Reduced measures for semilinear elliptic equations involving Dirichlet operators

We consider elliptic equations of the form (E) -Au=f(x,u)+μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-Au=f(x,u)+\mu $$\end{document}, where A is a negative definite self-adjoint Dirichlet operator, f is a function which is continuous and nonincreasing with respect to u and μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where A=Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=\Delta $$\end{document} and show basic properties of solutions of (E). We also prove Kato’s type inequality. Finally, we characterize the set of good measures in case f(u)=-up\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(u)=-u^p$$\end{document} for some p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}.


Introduction
Let E be a separable locally compact metric space and let m be a Radon measure on E such that supp[m] = E. In the present paper we study semilinear equations of the form − Au = f (x, u) + μ, (1.1) where μ is a Borel measure on E, f : E × R → R is a measurable function such that f (·, u) = 0, u ≤ 0, and f is nonincreasing and continuous with respect to u. As for the operator A, we assume that it is a negative definite self-adjoint Dirichlet operator on L 2 (E; m). Saying that A is a Dirichlet operator we mean that (Au, (u − 1) + ) ≤ 0, u ∈ D(A).

D(A) ⊂ D[E], E(u, v) = (−Au, v), u ∈ D(A), v ∈ D[E]
(1.2) (see [12,23]) The class of such operators is quite large. It contains many local as well as nonlocal operators. The model examples are Laplace operator (or uniformly elliptic divergence form operator) and the fractional Laplacian α with α ∈ (0, 1). Many other examples are to be found in [12,23]. Let Cap denote the capacity determined by (E, D [E]) (see Sect. 2). It is known (see [13]) that any Borel signed measure μ on E admits a decomposition μ = μ c + μ d into the singular (concentrated) part μ c with respect to Cap and the absolutely continuous (diffuse, smooth) part μ d with respect to Cap. The smooth part μ d is fully characterized in [20].
The study of semilinear equations of the form (1.1) in case μ is smooth, i.e. when μ c = 0, goes back to the papers by Brezis and Strauss [7] and Konishi [21] In [7,21] the existence of a solution of (1.1) is proved for μ ∈ L 1 (E; m). At present existence, uniqueness and regularity results are available for equation (1.1) involving general bounded smooth measure μ and operator corresponding to Dirichlet form (see Klimsiak and Rozkosz [17] for the case of symmetric regular Dirichlet form and [19] for the case of quasi-regular, possibly non-symmetric Dirichlet form). The case μ c = 0 is much more involved. Ph. Bénilan and H. Brezis [2] has observed that in such a case equation (1.1) need not have a solution even if A = . In [5] (see also [4]) H. Brezis, M. Marcus and A.C. Ponce introduced the concept of good measure, i.e. a bounded measure for which (1.1) has a solution, and the concept of reduced measure, i.e. the largest good measure, which is less then or equal to μ. In case A = these concepts are by now quite well investigated (see [2,5]). The situation is entirely different in case of more general local operators or nonlocal operators. There are known, however, some existence and uniqueness results for (1.1) in case A is a diffusion operator (see Véron [28]) and in case A = α with α ∈ (0, 1) (see Chen and Véron [8]).
The main purpose of the paper is to present a new approach to (1.1) that provides a unified way of treating (1.1) for the whole class of negative defined self-adjoint Dirichlet operators A and for μ from some class of measures M including the class M b of bounded signed Borel measures on E. In particular, we give a new definition of a solution of (1.1) and investigate the structure of good and reduced measures relative to (1.1). In case A = our definition is equivalent to the definition of a solution adopted in [2,5], so our results generalize the results of [2,5] to wide class of operators. In fact, they generalize the existing results even in case A = , because in this case M b M and M contains important in applications unbounded measures. The second purpose of our paper is to give a probabilistic interpretation for solutions of (1.1).
First, some remarks concerning our definition of a solution and the class M are in order. Suppose we want to consider problem (1.1) for some class of measures M including L 1 (E; m). Considering f ≡ 0 in (1.1) we see that then G := −A −1 should be well defined on L 1 (E; m), i.e. the following condition should be satisfied: T t g dt < ∞, m-a.e., g ∈ L 1,+ (E; m). (1.3) Condition (1.3) is nothing but the statement that the semigroup {T t , t ≥ 0} generated by A (or, equivalently, the Dirichlet form (E, D[E])) is transient (see [12,Section 1.5]). It is well known that then there exists a kernel {R(x, dy), x ∈ E} such that for every g ∈ L 1,+ (E; m), E g(y)R(·, dy) = Gg, m-a.e.
If u is a solution of (1.1) with f ≡ 0 then Therefore R • μ must be absolutely continuous with respect to the measure m for every bounded Borel measure μ. This condition is known in the literature as the Meyer hypothesis (L) (see [3]) or the condition of absolute continuity of the resolvent {G α , α > 0} (see [12]). For the reasons explained above in the paper we assume that {T t , t ≥ 0} is transient and hypothesis (L) is satisfied. It is known that under these assumptions there exists a Borel function r : Using the kernel r we can give our first, purely analytical definition of a solution of (1.1). Namely, we say that a Borel function u on E is a solution of (1.1) if . We also show that in case μ ∈ M b and A is a uniformly elliptic divergence form operator on a bounded domain in R d definition (1.4) is equivalent to Stampacchia's definition by duality (see [27]). Unfortunately, definition (1.4) is rather inconvenient for studying (1.1). One of the main results of the paper says that (1.4) is equivalent to our second, probabilistic in nature definition of a solution. At first glance the probabilistic definition seems to be more complicated than (1.4), but as a matter of fact suits much better to the purposes of the present paper. Let be a Hunt process with life time ζ associated with the form (E, D[E]). We say that u is a probabilistic solution of (1.1) if (a) f (·, u) · m ∈ M and there exists a local martingale additive functional M of X such that for quasi every (q.e. for short) x ∈ E (Here A μ d denotes a continuous additive functional of X of finite variation in the Revuz correspondence with μ d ), (b) for every polar set N ⊂ E, every stopping time T ≥ ζ and every sequence of stopping times {τ k } such that τ k T and E x sup t≤τ k |u(X t )| < ∞ for x ∈ E\N and k ≥ 1 we have where E x denotes the integration with respect to probability P x and The above probabilistic definition allows us to develop a general theory of equations of the form (1.1). Moreover, in our opinion, the theory based on the probabilistic definition is elegant and simple.
We first prove some regularity results. We show that if u is a solution of (1.1) and and is an extension of the domain of the form E such that the pair (E, D e [E]) is a Hilbert space (see [12]). We also prove Stampacchia's type inequality which says that for every strictly positive excessive function ρ (for ρ ≡ 1 for instance) and We next study the structure of the set G of good measures and the set of reduced measures relative to A, f . Let us recall that the reduced measure is the largest measure μ * ∈ M such that μ * ≤ μ and there exists a solution of (1.1) with μ replaced by μ * . A measure μ ∈ M is good, if μ * = μ. By results of [17,19], if μ c = 0, then μ is good. In the present paper we first show that μ − μ * ⊥Cap.
Then we show that, as in the case of Laplace operator, the set G is convex and closed under the operation of taking maximum of two measures. We also show that μ ∈ G if and only if μ = g − Av for some functions g, v on E such that g · m, f (·, v) · m ∈ M and Av ∈ M. From this characterization of G we deduce that for every strictly positive excessive function ρ, We also show that under some additional assumption on the growth of f (it is satisfied for instance if | f (x, u)| ≤ c 1 + c 2 e u 2 ), for every strictly positive excessive function ρ, where the closure is taken in the space (M ρ , · T V,ρ ).
In Sect. 6 we prove the so-called inverse maximum principle and Kato's type inequality. In our context Kato's inequality says that if u is a solution of (1.1) then Au + ∈ M and This form of Kato's inequality for Laplace operator was proved by H. Brezis and A.C. Ponce in [6].
In the last section we study the set of good measures G for problem (1.1) with f having at most polynomial growth, i.e. for f satisfying for some p > 1. For this purpose, we introduce a new capacity Cap A, p , which in the special case, when A = α on an open bounded set D ⊂ R d with zero boundary condition is equivalent to the Bessel capacity defined as for compact sets K ⊂ D. We prove that if μ ∈ M and μ + is absolutely continuous with respect to Cap A, p , where p denotes the Hölder conjugate to p, then a solution of (1.1) exists, i.e. μ ∈ G. For f of the form we fully characterize the set G. Namely, we prove that the absolute continuity of μ + with respect to Cap A, p is also necessary for the existence of a solution of (1.1). Thus, in case f is given by (1.6), Moreover, where μ + Cap A, p denotes the absolutely continuous part of μ + with respect to Cap A, p .

Preliminaries
In the paper E is a locally compact separable metric space and m is a positive Radon measure on E such that supp[m] = E. By (E, D[E]) we denote a symmetric regular Dirichlet form on L 2 (E; m) (see [12] or [23] for the definitions). We will always assume that (E, there exists a strictly positive function g on E such that By Riesz's theorem, for every α > 0 and f ∈ L 2 (E; m) there exists a unique function It is an elementary check that {G α , α > 0} is a strongly continuous contraction resolvent on L 2 (E; m). By {T t , t ≥ 0} we denote the associated semigroup and by (A, D(A)) the self-adjoint negative definite Dirichlet operator generated by {T t }. It is well known that A satisfies (1.2) (see [12,Section 1.3]). Conversely, one can prove (see [23, page 39 By u · μ w denote the Borel measure on E defined as whenever the integrals exist.

With a regular symmetric Dirichlet form (E, D[E]) one can associate uniquely a symmetric Hunt process
where E x stands for the expectation with respect to the measure P x . For α, t ≥ 0 and Observe that for α, t > 0 and f ∈ L 2 (E; m), For simplicity we denote R 0 by R. We say that some function on E is measurable if it is universally measurable, i.e. measurable with respect to the σ -algebra By T we denote the set of all stopping times with respect to the filtration (F t ) t≥0 and by D the set of all measurable functions u on E for which the family is uniformly integrable with respect to the measure P x for q.e. x ∈ E.
For a Borel measure μ on E and α ≥ 0 we denote by μ • R α the measure defined as and by P μ we denote the measure In the whole paper we assume that m is the reference measure for X, i.e. for all x ∈ E and It is also clear that by symmetry of X, r α (x, y) = r α (y, x) for x, y ∈ E, α ≥ 0. In what follows we put r (x, y) = r 0 (x, y), x, y ∈ E. Thanks to the existence of r α we may define R α μ for arbitrary positive Borel measure μ by putting It is well known (see [12,Section 5.1] and [3, Theorem V.2.1] that for each μ ∈ S there exists a unique perfect positive continuous additive functional A μ in the Revuz duality with μ, and moreover,

Linear equations
In this section we give some definitions of a solution of the linear problem The class of such measures will be denoted by M.
In the whole paper we adopt the convention that E r (x, y) dμ(y) = 0 for every Borel

Solutions defined via the resolvent kernel and regularity results
Definition 3. 1 We say that a measurable numerical function u on E is a solution of (3.1) if Let us note that by [3, Proposition V.1.4], if the above equality holds for every x ∈ E, then u is Borel measurable. Since μ ∈ M, u is finite q.e.

Proposition 3.2 M b ⊂ M.
Proof Since the form E is assumed to be transient, there exists a strictly positive Borel function f on E such that R f < ∞, q.e. From this we conclude that f · m is a smooth measure. Hence, by [12,Theorem 2.2.4], there exists an increasing sequence {F n } of closed subsets of E such that n≥1 F n = E, q.e. and sup x∈E R(1 F n f )(x) < ∞ (see also comments following [12, Corollary 2.2.2]). As a matter of fact, in [12] in the last condition sup is replaced by ess sup with respect to m, however in view of [3, Proposition II.3.2], it holds true also with supremum norm. We have Hence R|μ| is finite q.e., i.e. μ ∈ M.
Using Definition 3.1 we can easily prove some regularity result for solutions of (3.1). For this purpose, for k ≥ 0 set

Theorem 3.3 Let μ ∈ M b (E) and let u be a solution of (3.1). Then T k (u) ∈ D e [E]
and for every k ≥ 0, whenever the integral exists. By the definition of a solution of (3.1), u · m = μ • R. Hence On the other hand, since α R α is Markovian, we have Consequently,

Probabilistic solutions
In this subsection we give an equivalent definition of solution of (3.1) using stochastic equations involving a Hunt process X associated with the Dirichlet operator A. We begin with the following lemma.
Proof Since μ, ν ∈ M, there exists a strictly positive Borel function ψ on E such that Hence, by the Lebesgue dominated convergence theorem, which completes the proof. Theorem 3.6 Assume that μ ∈ M + and μ⊥Cap. Then u = Rμ is quasi-continuous and the process [0, ∞) t → u(X t ) is a cádlág local martingale under the measure P x for q.e.
x ∈ E.
By the Markov property, for every t ≥ 0 and x ∈ E we have and let N α,x denote a cádlág modification of the martingaleN α,x . Then for every x ∈ E, By the integration by parts formula applied to the processes e αt and e −αt u α (X ) we get Since u is an excessive function, A α is an increasing process and Let ζ i , ζ p denote the totally inaccessible and the predictable part of ζ , respectively. From [12,Theorem 4.2.2] it also follows that while by the fact that u α , u are potentials, By what has already been proved, By the generalized Dini theorem (see [10, p. 185]), u α (X ) u(X ) uniformly on compact subsets of [0, ∞). Observe that for every t ≥ 0 and q.e. x ∈ E, Hence u(X ) is a supermartingale and lim t→∞ E x u(X t ) < ∞. Therefore by [25,Theorem III.13], for q.e. x ∈ E there exists an increasing predictable process C x with E x C x ζ < ∞ and a cádlág local martingale M x such that Since the filtration is quasi-left continuous, M x has no predictable jumps. Since X is quasileft continuous, it also has no predictable jumps, which implies that u(X ) has no predictable jumps, because u is quasi-continuous. Thus C x is continuous. Since u(X ) is a special semimartingale, there exists a localizing sequence {τ x n } ⊂ T such that for every n ≥ 1, By [16,Proposition 3.2], {u(X ) − u α (X )} satisfies the so-called condition UT. Therefore by [16,Corollary 2.8 In fact, by (3.2), for every n ≥ 1 we have By [12,Lemma A.3.3] there exists a process A such that A = C x for q.e. x ∈ E. Of course, A is a positive continuous additive functional. Putting Observe that by the resolvent identity, for every α ≥ 0 we have On the other hand, by (3.2) and the integration by parts formula applied to the processes e −αt and u(X t ), It is clear that as k → ∞. From this, (3.4) and (3.5) we conclude that for q.e x ∈ E, By this and [3, Proposition II.3.2], R α (μ − ν) ≥ 0. Since α ≥ 0 was arbitrary, applying Lemma 3.5 shows that μ ≥ ν. Since μ⊥Cap, it follows that ν ≡ 0 or, equivalently, that A ν ≡ 0. Therefore from (3.3) it follows that u(X ) is a local martingale.
Let us recall that a process M is called a local martingale additive functional (MAF) if it is an additive functional and M is an (F, P x )-local martingale for q.e. x ∈ E. Theorem 3.7 Assume that μ ∈ M + and let u = Rμ. Then u is quasi-continuous and there exists a local MAF M such that for q.e. x ∈ E. Moreover, for every polar set N ⊂ E, every stopping time T ≥ ζ and sequence {τ k } ⊂ T such that τ k T and E x sup t≤τ k u(X t ) < ∞ for x ∈ E\N and k ≥ 1 we have lim Proof Let w = Rμ c and v = Rμ d . It is well known (see [17,Lemma4.3]) that v is quasicontinuous and that there exists a uniformly integrable MAF for q.e. x ∈ E. By Theorem 3.6, w is quasi-continuous and there exists a local MAF M w such that for q.e. x ∈ E. Let N ⊂ E be a polar set such that (3.8), (3.9) hold for x ∈ E\N . Let {τ k } be as in the formulation of the theorem. Then M v,τ k , M w,τ k are both uniformly integrable and by (3.8) and (3.9), Letting k → ∞ in the above equation yields which proves (3.7). Adding (3.8) to (3.9) gives (3.6).

Remark 3.8
Under the assumptions of Theorem 3.7, for every α > 0, To see this we use (3.4) and arguments following it.
We are now ready to introduce the second definition of a solution of (3.1) making use of the Hunt process X associated with operator A. Solutions of (3.1) in the sense of this definition will be called probabilistic solutions or simply solutions, because we will show that our second definition is equivalent to the definition via the resolvent kernel. Definition 3. 9 We say that a measurable numerical function u on E is a probabilistic solution of (3.1) if (a) there exists a local MAF M such that for q.e. x ∈ E, for every polar set N ⊂ E, every stopping time T ≥ ζ and every sequence {τ k } ⊂ T such that τ k T and E x sup t≤τ k |u(X t )| < ∞ for every x ∈ E\N and k ≥ 1 we have Any sequence {τ k } with the properties listed in (b) will be called the reducing sequence for u, and we will say that {τ k } reduces u.

Remark 3.10
Since u(X ) in the above definition is a special semimartingale, there exists at least one reducing sequence {τ k } for u. In fact, the stopping times defined as form a reducing sequence (see the reasoning in the proof of [26, Theorem 51.1]).

Remark 3.11
If μ is a smooth measure then Definition 3.9 reduces to the definition of a solution introduced in [17]. Indeed, by condition (a), x ∈ E. Therefore letting k → ∞ and using (b) we see that for q.e. x ∈ E, Note that if A is a uniformly elliptic divergence form operator then by [17,Proposition 5.3], u is also a solution of (3.1) in the sense of Stampacchia (see [27]). In the sequel we will show that this holds true for general Borel measures and wider class of operators.

Proposition 3.12 A measurable function u on E is a probabilistic solution of (3.1)if and only if it is a solution of (3.1) in the sense of Definition 3.1.
Proof Assume that u is a solution of (3.1) in the sense of Definition 3.1. Then by Theorem 3.7, u is a probabilistic solution. Now suppose that u is a probabilistic solution of (3.1). Then using (a) and (b) of the definition of a probabilistic solution of (3.1) we obtain for q.e. x ∈ E.

Semilinear equations
In what follows μ ∈ M and f : E × R → R is a function satisfying the following conditions: In this section we consider semilinear equation of the form Definition 4. 1 We say that a measurable numerical function u on E is a solution of (4.1) if f (·, u) · m ∈ M and u is a solution of (3.1) with μ replaced by f (·, u) · m + μ.
Proof Let {τ k } be a common reducing sequence for u 1 and u 2 . We assume that f 1 satisfies (H1). By the Tanaka-Meyer formula (see [25, Theorem IV.66]), for every k ≥ 1, for q.e. x ∈ E. From the assumption μ 1 ≤ μ 2 and properties of the Revuz duality it follows s. for q.e. x ∈ E. By (H1) and the assumptions on f 1 and f 2 , for q.e. x ∈ E, which proves the proposition.

Proposition 4.4
Let u 1 , u 2 be solutions of (4.1) with μ 1 ∈ M and μ 2 ∈ M, respectively. If f satisfies (H1), then Proof Let {τ k } be a common reducing sequence for u 1 and u 2 . By the Tanaka-Meyer formula, for q.e. x ∈ E. By (H1) the second term on the right-hand side of (4.2) is nonpositive. Therefore from (4.2) it follows that for q.e. x ∈ E (see the reasoning at the end of the proof of Proposition 4.2). From this and [3, Proposition II.3.2] we get the desired result.

Proposition 4.5
Let u be a solution of (4.1) with f satisfying (H1), (H3). Then Proof We apply Proposition 4.4 to u 1 = u, u 2 = 0, , because by [22] there exists c, C > 0 such that In the rest of the paper we assume that ρ ∈ S and ρ is strictly positive. Proposition 4.7 Let u 1 , u 2 be solutions of (4.1) with μ 1 ∈ M ρ and μ 2 ∈ M ρ , respectively. If f satisfies (H1) then Proof Follows from Proposition 4.4 and Lemma 4.6.

Theorem 4.9
Let u be a solution of (4.1) with μ ∈ M b and f satisfying (H1) and such that f (·, 0) ∈ L 1 (E; m). Then for every k ≥ 0, T k (u) ∈ D e [E] and Proof Follows from Theorem 3.3 and Proposition 4.8.

Stampacchia's definition by duality
In [27] Stampacchia introduced a definition of a solution of (3.1) in case μ ∈ M b and A is uniformly elliptic operator of the form on a bounded open set D ⊂ R d . According to this definition, now called Stampacchia's definition by duality, a measurable function u ∈ L 1 (D; m), where m is the Lebesgue measure on R d , is a solution of (3.1) if The above definition has sense, because it is well known that for A as above Gη has a bounded continuous version. In the general case considered in the paper the original Stampacchia's definition has to be modified, because the measure μ is not assumed to be bounded, Gη may be not continuous for η ∈ L ∞ (E; m) and moreover, the solution of (3.1) may be not locally integrable (see [17,Example 5.7]). In [17] we introduced a generalized Stampacchia's definition for solutions of (4.1) with Dirichlet operator A and bounded measure μ such that μ Cap. Here we give a definition for general measures of the class M.

Definition 4.11
We say that a measurable numerical function u on E is a solution of (4.1) in the sense of Stampacchia if for every η ∈ B(E) such that (|μ|, R|η|) < ∞ the integrals (u, η), ( f u , Rη) are finite and we have

Proposition 4.12 Let μ ∈ M. A measurable function u on E is a solution of (4.1) in the sense of Definition 4.11 if and only if it is a solution of (4.1) in the sense of Definition 3.1.
Proof Let u be a solution of (4.1) in the sense od Definition 3.1. Then by Proposition 4.5, |u| + R| f u | ≤ R|μ|, it is clear that u is a solution of (4.1) in the sense of Stampacchia. Now assume that u is a solution of (4.1) in the sense of Stampacchia. By Lemma 4.10 there exists a strictly positive ρ ∈ S such that μ ∈ M ρ . In fact, from the proof of Lemma 4.10 it follows that we may take ρ = Rg for some strictly positive Borel function g on E. We have for every B ∈ B(E). Hence u = R f u + Rμ, m-a.e., and the proof is complete. Remark 4.14 In [18] renormalized solutions of (4.1) are defined in case μ is a bounded smooth measure. It is also proved there that u is a renormalized solution of (4.1) if and only it is a probabilistic solution. Thus, in case μ is smooth, all the definitions (renormalized, Stampacchia's by duality, probabilistic, via the resolvent kernel) are equivalent.

Remark 4.15
In case A is the Laplace operator on an open bounded set D ⊂ R d , also the socalled weak solutions of (4.1) are considered in the literature (see, e.g., [5]). A weak solution of (4.1) is a function u ∈ L 1 (D; dx) such that f u ∈ L 1 (D; dx) and for every η ∈ C ∞ 0 (D), It is clear that the definition of weak solution is equivalent to Stampacchia's definition by duality. It is worth pointing out that in fact the concept of weak solutions is also due to Stampacchia (see [27,Definition 9.1]).

Existence of solutions
In [17] (see also [19] for the case of operator corresponding to general nonsymmetric quasiregular form) it is proved that if μ is smooth then under conditions (H1)-(H3) there exists a solution of (4.1). It is well known that if A = and μ is not smooth, i.e. μ c = 0, then in general assumptions (H1)-(H3) are not sufficient for the existence of a solution of (4.1). In this section we give an existence result for (4.1) under the following additional hypothesis: (H4) there exists a positive Borel measurable function g on E such that g · m ∈ M and | f (x, y)| ≤ g(x), x ∈ E, y ∈ R.
Let us observe that (H4) implies (H2), (H3). In the paper we have assumed Meyer's hypothesis (L), so we may also drop (H1). Hypothesis (H4) imposes rather restrictive assumption on the growth of f but allows us to prove the existence of solutions for any μ ∈ M and any Dirichlet operator A.

Theorem 4.16 Assume (H4). Then there exists a solution of (4.1).
Proof Let be a strictly positive Borel function on E such that Observe that for every u ∈ L 1 (E, · m), (u) L 1 (E; ·m) ≤ r . It is an elementary check that is continuous. Let {u n } ⊂ L 1 (E; · m) and let v n = (u n ). By [11, Lemma 94, page 306], {v n } has a subsequence convergent m-a.e., which when combined with the fact that |v n |(x) ≤ Rg(x) + R|μ|(x) for x ∈ E implies that, up to a subsequence, {v n } converges in L 1 (E; · m). Therefore by the Schauder fixed point theorem there exists u ∈ L 1 (E; · m) such that (u) = u, which proves the theorem.

Good measures and reduced measures
In this section we develop the theory of reduced measures for (1.1) in case of general Dirichlet operator A and general measure μ of the class M. Our results generalize the corresponding results from H. Brezis, M. Marcus and A.C. Ponce [5] proved in the case where A is the Laplace operator on a bounded domain in R d and μ is a bounded measure. Also note that in [5] it is assumed that f does not depend on x.
In the whole section in addition to (H1)-(H3) we assume that f (x, y) = 0 for y ≤ 0.
Definition 5. 1 We say that a measurable numerical function v on E is a subsolution of (4.1) if f v · m ∈ M and there exists a measure ν ∈ M such that ν ≤ μ and
Proof Let {τ k } be a reducing sequence for u n . By the Tanaka-Meyer formula, Letting k → ∞ in the above inequality we get for q.e. x ∈ E. Let v n , w n be solutions of the following equations −Av n = f + n (·, u n ) + μ + , −Aw n = f − n (·, u n ) + μ − . Of course, v n , w n are excessive functions and by (5.1), We may assume that τ k = δ 1 k = δ 2 k . Since by Proposition 4.2, u n (x) ≥ u n+1 (x), n ≥ 1, for q.e. x ∈ E, there exists u * such that u n u * , q.e. Therefore letting n → ∞ in the equation and using (H1)-(H3), (5.1) (and the fact that for q.e. x ∈ E. By (5.1) and Fatou's lemma, f (·, u * ) · m ∈ M. Hence u * is a solution of (4.1) with μ replaced by μ * := μ d + ν c . What is left is to show that u * is the maximal subsolution of (4.1). By the construction of u * , u n ≥ u * . Therefore by condition (b) of the definition of a probabilistic solution of (4.1) and Lemma 3.5 (see also Remark 3.8) we have μ * c ≤ μ c , which when combined with the fact that μ * d = μ d shows that μ * ≤ μ, i.e. that u * is subsolution of (4.1). Suppose that v is another subsolution of (4.1). Then there exists β ∈ M such that β ≤ μ and v is a solution of (4.1) with μ replaced by β. Since β ≤ μ and f n ≥ f , applying Proposition 4.2 shows that u n ≥ v q.e., hence that u * ≥ v q.e., which completes the proof.
Let μ ∈ M. From now on by μ * , u * we denote the objects constructed in Theorem 5.2. By Theorem 5.2, μ * ≤ μ. It is known (see [2]) that it may happen that μ * = μ, i.e. that there is no solution of (4.1) under assumptions (H1)-(H3). In what follows we denote by G the set of all good measures relative to A and f . Of course, μ * ∈ G.
To prove (v), let us observe that −μ − ∈ G, because −Rμ − is a solution of (4.1) with μ replaced by −μ − . Hence, by (iv), −μ − ≤ μ * , from which we easily get (v). To show (vi), let us observe that μ * ∈ G, and by (i), μ * ≤ ν. Hence μ * ≤ ν * by (iv). Proposition 5.5 A measure μ ∈ M is good if and only if the sequence { f n (X, u n (X ))} considered in the proof of Theorem 5.2 is uniformly integrable under the measure dt ⊗ P x for m-a.e. x ∈ E.
Proof From the proof of Theorem 5.2 we know that f n (X, u n (X )) → f (X, u * (X )), dt ⊗ P xa.e. for m-a.e. x ∈ E and for q.e. x ∈ E. If { f n (X, u n (X ))} is uniformly integrable then letting n → ∞ in (5.4) shows that for q.e. x ∈ E, i.e. μ is a good measure. If μ ∈ G then there exists a solution u of (4.1), i.e.
Of course, u is a subsolution of (4.1), so by Theorem 5.2, u = u * and u n u. By this and (5.4), for q.e. x ∈ E. Since f n (X, u n (X )) → f (X, u(X )), dt ⊗ d P x -a.e. for q.e.
x ∈ E and f n (X, u n (X )) ≤ 0, applying Vitali's theorem shows that the sequence { f n (X, u n (X ))} is uniformly integrable under the measure dt ⊗ P x for q.e. x ∈ E, and hence for m-a.e. x ∈ E. Proposition 5.6 If ν ∈ M, μ ∈ G and ν ≤ μ, then ν ∈ G.
Proof Let {u n } be the sequence of functions of Theorem 5.2 associated with μ and let {v n } be a sequence constructed as {u n } but for μ replaced by ν. By Proposition 4.2, v n ≤ u n q.e. Consequently, f n (·, u n ) ≤ f (·, v n ) ≤ 0 q.e. Since μ ∈ G, we know from Proposition 5.5 that the sequence { f n (X, u n (X ))} is uniformly integrable under the measure dt ⊗ P x for m-a.e.
Proof Follows immediately from Proposition 5.6 and the fact that μ ≤ μ + .

Corollary 5.9
The set G is convex.
Theorem 5. 10 We have Proof It suffices to repeat step by step the reasoning from the proofs of Corollary 6 and Theorems 8-10 in [5].
Since it is clear that μ n − μ ρ → 0, we have μ ∈ A ρ ( f ), which completes the proof.

Inverse maximum principle and Kato's inequality
In this section we consider the linear equation (3.1). The following theorem generalizes the inverse maximum principle proved by H. Brezis and A.C. Ponce in [6] in case A is the Laplace operator on a bounded domain in R d .
Theorem 6.1 Let μ ∈ M and u be a solution of (3.1). If u ≥ 0 then μ c ≥ 0.
Proof Assume that u ≥ 0. Let {τ k } be a reducing sequence for u. By the definition of a solution of (3.1), for every α ≥ 0, lim k→∞ E x e −ατ k u(X τ k ) = R α μ c (x) for q.e. x ∈ E. In particular, R α μ c (x) ≥ 0 for q.e. x ∈ E, and hence, by [3,PropositionII.3.2], R α μ c ≥ 0 everywhere. That μ c ≥ 0 now follows from Lemma 3.5. We define the capacity of B ⊂ E as It is an elementary check that Cap A, p is subadditive and increasing (see, e.g., [1, Proposition 2.3.6]). We say that μ ∈ V p ∩ M + if for every η ∈ V + p , (η, μ) ≤ c η V p .
In the rest of the section we assume that p > 1. By p we denote the Hölder conjugate to p.  Proof Let η ∈ V + p . By our assumptions on μ and Lemma 7.2, for any η ∈ V + p with η V p = 1 we have which proves the lemma. Cap A, p (G n ) = 0, lim n→∞ μ(G n ) = 0, 1 E\G n · μ ≤ 2 n Cap A, p , n ≥ 1.
Proof It is enough to repeat step by step the proof of [12, Lemma 2.2.9], the only difference being in the fact that we choose the sets B n appearing in the proof of [12, Lemma 2.2.9] as Borel sets.
As a corollary to Lemma 7.4 we get the following proposition.

Proposition 7.5 A measure μ ∈ M + satisfies μ
Cap A, p if and only if there exists an increasing sequence {E n } of Borel subsets of E such that 1 E n · μ ∈ V p ∩ M + for n ∈ N and μ(E \ n≥1 E n ) = 0. Theorem 7.6 Assume (7.1). If μ ∈ M and μ + Cap A, p then μ ∈ G.
Proof By Theorem 5.11 we may assume that μ ≥ 0. By Lemma 4.10 there exists a strictly positive bounded excessive function ρ such that μ ∈ M + ρ , and by Proposition 7.5 there exists a sequence {μ n } ⊂ V p ∩ M + such that lim n→∞ μ n − μ ρ = 0. Therefore it is enough to show that μ n ∈ G. But this follows from Proposition 7.1.

Corollary 7.7
Assume that μ ∈ M and an let f (x, u) = −u p , x ∈ E, u ≥ 0. Then μ ∈ G if and only if μ + Cap A, p .
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