x) > 0 in Ω∗2 . Since φ is a positive solution of the equation P u = 0 in Ω∗2 which has a minimal growth in a neighborhood of infinity in Ω, it follows that there exists δ = δ(²) > 0 such that ²u² (x) + C0,² φ(x) − v(x) > δφ(x) in Ω∗2 . By the minimality of C0,² , we infer that for every ² > 0 C0,² = 0. Hence, ²u² (x) − v(x) > 0 in Ω∗2 for every ² > 0, contradicting the hypothesis that v(x1 ) > 0. ¤ Lemma 3.7. Let P be a critical or subcritical operator in Ω. Suppose that K,φ v ∈ C(Ω∗2 ) ∩ B∞ is a solution of the equation P u = 0 in Ω∗2 such that v 0 on ∂Ω2 . Then v 0 and v/φ is bounded in Ω∗2 . Moreover, if v > 0 on ∂Ω2 then v is a positive solution of minimal growth in a neighborhood of infinity in Ω.

ON THE MAXIMUM AND ANTI-MAXIMUM PRINCIPLES

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Proof. Denote by wk , k > 2 the solutions of the following Dirichlet problems Pu = 0 u = v u = 0

in Ωk,2 , on ∂Ω2 , on ∂Ωk .

By the generalized maximum principle, the sequence {wk }k>2 is a nondecreasing bounded sequence of nonnegative solutions which converges to a function w ≤ C1 φ. Recall that v 6= 0 on ∂Ω2 . It follows that w is a nonzero, nonnegative solution of the equation P u = 0 in Ω∗2 (in fact, w > 0 on at least one component of Ω∗2 ). Moreover, if v > 0 on ∂Ω2 , then w is a positive solution of minimal growth in a neighborhood of infinity in Ω. Let vk , k > 2 be the solutions of the Dirichlet problems Pu = u = u =

0 0 v

in Ωk,2 , on ∂Ω2 , on ∂Ωk .

Then v = wk + vk and therefore, the sequence {vk }k>2 converges to a function v0 . K,φ On the other hand, since v ∈ B∞ , it follows that for every ² > 0 there exist n ∈ N, C² ≥ 0 and u² ∈ KP (Ω) such that |v(x)| ≤ ²u² + C² φ in Ω∗n . By the generalized maximum principle, for every k ≥ n we have |vk (x)| ≤ ²u² + C² φ in Ω∗k,2 . Hence, K,φ . Lemma 3.6 implies that v0 = 0. Hence, |v0 (x)| ≤ ²u² + C² φ in Ω∗2 and v0 ∈ B∞ ¤ v = w ≥ 0 and v is a nonnegative solution such that v/φ is bounded in Ω∗2 . Lemma 3.8. Let P be a critical or subcritical operator in Ω. Let f ∈ C α (Ω∗2 ) be K,φ is a solution of the equation a nonnegative function. Suppose that v ∈ C(Ω∗2 )∩B∞ P u = f in Ω∗2 such that v ≥ 0 on ∂Ω2 . Then v ≥ 0. Moreover, Z Ω∗ GP 2 (x, y)f (y)dy, (3.4) v(x) = h(x) + Ω∗ 2

where h ∈ C(Ω∗2 ) is a nonnegative solution of the equation P u = 0 in Ω∗2 which is bounded by Cφ (for some constant C > 0) and satisfies the boundary condition R Ω∗ h = v on ∂Ω2 . In particular, Ω∗ GP 2 (x, y)f (y)dy < ∞. 2

+

−

K,φ Proof. Since |v| = v + v , it follows that v ± ∈ B∞ . Let wk,± , k > 2 be the nonnegative solutions of the following Dirichlet problems

Pu

= f±

in Ωk,2 ,

u u

±

on ∂Ω2 , on ∂Ωk .

= v = v±

K,φ By the generalized maximum principle and the definition of B∞ it follows that for every ² > 0 there exist n ∈ N, C² ≥ 0 and u² ∈ KP (Ω) such that for every k ≥ n we have 0 ≤ wk,− (x) ≤ ²u² + C² φ in Ω∗k,2 . By a standard elliptic argument, the sequence {wk,− } has a converging subsequence to a nonnegative solution of the equation P u = 0 in Ω∗2 which takes the value zero on ∂Ω∗2 . Since any such K,φ a solution is in B∞ , it follows from Lemma 3.6 that it is the zero solution and limk→∞ wk,− = 0. On the other hand, wk,+ ≥ 0. and since

(3.5)

wk,+ − wk,− = v,

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YEHUDA PINCHOVER

it follows that limk→∞ wk,+ = v ≥ 0. Note that

Z Ω

wk,+ (x) = hk (x) + gk (x) + Ωk,2

GP k,2 (x, y)f (y)dy,

where hk satisfies Pu = 0 u = v u = 0

in Ωk,2 , on ∂Ω2 , on ∂Ωk ,

Pu = u = u =

in Ωk,2 , on ∂Ω2 , on ∂Ωk .

and gk satisfies 0 0 v

Clearly, 0 ≤ hk < Cφ, and {hk } converges to a nonnegative solution h of the equation P u = 0 in Ω∗2 which is bounded by Cφ and satisfies the boundary condition h = v on ∂Ω2 . On the other hand, for every ² > 0 there exist n ∈ N, C² ≥ 0 and u² ∈ KP (Ω) such that for every k ≥ n we have 0 ≤ gk (x) ≤ ²u² + C² φ in Ω∗k,2 . The same argument used for wk,− → 0 demonstrates now that the sequence {gk } converges to 0. R Ω Moreover, the sequence { Ωk,2 GP k,2 (x, y)f (y)dy} is a nondecreasing locally bounded sequence of nonnegative solutions of the equation P u = f . By monoR Ω∗ tone convergence, this sequence converges to Ω∗ GP 2 (x, y)f (y)dy and the lemma 2 follows. ¤ Lemma 3.9. Let P be a critical or subcritical operator in Ω and let ψ ∈ C(∂Ω2 ) be a real function. Let f ∈ C α (Ω∗2 ) be a real function such that Z Ω∗ K,φ (3.6) GP 2 (x, y)|f (y)|dy ∈ B∞ . Ω∗ 2

K,φ Then there exists a unique solution v ∈ B∞ ∩ C(Ω∗2 ) of the equation P u = f ∗ ∗ in Ω2 which satisfies v = ψ on ∂Ω2 . Moreover, Z Ω∗ v(x) = h(x) + GP 2 (x, y)f (y)dy, Ω∗ 2

where h(x) is a solution of the homogeneous equation P u = 0 in Ω∗2 which satisfies h = ψ on ∂Ω∗2 and |h(x)| ≤ Cφ(x) in Ω∗2 for some C > 0. Proof. Let h± be the unique nonnegative solution of the equation P u = 0 in Ω∗2 which is bounded by Cφ for some constant C > 0, and satisfies h± = ψ ± on ∂Ω∗2 . Consider the function Z Ω∗ v± (x) = h± (x) + GP 2 (x, y)f ± (y)dy. Ω∗ 2

K,φ Clearly, v± ∈ B∞ ∩ C(Ω∗2 ) and satisfies the equation P v± = f ± in Ω∗2 and the boundary condition v± = ψ ± on ∂Ω∗2 . Therefore, the function v = v+ − v− is a desired solution. The uniqueness follows from Lemma 3.6 ¤

ON THE MAXIMUM AND ANTI-MAXIMUM PRINCIPLES

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We now establish the refined maximum principle for solutions which do not grow too fast at infinity. Theorem 3.10. Let P be a subcritical operator in Ω. Suppose that P v = f ≥ 0 in Ω, where f ∈ C α (Ω) and v ∈ B0 . Then v ≥ 0 in Ω. Proof. Suppose that there exists x1 ∈ Ω such that v(x1 ) > 0. Then there exists a ball B² = B(x1 , ²) ⊂ Ω such that v > 0 in B² . Lemma 3.8 implies that v ≥ 0 in B²∗ := Ω \ B² and therefore, v > 0 in Ω. So, we may assume that v ≤ 0 in Ω and suppose that there exists x1 ∈ Ω such that v(x1 ) < 0. Then there exists a ball B² = B(x1 , ²) ⊂ Ω such that v < 0 in B² . Without loss of generality, we may assume that B² ⊂ Ω1 . Let u²,k be the solution of the following Dirichlet problem Pu = u = u =

0 0 v

in Ωk \ B² , on ∂B² , on ∂Ωk .

Using again the generalized maximum principle and Lemma 3.6 it follows that limk→∞ u²,k = 0. Consider also the Dirichlet problem Pu = u = u =

0 v 0

in Ωk \ B² , on ∂B² , on ∂Ωk ,

and denote its negative solution by v²,k . Set, v² := limk→∞ v²,k . Clearly, 0 < Ω −v² ≤ C² GΩ P (·, x1 ). Using the well known asymptotic behavior of GP (·, x1 ) near the pole x1 , it follows that there exists C > 0 such that for ² > 0 small enough −1 Ω 0 < −v² ≤ C²d−2 GΩ GP (·, x1 ) if d = 2. P (·, x1 ) if d ≥ 3, and 0 < −v² ≤ −C(log ²) Therefore, lim²→0 v² = 0. Finally, let w²,k be the solution of the Dirichlet problem Pu = f u = 0 u = 0

in Ωk \ B² , on ∂B² , on ∂Ωk .

Then w²,k ≥ 0. On the other hand, (3.7)

v = u²,k + v²,k + w²,k .

Letting first k → ∞ and then ² → 0 in Equation (3.7), we obtain that µ ¶ v = lim lim w²,k ≥ 0 ²→0

k→∞

contradicting the hypothesis that v ≤ 0, v 6= 0.

¤

The next proposition deals with the critical case and extends Proposition 3.2. Proposition 3.11. Let P be a critical operator in Ω and let φ0 be a ground state of the operator P in Ω. Let W ∈ C0α (Ω) be a nonzero, nonnegative function. K Suppose that P u ≥ 0 (P u ≤ 0) in Ω, where u ∈ B0 P +W . Then u = C1 φ0 , where C1 is a real constant.

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Proof. Suppose first that P u 0. Then there exists V ∈ C0α (Ω) a nonzero nonnegative function such that (P +V )u ≥ 0 in Ω. Since the cones CP +W and CP +V K ,φ are equivalent [8], it follows that u ∈ B∞P +V . Applying the refined maximum principle (Theorem 3.10) it follows that u ≥ 0. Therefore, u is a nonnegative supersolution of the critical operator P and hence u = cφ0 , where c ≥ 0. Assume now that P u = 0. If u ≥ 0, then u = cφ0 . Suppose that u(x1 ) < 0. we may assume that u < 0 in Ω1 . Let V ∈ C0α (Ω1 ) be a nonzero, nonnegative function. K ,φ Using the equivalence of the cones CP +W and CP +V , it follows that u ∈ B∞P +V . Therefore, Lemma 3.7 implies that u < 0 in Ω∗1 . Hence, −u is a global nonnegative solution of the critical operator P . Thus u = cφ0 , where c is a real constant. ¤ In the following theorem we prove existence, uniqueness and integral representation for solutions in B0 of the nonhomogeneous equation. Theorem 3.12. Let P be a subcritical operator in Ω. (i) Let f ∈ C α (Ω), 0 < α ≤ 1, be a real function such that Z (3.8) GΩ P (x, y)|f (y)|dy ∈ B0 . Ω

Then there exists a unique solution u ∈ B0 of the equation P u = f in Ω. Moreover, Z u(x) = GΩ P (x, y)f (y)dy. Ω

α

(ii) Suppose that f ∈ C (Ω), 0 < α ≤ 1, and f ≥ 0. Then Z GΩ P (x, y)f (y)dy ∈ B0 Ω

if and only if there exists a solution u ∈ B0 of the equation P u = f in Ω. In this case, u is the minimal nonnegative solution of the equation P v = f in Ω. Proof. (i) Let

Z un (x) = Ωn

n GΩ P (x, y)f (y)dy.

By the Lebesgue dominated convergence theorem and a standard elliptic argument, Z u(x) = lim un (x) = GΩ P (x, y)f (y)dy n→∞

Ω

is a solution of P u = f in Ω. It follows that u ∈ B0 . The uniqueness follows from Theorem 3.10. (ii) Suppose that u ∈ B0 is a solution of the equation P u = f ≥ 0 in Ω. Theorem 3.10 implies that u ≥ 0. Consider again the sequence Z n un (x) = GΩ P (x, y)f (y)dy. Ωn

Clearly, 0 ≤ un ≤ u in Ωn and therefore, Z Z 0 ≤ w(x) := GΩ (x, y)f (y)dy = lim P Ω

n→∞ Ω n

n GΩ P (x, y)f (y)dy ≤ u(x) ∈ B0 .

Part (i) implies now that w = u ∈ B0 . Let u ˜ ≥ 0 be a solution of the equation P v = f in Ω and consider again the sequence {un }. By the generalized maximum principle 0 ≤ un ≤ u ˜. Hence,

ON THE MAXIMUM AND ANTI-MAXIMUM PRINCIPLES

11

u(x) = limn→∞ un (x) ≤ u ˜(x) and u is the minimal nonnegative solution of the equation P v = f in Ω. ¤ Remark 3.13. Condition (3.8) holds true if f /u is a small perturbation of P in Ω for some u ∈ CP (Ω). In particular, it holds for all f ∈ B if 1 is a small perturbation. Corollary 3.14. Let P be a subcritical operator in Ω. Let f ∈ C α (Ω), 0 < α ≤ 1, be a real function such that f /u is a small perturbation of the operator P in Ω for some u ∈ CP (Ω). Then there exists a unique solution v ∈ B0 of the equation P v = f in Ω. In particular, if 1 is a small perturbation of the operator P in Ω, then for every f ∈ B there exists a unique solution v ∈ B0 of the equation P v = f in Ω. Remark 3.15. A function f ∈ C(Rd ), d ≥ 3 is said to be in Kd∞ , the Kato class at infinity, if Z |f (y)| lim sup dy = 0 . R→∞ x∈Rd |y|>R |x − y|d−2 It is well known that f ∈ Kd∞ implies that f is a small perturbation of −∆ in Rd . It follows from Corollary 3.14 that if f is a small perturbation of −∆ in Rd , then f ∈ Kd∞ . The following example demonstrates that in Corollary 3.14 we really need that f /u is a small perturbation. Recall that the function (1 + |x|)−2−² is a small perturbation of −∆ in Rd , if and only if ² > 0. Example 3.16. Let S d−1 be the unit sphere, and let g ∈ C ∞ (S d−1 ), d ≥ 3 be a nonconstant, positive function such that 1 ≤ g(ω) ≤ 2, for all ω ∈ S d−1 . Let v be a positive, smooth function in Rd such that v(x) = g(x/|x|), for all |x| ≥ 1. Denote by f = −∆v. Clearly |f (x)| ≤ C(1 + |x|)−2 and v is a solution in B of the equation −∆u = f in Rd . Note that if w is a bounded solution of the equation −∆u = f in Rd , then there exists c ∈ R such that w(x) = v(x) + c, for all x ∈ Rd . Therefore, the equation −∆u = f does not admit a solution in B0 . Remark 3.17. The results of this section continue to hold if we suppose that f ∈ C(Ω) and we consider strong solutions. This follows by a standard approximation argument. We use this observation in the next sections. 4. Anti-maximum principle and semismall perturbations In this section we extend the anti-maximum principle proved in [10, Theorem 5.3] which deals with supersolutions (f ∈ B, f ≥ 0) to a more general case. Throughout this section we assume that P is subcritical in Ω and 1 is a semismall perturbation of P ∗ in Ω. Denote by λ0 = λ0 (1, P, Ω) and φ0 the generalized principal eigenvalue and the corresponding normalized ground state of the operator P in Ω. So, (P − λ0 )φ0 = 0 in Ω, λ0 > 0 and φ0 (x0 ) = 1. We need the following Theorem (see [10, Theorem 5.1]) Theorem 4.1. Let P be a subcritical operator in Ω and assume that 1 is a semismall perturbation of the operator P ∗ in Ω. Consider the integral operator Z Gf (x) = GΩ P (x, y)f (y)dy. Ω

12

YEHUDA PINCHOVER

Then G is a compact positive operator on B and λ−1 0 ∈ σ(G) is the principal eigenvalue of G with a principal eigenfunction φ0 . Moreover, λ−1 is simple and is also the principal eigenvalue of the compact 0 operator G ∗ with a unique principal eigenfunction φ˜0 satisfying hφ˜0 , φ0 i = 1. Our main result of this section is the following anti-maximum principle for solutions in B. It extends Theorem 5.3 in [10]. Theorem 4.2. Let P be a subcritical operator in Ω and assume that 1 is a semismall perturbation of the operator P ∗ in Ω. Denote by λ0 = λ0 (1, P, Ω) and φ0 the generalized principal eigenvalue and the corresponding normalized ground state of the operator P . Let φ˜0 be the unique principal eigenfunction of the operator G ∗ satisfying hφ˜0 , φ0 i = 1. Suppose that f ∈ C α (Ω) satisfies Z Z Ω ˜ GP (x, y)|f (y)|dy ∈ B, and hφ0 , GΩ P (·, y)f (y)dyi > 0 . Ω

Ω

Then there exists ²0 > 0 (which may depend on f ) such that for every λ ∈ (λ0 , λ0 + ²0 ), any solution u ∈ B of the equation (P − λ)u = f satisfies u ≤ −Cφ0 < 0, where C is a positive constant depending on f and λ. In particular, if f ∈ B and hφ˜0 , f i > 0 then the above anti-maximum principle holds true. Proof. Let f satisfy the assumptions of the theorem and denote Z h(x) := GΩ P (x, y)f (y)dy ∈ B. Ω

−1

Set µ0 = (λ0 )

. Suppose that u ∈ B satisfies the differential equation

(4.1)

(P − λ)u = f

in Ω.

Note that GP u is a well defined function in B. By Equation (4.1) and Theorem 3.3, we have P (GP u) = λP (Gu) + P (Gf ) = λu + f = P u. It follows from Theorem 3.1 that GP u = u. Therefore, u satisfies the equation (4.2)

(G − µI)u = −µh, −1

where µ = (λ) . On the other hand, by the compactness of G, there exists ²0 > 0 such that (µ0 − ²0 , µ0 ) ∩ σ(G) = ∅. Therefore, for every µ ∈ (µ0 − ²0 , µ0 ) there exists a unique solution uµ ∈ B of Equation (4.2). Moreover, using theorems 3.1 and 3.3 and elliptic regularity, it follows that uµ is the unique solution in B of Equation (4.1). Note also that hφ˜0 , hi = hφ˜0 , Gf i > 0. Therefore, it is enough to show that for every h ∈ B satisfying hφ˜0 , hi > 0 there exists 0 < ² ≤ ²0 such that for every µ ∈ (µ0 − ², µ0 ) the equation (G − µI)u = µh admits a positive solution vµ ∈ B, vµ ≈ φ0 . We first claim that B admits the following topological direct decomposition B = span {φ0 } ⊕ R := span {φ0 } ⊕ R(G − µ0 I) . By Theorem 4.1, the operator G is compact, therefore, the operator G − µ0 I is a Fredholm operator in B with index 0. Hence, R is a closed subspace and codim R = dim N (G − µ0 I) = 1 .

ON THE MAXIMUM AND ANTI-MAXIMUM PRINCIPLES

13

Therefore, R is complemented in B. It is thus enough to prove that φ0 6∈ R. Suppose to the contrary that (4.3)

(G − µ0 I)u = φ0

for some u ∈ B.

Then v := −u is a classical solution of the equation (P − λ0 )v = (λ0 )2 φ0 ≥ 0. It follows from Proposition 3.2 that u = C1 φ0 which contradicts Equation (4.3). In particular, we see that the algebraic multiplicity of µ0 equals 1. Let Q be the projection operator in B such that QB = span {φ0 } and (I − Q)B = R. We claim that for every f ∈ B we have Qf = hφ˜0 , f iφ0 . Let f ∈ B and decompose f as f = αφ0 + f1 , where f1 ∈ R and α is a scalar. Let u1 ∈ B such that (4.4)

(G − µ0 I)u1 = f1 .

Applying φ˜0 on both sides of Equation (4.4), we obtain that hφ˜0 , f1 i = 0. Therefore, hφ˜0 , f i = hφ˜0 , αφ0 + f1 i = α and the claim is proved. Let µ ∈ (µ0 − ²0 , µ0 ) and consider a solution vµ ∈ B of the equation (4.5)

(G − µI)vµ = µh ,

where h ∈ B satisfies hφ˜0 , hi > 0. Writing the unique decompositions of h and vµ as h = vµ

=

hφ˜0 , hiφ0 + h1 , hφ˜0 , vµ iφ0 + wµ := βµ φ0 + wµ ,

where h1 , wµ ∈ R, we obtain that (G − µI)vµ = (G − µ0 I)βµ φ0 + (µ0 − µ)βµ φ0 + (G − µI)wµ = (µ0 − µ)βµ φ0 + (G − µ0 I)wµ + (µ0 − µ)wµ = µ(hφ˜0 , hiφ0 + h1 ). By the unique decomposition we have, βµ = µ(µ0 − µ)−1 hφ˜0 , hi , and (G − µI)wµ = µh1 . Since the restriction of the operator (G − µ0 I) to the closed subspace R is an isomorphism on R, it follows that there exists 0 < δ < µ0 /2 such that the operator (G − µI) is invertible in R, for every |µ − µ0 | < δ. Moreover, for |µ − µ0 | < δ the operators (G − µI) have uniformly bounded inverse operators in R. It follows that there exists a positive constant M such that kwµ kB ≤ M , for all µ ∈ (µ0 − δ, µ0 ). In other words, |wµ (x)| ≤ M φ0 (x), for all x ∈ Ω and all µ ∈ (µ0 − δ, µ0 ). Therefore, if 0 < µ0 − µ < min{δ, (2M )−1 µ0 hφ˜0 , hi}, then for some C > 0 vµ (x) = βµ φ0 + wµ ≥ µ(µ0 − µ)−1 hφ˜0 , hiφ0 (x) − M φ0 (x) ≥ Cφ0 (x). ¤ Remark 4.3. (i) Let vµ be the function defined by Equation (4.5). Then uλ = −v1/µ is the unique solution in B of the equation (P − λ)u = f . It follows that lim k(λ0 − λ)uλ − hφ˜0 , Gf iφ0 (x)kB = lim k(λ − λ0 )w1/λ kB = 0.

λ→λ0

λ→λ0

14

YEHUDA PINCHOVER

(ii) Suppose that µ1 = max{|µ| : µ ∈ σ(G), |µ| < µ0 } is a real eigenvalue of G with an eigenfunction φ1 . Set λ1 = 1/µ1 and f = φ0 . Then ²0 = ²0 (φ0 ) of Theorem 4.2 equals to the spectral gap λ1 − λ0 . On the other hand, in the selfadjoint case, φ˜0 = φ0 and hφ0 , φ1 i = 0. Set fn := (n)−2 φ0 + (λ1 − λ0 − 1/n)φ1 , and un = −(n)−1 φ0 + φ1 . Then hφ0 , fn i > 0, the function un changes its sign for sufficiently large n, and satisfies the equation (P − λ0 − 1/n)un = fn . Thus, ²0 (fn ) < 1/n and limn→∞ ²0 (fn ) = 0. 5. Weak anti-maximum principle and small perturbations In Section 4, we proved an anti-maximum principle for functions in B. The main aim of this section is to prove an anti-maximum principle for solutions uλ ∈ B0 of the equation (5.1)

(P − λ)u = f 0

in Ω,

λ0 < λ.

In this space we prove a weaker version of the anti-maximum principle. Namely, for every compact K ⊂ Ω and 0  Gf ∈ B0 , there exists ² > 0 which may depend on K and f such that uλ < 0 in K, for all λ ∈ (λ0 , λ0 + ²). Throughout this section we assume that P is subcritical in Ω and 1 is a small perturbation of P in Ω. Denote by λ0 and φ0 the generalized principal eigenvalue and the corresponding normalized ground state of the operator P in Ω. We first prove the compactness of the operator G in B0 . Theorem 5.1. Let P be a subcritical operator in Ω and suppose that 1 is a small perturbation of the operator P in Ω. Consider the integral operator Z GB f (x) := GΩ P (x, y)f (y)dy. Ω

Then GB is a compact positive operator on B0 (and also from B into B0 ). Furthermore, λ−1 0 ∈ σ(GB ) is the principal eigenvalue of GB with a principal eigenfunction φ0 . Moreover, λ−1 is simple and is also the principal eigenvalue of the compact 0 operator (GB )∗ with a unique principal eigenfunction φ˜0 satisfying hφ˜0 , φ0 i = 1. Proof. It follows from Corollary 3.14 that GB : B −→ B0 . Let {fn } be a bounded sequence in B. Since 1 is a small perturbation of P , it follows from Corollary 3.14 that the function un (x) := GB fn (x) is well defined, un ∈ B0 and un satisfies Z |un (x)| ≤ Ω

GΩ P (x, y)|fn (y)|dy ≤ Chn (x),

where hn ∈ K, and C is a positive constant independent of n. Since un ∈ B0 , Theorem 3.12 and Remark 3.17 imply that un is the unique function in B0 which is a solution of the equation P u = fn in Ω. By a standard elliptic argument, it follows that the sequence {un } admits a subsequence which converges in the compact open topology to a function u (for simplicity, we denote this subsequence as {un }). For any given ² > 0 there exists k0 such that for any k ≥ k0 and n, m ∈ N Z GΩ P (x, y)|fn (y) − fm (y)|dy Z (5.2)

≤ 2C Ω∗ k

Ω∗ k

GΩ P (x, y)(hn (y) + hm (y))dy < ²(hn (x) + hm (x))

ON THE MAXIMUM AND ANTI-MAXIMUM PRINCIPLES

15

for all x ∈ Ω∗k . Let C > 0 be a constant such that φ ≤ Cv for some v ∈ K, and let δ = ²/C. Take n0 ∈ N such that |un (x) − um (x)| ≤ δφ(x) on Ωk0 , for all n, m ≥ n0 . Lemma 3.9 and Estimate (5.2) imply that |un (x) − um (x)| ≤ ²v(x) + ²(hn (x) + hm (x)) in Ω∗k0 . Therefore, kun − um kB < 3², for all n, m ≥ n0 . Thus, {un } is a converging sequence in B. The other statements of the theorem follows from the Krein-Rutman theorem and Theorem 1.5 in [7]. ¤ It is now possible to imitate the proof of the anti-maximum principle (Theorem 4.2) and to extend the results obtained in Section 4 for Gf, u ∈ B to the case where Gf, u ∈ B0 . We have the following weak anti-maximum principle. Theorem 5.2. Let P be a subcritical operator in Ω and assume that 1 is a small perturbation of the operator P in Ω. Denote by λ0 and φ0 the generalized principal eigenvalue and the corresponding normalized ground state of the operator P . Let φ˜0 be the unique principal eigenfunction of the operator (GB )∗ satisfying hφ˜0 , φ0 i = 1. Suppose that f ∈ C α (Ω) satisfies Z Z ˜0 , GΩ (x, y)|f (y)|dy ∈ B , and h φ GΩ 0 P P (·, y)f (y)dyi > 0 . Ω

Ω

Then for every compact set K ⊂ Ω there exists ²0 > 0 (which may depend on f and K) such that for every λ ∈ (λ0 , λ0 + ²0 ), if u ∈ B0 is a solution of the equation (P − λ)u = f , then u < 0 in K. In particular, if f ∈ B0 and hφ˜0 , f i > 0, then the above weak anti-maximum principle holds true. Proof. Since the proof of the theorem is similar to the proof of Theorem 4.2, we only indicate the main steps without proofs. Observe that Theorem 5.1 and Proposition 3.11 replace Theorem 4.1 and Proposition 3.2, respectively. R Step 1: The equation (P − λ)u = f , where Ω GΩ P (·, y)f (y)dy, u ∈ B0 is equivalent to the equation (G − µI)u = −µh, where µ = (λ)−1 and B R Ω h(x) := Ω GP (x, y)f (y)dy ∈ B0 . Step 2: Let µ0 = (λ0 )−1 , then B0 admits the topological direct decomposition B0 = span {φ0 } ⊕ R := span {φ0 } ⊕ R(GB − µ0 I) . Step 3: The projection operator Q in B0 such that QB0 = span {φ0 } and (I − Q)B0 = R is given by Qf = hφ˜0 , f iφ0 . Step 4: Suppose that B0 3 vµ = hφ˜0 , vµ iφ0 + wµ := βµ φ0 + wµ is the solution of the equation (GB − µI)vµ = µh , where B0 3 h = hφ˜0 , hiφ0 + h1 ,

16

YEHUDA PINCHOVER

h1 , wµ ∈ R, µ ∈ (µ0 − ², µ0 ) and ² > 0 is small enough. Then βµ = µ(µ0 − µ)−1 hφ˜0 , hi ,

and (GB − µI)wµ = µh1 .

Step 5: There exist positive constants δ < ² and M such that kwµ kB ≤ M , for all µ ∈ (µ0 − δ, µ0 ). Step 6: Let K ⊂⊂ Ω. There exist δ1 , C > 0 such that for all x ∈ K and 0 < µ0 −µ < δ1 , we have vµ (x) = µ(µ0 −µ)−1 hφ˜0 , hiφ0 (x)+wµ ≥ Cφ0 (x) > 0. ¤ Example 5.3 (Smooth bounded domain). Let Ω ⊂ Rd , d ≥ 3 be a bounded C domain, and let P be a uniformly elliptic subcritical operator with (up to the boundary) smooth coefficients. Denote by δ(x) the distance function to the boundary. Then 1 is a small perturbation [1], and (see [13, 5]), 1,1

GΩ P (x, y) ≈

δ(x)δ(y) . |x − y|d−2 (|x − y|2 + δ(x)2 + δ(y)2 )

If f ∈ Lp (Ω) ∩ C(Ω), p > d, then by the H¨older inequality |Gf (x)| ≤ Cd(x) in Ω for some C > 0. Consequently, Gf ∈ B and the anti-maximum holds for f ∈ Lp (Ω), p > d. Thus, our anti-maximum principle extends [4] for the Dirichlet problem. Note that p > d is sharp, since G. Sweers has recently shown that the anti-maximum principle does not hold true for f ∈ Ld (Ω) ∩ C(Ω) [12]. Example 5.4 (Nonsmooth bounded domain). Let P = −∆ and Ω = (0, π) × (0, π) ⊂ R2 . It follows from [1, Corollary 1.2] that 1 is a small perturbation of −∆ in Ω. Clearly, λ0 = 2 and φ0 (x, y) = sin x sin y. Consider the function σ(x, y) = sin x sin y log(x2 + y 2 ), and let φ := −∆σ. Then φ is bounded in Ω and σ = Gφ ∈ B0 \ B. It follows that 1 ∈ B \ B and G1 ∈ B0 \ B. Moreover, I. Birindelli proved that the anti-maximum does not hold in Ω for P = −∆, and f = 1 [3, Proposition 3.2]. On the other hand, by Theorem 5.2 the weak anti-maximum does hold. References [1] A. Ancona, First eigenvalues and Comparison of Green’s functions for elliptic operators on manifolds or domains, J. Analyse. Math. 72 (1997), 45–92. [2] A. Ancona, Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators, preprint, August 1999. [3] I. Birindelli, Hopf ’s lemma and anti-maximum principle in general domains, J. Differential Equations 119 (1995), 450–472. [4] Ph. Cl´ ement, and L. A. Peletier, An anti-maximum principle for second-order elliptic operators. J. Differential Equations 34 (1979), 218–229. [5] N. J. Kalton, and I. E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc. 351 (1999), 3441–3497. [6] M. Murata, Structure of positive solutions to (−∆ + V )u = 0 in IRn , Duke Math. J. 53 (1986), 869–943. [7] M. Murata, Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math. 102 (1997), 29–60. [8] Y. Pinchover, On positive solutions of second-order elliptic equations, stability results and classification, Duke Math. J. 57 (1988), 955–980. [9] Y. Pinchover, Criticality and ground states for second-order elliptic equations, J. Differential Equations 80 (1989), 237–250. [10] Y. Pinchover, Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann. 314 (1999), 555–590.

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17

[11] Y. Pinchover, Anti-maximum principles for indefinite-weight elliptic problems, preprint, August 1999. [12] G. Sweers, Ln is sharp for the anti-maximum principle, J. Differential Equations 134 (1997), 148–153. [13] Z. Zhao, Green function for Schr¨ odinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), 309–334. Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, ISRAEL E-mail address: [email protected]