# Reduced measures for semilinear elliptic equations involving Dirichlet operators

## Abstract

We consider elliptic equations of the form (E) \(-Au=f(x,u)+\mu \), where *A* is a negative definite self-adjoint Dirichlet operator, *f* is a function which is continuous and nonincreasing with respect to *u* and \(\mu \) 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 \(A=\Delta \) 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 \(f(u)=-u^p\) for some \(p>1\).

### Mathematics Subject Classification

35J75 60J45## 1 Introduction

*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

*E*, \(f:E\times {\mathbb R}\rightarrow {\mathbb R}\) is a measurable function such that \(f(\cdot ,u)=0\), \(u\le 0\), 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

*A*corresponds to some symmetric Dirichlet form \(({\mathcal {E}},D[{\mathcal {E}}])\) on \(L^2(E;m)\) in the sense that

*E*admits a decomposition

The study of semilinear equations of the form (1.1) in case \(\mu \) is smooth, i.e. when \(\mu _c=0\), 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 \(\mu \in L^1(E;m)\). At present existence, uniqueness and regularity results are available for equation (1.1) involving general bounded smooth measure \(\mu \) 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 \(\mu _c\ne 0\) 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 \(A=\Delta \). 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 \(\mu \). In case \(A=\Delta \) 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 \(A=\Delta ^{\alpha }\) with \(\alpha \in (0,1)\) (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 \(\mu \) from some class of measures \({\mathbb M}\) including the class \({\mathcal {M}}_b\) 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 \(A=\Delta \) 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 \(A=\Delta \), because in this case \({\mathcal {M}}_b\varsubsetneq {\mathbb M}\) and \({\mathbb M}\) contains important in applications unbounded measures. The second purpose of our paper is to give a probabilistic interpretation for solutions of (1.1).

*A*(or, equivalently, the Dirichlet form \(({\mathcal {E}},D[{\mathcal {E}}]))\) is transient (see [12, Section 1.5]). It is well known that then there exists a kernel \(\{R(x,dy),x\in E\}\) such that for every \(g\in L^{1,+}(E;m)\),

*u*is a solution of (1.1) with \(f\equiv 0\) then

*m*for every bounded Borel measure \(\mu \). This condition is known in the literature as the Meyer hypothesis (L) (see [3]) or the condition of absolute continuity of the resolvent \(\{G_\alpha ,\alpha >0\}\) (see [12]).

*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

*m*-a.e. \(x\in E\). Of course, to make this definition correct we have to assume that the integrals in (1.4) exist. Therefore the class \({\mathbb M}\) we consider consists of Borel measures \(\mu \) on

*E*such that \(\int _Er(x,y)\,|\mu |(dy)<\infty \) for

*m*-a.e. \(x\in E\). We will show that \({\mathcal {M}}_b(E)\subset \mathbb {M}\). In general, the inclusion is strict. For instance, if \(A=\Delta ^\alpha \), \(\alpha \in (0,1]\), on an open set \(D\subset {\mathbb R}^d\), then \({\mathbb M}\) includes the set of all Borel measures \(\mu \) on

*E*such that \(\delta ^\alpha \cdot \mu \in {\mathcal {M}}_b\), where \(\delta (x)=\text{ dist }(x,\partial D)\). We also show that in case \(\mu \in {\mathcal {M}}_b\) and

*A*is a uniformly elliptic divergence form operator on a bounded domain in \({\mathbb R}^d\) definition (1.4) is equivalent to Stampacchia’s definition by duality (see [27]).

*u*is a probabilistic solution of (1.1) if

- (a)\(f(\cdot ,u)\cdot m\in \mathbb {M}\) and there exists a local martingale additive functional
*M*of \({\mathbb X}\) such thatfor quasi every (q.e. for short) \(x\in E\) (Here \(A^{\mu _d}\) denotes a continuous additive functional of \({\mathbb X}\) of finite variation in the Revuz correspondence with \(\mu _d\)),$$\begin{aligned} u(X_t)=u(X_0)-\int _0^tf(X_r,u(X_r))\,dr-\int _0^t\,dA^{\mu _d}_r +\int _0^t\,dM_r,\quad t\ge 0,\quad P_x\text{-a.s. } \end{aligned}$$ - (b)for every polar set \(N\subset E\), every stopping time \(T\ge \zeta \) and every sequence of stopping times \(\{\tau _k\}\) such that \(\tau _k\nearrow T\) and \(E_x\sup _{t\le \tau _k}|u(X_t)|<\infty \) for \(x\in E{\setminus } N\) and \(k\ge 1\) we havewhere \(E_x\) denotes the integration with respect to probability \(P_x\) and$$\begin{aligned} E_xu(X_{\tau _k})\rightarrow R\mu _c(x),\quad x\in E{\setminus } N, \end{aligned}$$$$\begin{aligned} R\mu _c(x)=\int _E r(x,y)\mu _c(dy),\quad x\in E. \end{aligned}$$

*u*is a solution of (1.1) and \(\mu \in {\mathcal {M}}_b\) then \(T_k(u)\in D_e[{\mathcal {E}}]\) and

*A*,

*f*. Let us recall that the reduced measure is the largest measure \(\mu ^*\in \mathbb {M}\) such that \(\mu ^*\le \mu \) and there exists a solution of (1.1) with \(\mu \) replaced by \(\mu ^*\). A measure \(\mu \in {\mathbb M}\) is good, if \(\mu ^*=\mu \). By results of [17, 19], if \(\mu _c=0\), then \(\mu \) is good. In the present paper we first show that

*g*,

*v*on

*E*such that \(g\cdot m, f(\cdot ,v)\cdot m\in \mathbb {M}\) and \(Av\in \mathbb {M}\). From this characterization of \({\mathcal {G}}\) we deduce that for every strictly positive excessive function \(\rho \),

*f*(it is satisfied for instance if \(|f(x,u)|\le c_1+c_2 e^{u^2}\)), for every strictly positive excessive function \(\rho \),

*u*is a solution of (1.1) then \(Au^+\in \mathbb {M}\) and

*f*having at most polynomial growth, i.e. for

*f*satisfying

*p*, then a solution of (1.1) exists, i.e. \(\mu \in {\mathcal {G}}\). For

*f*of the form

*f*is given by (1.6),

## 2 Preliminaries

*E*is a locally compact separable metric space and

*m*is a positive Radon measure on

*E*such that supp\([m]=E\). By \(({\mathcal {E}},D[{\mathcal {E}}])\) we denote a symmetric regular Dirichlet form on \(L^2(E;m)\) (see [12] or [23] for the definitions). We will always assume that \(({\mathcal {E}},D[{\mathcal {E}}])\) is transient, i.e. there exists a strictly positive function

*g*on

*E*such that

*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]) that for every self-adjoint negative definite Dirichlet operator

*A*there exists a unique Dirichlet form \(({\mathcal {E}},D[{\mathcal {E}}])\) such that (1.2) holds.

*E*is called nest if Cap\((E{\setminus } F_n)\rightarrow 0\) as \(n\rightarrow \infty \). A subset \(N\subset E\) is called exceptional if Cap\((N)=0\). We say that some property

*P*holds quasi everywhere (q.e. for short) if a set for which it does not hold is exceptional.

We say that a function *u* on *E* is quasi-continuous if there exists a nest \(\{F_n\}\) such that \(u_{|F_n}\) is continuous for every \(n\ge 1\). It is known that each function \(u\in D[{\mathcal {E}}]\) has a quasi-continuous *m*-version.

A Borel measure \(\mu \) on *E* is called smooth if it does not charge exceptional sets and there exists a nest \(\{F_n\}\) such that \(|\mu |(F_n)<\infty ,\, n\ge 1\). By *S* we denote the set of all smooth measures on *E*.

*m*-measurable functions on

*E*for which there exists an \({\mathcal {E}}\)-Cauchy sequence \(\{u_n\}\subset D[{\mathcal {E}}]\) convergent

*m*-a.e. to

*u*(the so-called approximating sequence). One can show that for \(u\in D_e[{\mathcal {E}}]\) the limit \({\mathcal {E}}(u,u)=\lim _{n\rightarrow \infty }{\mathcal {E}}(u_n,u_n)\) exists and does not depend on the approximating sequence \(\{u_n\}\) for

*u*. Each element \(u\in D_e[{\mathcal {E}}]\) has a quasi-continuous version. It is known that \(({\mathcal {E}},D[{\mathcal {E}}])\) is transient if and only if \(({\mathcal {E}},D_e[{\mathcal {E}}])\) is a Hilbert space. In the latter case for a given measure \(\mu \in S_{0}^{(0)}\) inequality (2.1) holds for every \(u\in D_e[{\mathcal {E}}]\).

By \({\mathcal {M}}_b\) we denote the set of all bounded Borel measures on *E* and by \({\mathcal {M}}_{0,b}\) the subset of \({\mathcal {M}}_b\) consisting of smooth measures.

*E*and a Borel measure \(\mu \) on

*E*we write

*E*defined as

*R*. We say that some function on

*E*is measurable if it is universally measurable, i.e. measurable with respect to the \(\sigma \)-algebra

*E*and \(\mathcal {B}^\mu (E)\) is the completion of \(\mathcal {B}(E)\) with respect to the measure \(\mu \).

A positive measurable function *u* on *E* is called \(\alpha \)-excessive if for every \(\beta >0\), \((\alpha +\beta ) R_{\alpha +\beta } u\le u\) and \(\alpha R_\alpha u\nearrow u\) as \(\alpha \rightarrow \infty \). By \(\mathcal {S}_\alpha \) we denote the set of \(\alpha \)-excessive functions. We put \(\mathcal {S}=\mathcal {S}_0\).

*B*we set

*B*, \(D_A\) is the first debut time of

*B*and \(\tau _B\) is the first exit time of

*B*.

*u*on

*E*for which the family

*E*and \(\alpha \ge 0\) we denote by \(\mu \circ R_\alpha \) the measure defined as

*m*is the reference measure for \(\mathbb {X}\), i.e. for all \(x\in E\) and \(\alpha >0\) we have \(R_\alpha (x,\cdot )\ll m\). It is well known (see [12, Lemma 4.2.4]) that in this case for every \(\alpha \ge 0\) there exists a \({\mathcal {B}}(E)\otimes {\mathcal {B}}(E)\) measurable function

## 3 Linear equations

In the whole paper we adopt the convention that \(\int _E r(x,y)\,d\mu (y)=0\) for every Borel measure \(\mu \) on *E* such that \(\int _E r(x,y)\,d\mu ^+(y)=\int _E r(x,y)\,d\mu ^-(y)=\infty \). We call \(u:E\rightarrow {\mathbb R}\cup \{-\infty ,\infty \}\) a numerical function on *E*.

### 3.1 Solutions defined via the resolvent kernel and regularity results

### Definition 3.1

*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 \(x\in E\), then *u* is Borel measurable. Since \(\mu \in {\mathbb M}\), *u* is finite q.e.

### Proposition 3.2

\({\mathcal {M}}_b\subset {\mathbb M}.\)

### Proof

*f*on

*E*such that \(Rf<\infty \), q.e. From this we conclude that \(f\cdot m\) 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 \(\bigcup _{n\ge 1} F_n=E\), q.e. and \(\sup _{x\in E} R(\mathbf {1}_{F_n} f)(x)<\infty \) (see also comments following [12, Corollary 2.2.2]). As a matter of fact, in [12] in the last condition \(\sup \) is replaced by \(\text{ ess }\sup \) with respect to

*m*, however in view of [3, Proposition II.3.2], it holds true also with supremum norm. We have

### Theorem 3.3

*u*be a solution of (3.1). Then \(T_k(u)\in D_e[{\mathcal {E}}]\) and for every \(k\ge 0\),

### Proof

*u*,

*v*on

*E*set

### Remark 3.4

- (i)
By Theorem 3.3, \(T_k(u)\in D[{\mathcal {E}}]\) if \(m(E)<\infty \), because by [12, Theorem 1.5.2(iii)], \(D[{\mathcal {E}}]=D_e[{\mathcal {E}}]\cap L^2(E;m)\).

- (ii)
\(T_k(u)\in D[{\mathcal {E}}]\) if the form satisfies Poincaré type inequality \(c(u,u)\le {\mathcal {E}}(u,u)\) for every \(u\in D[{\mathcal {E}}]\) and some \(c>0\), because then \(D_e[{\mathcal {E}}]=D[{\mathcal {E}}]\).

### 3.2 Probabilistic solutions

In this subsection we give an equivalent definition of solution of (3.1) using stochastic equations involving a Hunt process \(\mathbb {X}\) associated with the Dirichlet operator *A*. We begin with the following lemma.

### Lemma 3.5

Assume that \(\mu ,\nu \in {\mathbb M}\) and there is \(\alpha _0\ge 0\) such that \( R_\alpha \mu \ge R_\alpha \nu \) for \(\alpha \ge \alpha _0.\) Then \(\mu \ge \nu \).

### Proof

*E*such that \((R\psi ,|\mu |+|\nu |)<\infty \). So, it is clear that it is enough to prove that \((\eta R\psi ,\mu )\ge (\eta R\psi ,\nu )\) for every \(\eta \in C^+_b(E)\). Let \(\eta \in C_b^+(E)\). An elementary calculus shows that \(\alpha R_\alpha (\eta R\psi )(x)\rightarrow \eta R\psi (x)\) for every \(x\in E\). On the other hand, \(\alpha R_\alpha (\eta R\psi )(x)\le \Vert \eta \Vert _\infty R\psi (x),\, x\in E\). Hence, by the Lebesgue dominated convergence theorem,

### Theorem 3.6

Assume that \(\mu \in {\mathbb M}^+\) and \(\mu \bot \)Cap. Then \(u=R\mu \) is quasi-continuous and the process \([0,\infty )\ni t\mapsto u(X_t) \) is a cádlág local martingale under the measure \(P_x\) for q.e. \(x\in E\).

### Proof

*u*is quasi-continuous. Let us put

*u*is an excessive function, \(A^\alpha \) is an increasing process and \(u_\alpha (x)\nearrow u(x)\) for every \(x\in E\) as \(\alpha \nearrow \infty \). Hence

*u*(

*X*)] denote the quadratic variations of processes \(u_{\alpha }(X)\) and

*u*(

*X*), respectively. By [12, Theorem 4.2.2] there exists an exceptional set \(N\subset E\) such that for every \(x\in E{\setminus } N\),

*u*(

*X*) is a supermartingale and \(\lim _{t\rightarrow \infty } E_xu(X_t)<\infty \). Therefore by [25, Theorem III.13], for q.e. \(x\in E\) there exists an increasing predictable process \(C^x\) with \(E_xC^x_\zeta <\infty \) and a cádlág local martingale \(M^x\) such that

*X*is quasi-left 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 \(\{\tau ^x_n\}\subset \mathcal {T}\) such that for every \(n\ge 1\),

*A*such that \(A=C^x\) for q.e. \(x\in E\). Of course,

*A*is a positive continuous additive functional. Putting

*M*is an additive functional and \(M^x=M\), \(P_x\)-a.s. for q.e. \(x\in E\). Thus

*M*is a local martingale additive functional. By [12, Theorem 5.1.4] there exists \(\nu \in S\) such that \(A=A^\nu \). In particular, for every \(\alpha \ge 0\),

*u*(

*X*) is a local martingale. \(\square \)

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 \(({\mathcal {F}},P_x)\)-local martingale for q.e. \(x\in E\).

### Theorem 3.7

*u*is quasi-continuous and there exists a local MAF

*M*such that

### Proof

*v*is quasi-continuous and that there exists a uniformly integrable MAF \(M^v\) such that

*w*is quasi-continuous and there exists a local MAF \(M^w\) such that

### Remark 3.8

We are now ready to introduce the second definition of a solution of (3.1) making use of the Hunt process \(\mathbb {X}\) 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

*u*on

*E*is a probabilistic solution of (3.1) if

- (a)there exists a local MAF
*M*such that for q.e. \(x\in E\),$$\begin{aligned} u(X_t)=u(X_0)-\int _0^t\,dA^{\mu _d}_r+\int _0^t\,dM_r, \quad t\ge 0,\quad P_x\text{-a.s. }, \end{aligned}$$ - (b)for every polar set \(N\subset E\), every stopping time \(T\ge \zeta \) and every sequence \(\{\tau _k\}\subset \mathcal {T}\) such that \(\tau _k\nearrow T\) and \(E_x\sup _{t\le \tau _k}|u(X_t)|<\infty \) for every \(x\in E{\setminus }N\) and \(k\ge 1\) we have$$\begin{aligned} E_xu(X_{\tau _k})\rightarrow R\mu _c(x),\quad x\in E{\setminus }N. \end{aligned}$$

*u*, and we will say that \(\{\tau _k\}\) reduces

*u*.

### Remark 3.10

*u*(

*X*) in the above definition is a special semimartingale, there exists at least one reducing sequence \(\{\tau _k\}\) for

*u*. In fact, the stopping times defined as

### Remark 3.11

*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

*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

## 4 Semilinear equations

In what follows \(\mu \in {\mathbb M}\) and \(f:E\times {\mathbb R}\rightarrow {\mathbb R}\) is a function satisfying the following conditions: \({\mathbb R}\ni y\mapsto f(x,y)\) is continuous for every \(x\in E\) and \(E\ni x\mapsto f(x,y)\) is measurable for every \(y\in {\mathbb R}\).

### Definition 4.1

We say that a measurable numerical function *u* on *E* is a solution of (4.1) if \(f(\cdot ,u)\cdot m\in {\mathbb M}\) and *u* is a solution of (3.1) with \(\mu \) replaced by \(f(\cdot ,u)\cdot m+\mu \).

- (H1)
for every \(x\in E\) the mapping \(y\mapsto f(x,y)\) in nonincreasing,

- (H2)
for every \(y\in {\mathbb R}\) the mapping \(x\mapsto f(x,y)\in qL^1(E;m)\),

- (H3)
\(f(\cdot ,0)\cdot m\in {\mathbb M}\).

### 4.1 Comparison results, a priori estimates and regularity of solutions

*u*on

*E*we write

### Proposition 4.2

Assume that \(\mu _1,\mu _2\in {\mathbb M}\), \(\mu _1\le \mu _2\), \(f^1(x,y)\le f^2(x,y)\) for \(x\in E,\, y\in {\mathbb R}\) and \(f^1\) or \(f^2\) satisfies (H1). Then \(u_1\le u_2\) q.e., where \(u_1\) (resp. \(u_2\)) is a solution of (4.1) with data \(f^1,\mu ^1\) (resp. \(f^2,\mu ^2\)).

### Proof

### Corollary 4.3

Under (H1) there exists at most one solution of (4.1).

### Proposition 4.4

*f*satisfies (H1), then

### Proof

### Proposition 4.5

*u*be a solution of (4.1) with

*f*satisfying (H1), (H3). Then

### Proof

We apply Proposition 4.4 to \(u_1=u, u_2=0, \mu _1=\mu , \mu _2=-f(\cdot ,0)\). \(\square \)

Given a positive function \(\rho \in \mathcal {S}\), we denote by \({\mathcal {M}}_\rho \) the set of all measures \(\mu \in {\mathcal {M}}\) such that \(\Vert \mu \Vert _\rho <\infty \), where \(\Vert \mu \Vert _\rho =\Vert \rho \cdot \mu \Vert _{TV}\).

*E*. Let us also note that if \(A=\Delta ^\alpha \) (with \(\alpha \in (0,1]\)) on an open bounded set \(D\subset {\mathbb R}^d\) (see Remark 4.13) then for \(\rho =R1\) we have \({\mathcal {M}}_\rho =\{\mu \in {\mathcal {M}}:\delta ^{\alpha }\cdot \mu \in {\mathcal {M}}_b\}\), where \(\delta (x)=\text{ dist }(x,\partial D)\), because by [22] there exists \(c,C>0\) such that

### Lemma 4.6

Assume that \(\mu ,\nu \in {\mathcal {M}}_\rho \) and \( R\mu (x)\le R\nu (x)\) for \(x\in E\). Then \(\Vert \mu \Vert _{\rho }\le \Vert \nu \Vert _{\rho }\).

### Proof

*E*such that \(Rh_n\nearrow \rho \). For \(n\ge \) we have

### Proposition 4.7

*f*satisfies (H1) then

### Proposition 4.8

*u*be a solution of (4.1) with \(\mu \in {\mathcal {M}}_\rho \) and

*f*satisfying (H1) and such that \(f(\cdot ,0)\in L^1(E;\rho \cdot m)\). Then

### Theorem 4.9

*u*be a solution of (4.1) with \(\mu \in {\mathcal {M}}_b\) and

*f*satisfying (H1) and such that \(f(\cdot ,0)\in L^1(E;m)\). Then for every \(k\ge 0\), \(T_k(u)\in D_e[{\mathcal {E}}]\) and

### 4.2 Stampacchia’s definition by duality

*A*is uniformly elliptic operator of the form

*m*is the Lebesgue measure on \({\mathbb R}^d\), is a solution of (3.1) if

*A*as above \(G\eta \) 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 \(\mu \) is not assumed to be bounded, \(G\eta \) may be not continuous for \(\eta \in L^\infty (E;m)\) 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 \(\mu \) such that \(\mu \ll \) Cap. Here we give a definition for general measures of the class \(\mathbb {M}\).

### Lemma 4.10

We have \( \mathbb {M}=\bigcup \mathcal {M}_\rho \), where the union is taken over all strictly positive excessive bounded functions.

### Proof

It is clear that \(\bigcup \mathcal {M}_\rho \subset \mathbb {M}\). To prove the opposite inclusion, let us assume that \(\mu \in \mathbb {M}\). Then \(R|\mu |<\infty \), *m*-a.e. Therefore there exists a strictly positive Borel function \(\eta \) on *E* such that \((R|\mu |,\eta )=(|\mu |, R\eta )<\infty \). On the other hand, since the form \(({\mathcal {E}},D[{\mathcal {E}}])\) is transient, there exists a strictly positive Borel function *g* on *E* such that \(\Vert Rg\Vert _\infty <\infty \) (see [24, Corollary 1.3.6]). Let us put \(\rho =R(\eta \wedge g)\). It is clear that \(\rho \) is*** a bounded strictly positive excessive function. \(\square \)

### Definition 4.11

*u*on

*E*is a solution of (4.1) in the sense of Stampacchia if for every \(\eta \in {\mathcal {B}}(E)\) such that \((|\mu |,R|\eta |)<\infty \) the integrals \((u,\eta )\), \((f_u,R\eta )\) are finite and we have

### Proposition 4.12

Let \(\mu \in \mathbb {M}\). 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

*u*be a solution of (4.1) in the sense od Definition 3.1. Then by Proposition 4.5, \(|u|+R|f_u|\le R|\mu |\), 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 \(\rho \in \mathcal {S}\) such that \(\mu \in \mathcal {M_\rho }\). In fact, from the proof of Lemma 4.10 it follows that we may take \(\rho =Rg\) for some strictly positive Borel function

*g*on

*E*. We have

*m*-a.e., and the proof is complete. \(\square \)

### Remark 4.13

Let \(\alpha \in (0,1]\) and let *D* be an open subset of \({\mathbb R}^d\). Denote by \(({\mathcal {E}},D[{\mathcal {E}}])\) the Dirichlet form associated with the operator \(\Delta ^\alpha \) on \({\mathbb R}^d\) (see [12, Example 1.4.1]), and by \(({\mathcal {E}}_D,D[{\mathcal {E}}_D])\) the part of \(({\mathcal {E}},D[{\mathcal {E}}])\) on *D* (see [12, Section 4.4]). By *A* denote the operator associated with \(({\mathcal {E}}_D,D[{\mathcal {E}}_D])\), i.e. the fractional Laplacian \(\Delta ^\alpha \) on *D* with zero boundary condition. If \(\mu \in {\mathcal {M}}^\alpha _\delta \) then in Definition 4.11 one can take any function \(\eta \in {\mathcal {B}}_b(E)\) as a test function. It follows in particular that in case of equations involving operator *A* Stampacchia’s definition is equivalent to the one introduced in [8, Definition 1.1].

### Remark 4.14

In [18] renormalized solutions of (4.1) are defined in case \(\mu \) 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 \(\mu \) is smooth, all the definitions (renormalized, Stampacchia’s by duality, probabilistic, via the resolvent kernel) are equivalent.

### Remark 4.15

*A*is the Laplace operator on an open bounded set \(D\subset {\mathbb R}^d\), also the so-called weak solutions of (4.1) are considered in the literature (see, e.g., [5]). A weak solution of (4.1) is a function \(u\in L^1(D;dx)\) such that \(f_u\in L^1(D;dx)\) and for every \(\eta \in C_0^\infty (\overline{D})\),

### 4.3 Existence of solutions

- (H4)
there exists a positive Borel measurable function

*g*on*E*such that \(g\cdot m\in {\mathbb M}\) and \(|f(x,y)|\le g(x)\), \(x\in E,y\in {\mathbb R}\).

Hypothesis (H4) imposes rather restrictive assumption on the growth of *f* but allows us to prove the existence of solutions for any \(\mu \in {\mathbb M}\) and any Dirichlet operator *A*.

### Theorem 4.16

Assume (H4). Then there exists a solution of (4.1).

### Proof

*E*such that

*m*-a.e., which when combined with the fact that \(|v_n|(x)\le Rg(x)+R|\mu |(x)\) for \(x\in E\) implies that, up to a subsequence, \(\{v_n\}\) converges in \(L^1(E;\varrho \cdot m)\). Therefore by the Schauder fixed point theorem there exists \(u\in L^1(E;\varrho \cdot m)\) such that \(\Phi (u)=u\), which proves the theorem. \(\square \)

## 5 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 \(\mu \) of the class \({\mathbb M}\). 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 \({\mathbb R}^d\) and \(\mu \) 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 \(y\le 0\).

### Definition 5.1

*v*on

*E*is a subsolution of (4.1) if \(f_v\cdot m\in {\mathbb M}\) and there exists a measure \(\nu \in {\mathbb M}\) such that \(\nu \le \mu \) and

### Theorem 5.2

### Proof

*m*-a.e. to excessive functions

*v*and

*w*, respectively. By (5.2) and [14], there exists \(\nu _1, \nu _2\in {\mathbb M}^+\) such that \(v=R\nu _1\), \(w=R\nu _2\). By Theorem 3.7 the function \(h=R|\mu |\) is quasi-continuous. Therefore if we put \(\delta ^1_k=\inf \{t\ge 0:h(X_t)\ge k\}\wedge \zeta \), then \(\delta ^1_k\nearrow \zeta ,\, P_x\)-a.s. for q.e. \(x\in E\). From Theorem 3.7 it also follows that

*h*(

*X*) is a special semimartingale. Therefore there exists a sequence \(\{\delta ^2_k\}\subset \mathcal {T}\) such that \(\delta ^2_k\nearrow \zeta \) and for q.e. \(x\in E\),

*v*is another subsolution of (4.1). Then there exists \(\beta \in {\mathbb M}\) such that \(\beta \le \mu \) and

*v*is a solution of (4.1) with \(\mu \) replaced by \(\beta \). Since \(\beta \le \mu \) and \(f_n\ge f\), applying Proposition 4.2 shows that \(u_n\ge v\) q.e., hence that \(u^*\ge v\) q.e., which completes the proof. \(\square \)

Let \(\mu \in {\mathbb M}\). From now on by \(\mu ^*\), \(u^*\) we denote the objects constructed in Theorem 5.2. By Theorem 5.2, \(\mu ^*\le \mu \). It is known (see [2]) that it may happen that \(\mu ^*\ne \mu \), i.e. that there is no solution of (4.1) under assumptions (H1)–(H3).

### Definition 5.3

- (a)
We call \(\mu ^*\) the reduced measure associated to \(\mu \).

- (b)
We call \(\mu \in {\mathbb M}\) a good measure (relative to

*A*and*f*) if there exists a solution of (4.1).

In what follows we denote by \({\mathcal {G}}\) the set of all good measures relative to *A* and *f*. Of course, \(\mu ^*\in {\mathcal {G}}\).

### Proposition 5.4

- (i)
\(\mu ^*\le \mu \),

- (ii)
\(\mu -\mu ^*\bot \text{ Cap }\), \((\mu ^*)_d=\mu _d\),

- (iii)
\(\mathcal {A}\cap S\subset {\mathcal {G}}\),

- (iv)
\(\mu ^*\) is the largest good measure less then or equal to \(\mu \),

- (v)
\(|\mu ^*|\le |\mu |\),

- (vi)
if \(\mu ,\nu \in {\mathbb M}\) and \(\mu \le \nu \), then \(\mu ^*\le \nu ^*\).

### Proof

*v*of (4.1) with \(\mu \) replaced by \(\nu \). Since \(\nu \le \mu \), the latter means that

*v*is a subsolution of (4.1). Therefore by Theorem 5.2, \(v\le u^*\) q.e. From this, condition (b) of the definition of a probabilistic solution and Remark 3.8,

### Proposition 5.5

A measure \(\mu \in {\mathbb M}\) 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 \(dt\otimes P_x\) for *m*-a.e. \(x\in E\).

### Proof

*m*-a.e. \(x\in E\) and

*u*of (4.1), i.e.

*u*is a subsolution of (4.1), so by Theorem 5.2, \(u=u^*\) and \(u_n\searrow u\). By this and (5.4),

*m*-a.e. \(x\in E\). \(\square \)

### Proposition 5.6

If \(\nu \in {\mathbb M},\, \mu \in {\mathcal {G}}\) and \(\nu \le \mu \), then \(\nu \in {\mathcal {G}}\).

### Proof

Let \(\{u_n\}\) be the sequence of functions of Theorem 5.2 associated with \(\mu \) and let \(\{v_n\}\) be a sequence constructed as \(\{u_n\}\) but for \(\mu \) replaced by \(\nu \). By Proposition 4.2, \(v_n\le u_n\) q.e. Consequently, \(f_n(\cdot ,u_n)\le f(\cdot ,v_n)\le 0\) q.e. Since \(\mu \in {\mathcal {G}}\), we know from Proposition 5.5 that the sequence \(\{f_n(X,u_n(X))\}\) is uniformly integrable under the measure \(dt\otimes P_x\) for *m*-a.e. \(x\in E\). Therefore \(\{f_n(X,v_n(X))\}\) has the same property. By Proposition 5.5, this implies that \(\nu \in {\mathcal {G}}\). \(\square \)

### Corollary 5.7

If \(\mu \in {\mathbb M}\) and \(\mu ^+\in {\mathcal {G}}\), then \(\mu \in {\mathcal {G}}\).

### Proof

Follows immediately from Proposition 5.6 and the fact that \(\mu \le \mu ^+\). \(\square \)

### Corollary 5.8

If \(\mu _1,\mu _2\in {\mathcal {G}}\), then \(\mu _1\vee \mu _2\in {\mathcal {G}}\).

### Proof

Let \(\mu =\mu _1\vee \mu _2\). Since \(\mu _1\le \mu \), \(\mu _2\le \mu \) and \(\mu _1,\mu _2\in {\mathcal {G}}\), it follows from Proposition 5.4(iv) that \(\mu _1\le \mu ^*\) and \(\mu _2\le \mu ^*\). Hence \(\mu \le \mu ^*\). On the other hand, by Proposition 5.4(i), \(\mu ^*\le \mu \), so \(\mu =\mu ^*\), i.e. \(\mu \in {\mathcal {G}}\). \(\square \)

### Corollary 5.9

The set \({\mathcal {G}}\) is convex.

### Proof

Let \(\mu _1,\mu _2\in {\mathcal {G}}\). Then \(\mu _1\vee \mu _2\in {\mathcal {G}}\) by Corollary 5.8. But for every \(t\in [0,1]\), \(t\mu _1+(1-t)\mu _2\le \mu _1\vee \mu _2\), so by Proposition 5.6, \(t\mu _1+(1-t)\mu _2\in {\mathcal {G}}\), \(t\in [0,1]\). \(\square \)

Set \({\mathcal {G}}_\rho ={\mathcal {G}}\cap {\mathcal {M}}_\rho \).

### Theorem 5.10

- (i)
\(\Vert \mu -\mu ^*\Vert _\rho =\min _{\nu \in {\mathcal {G}}_\rho }\Vert \mu -\nu \Vert _\rho \) for every \(\mu \in {\mathcal {M}}_\rho \),

- (ii)
if \(\mu _1,\mu _2\in {\mathbb M}\) and \(\mu _1\bot \mu _2\), then \((\mu _1+\mu _2)^*=\mu _1^*+\mu _2^*, \)

- (iii)
\((\mu \wedge \nu )^*=\mu ^*\wedge \nu ^*\) and \((\mu \vee \nu )^*=\mu ^*\vee \nu ^*\) for every \(\mu ,\nu \in {\mathbb M}\),

- (iv)
\((\mu ^*-\nu ^*)^+\le (\mu -\nu )^+ \) for every \(\mu ,\nu \in {\mathbb M}\).

### Proof

It suffices to repeat step by step the reasoning from the proofs of Corollary 6 and Theorems 8–10 in [5]. \(\square \)

### Theorem 5.11

- (i)
\(\mu \in {\mathcal {G}}\),

- (ii)
\(\mu ^+\in {\mathcal {G}}\),

- (iii)
\(\mu _c\in {\mathcal {G}}\),

- (iv)
\(\mu =g-Av\) for some functions

*g*,*v*on*E*such that \(g\cdot m\in {\mathbb M}\) and \(f(\cdot ,v)\cdot m\in {\mathbb M}\).

### Proof

### Corollary 5.12

- (i)
\({\mathcal {G}}+{\mathbb M}\cap S\subset {\mathcal {G}}\),

- (ii)
\(\mathcal {A}(f)+L(E;m)={\mathcal {G}}\),

- (iii)
\(\mathcal {A}_\rho (f)+L^1(E;\rho \cdot m)={\mathcal {G}}_\rho \).

- (A)for every \(\theta \in [0,1)\), \(c\ge 0\) there exist \(\alpha (c,\theta ),\,\beta (c,\theta )\ge 0\) such that$$\begin{aligned} |f(x,\theta u+c)|\le \alpha (c,\theta )|f(x,u)|+\beta (c,\theta ), \quad x\in E,\, u\in {\mathbb R}. \end{aligned}$$

### Theorem 5.13

Let \(\rho \in L^1(E;m)\). If (A) is satisfied then \(\overline{\mathcal {A}_\rho (f)}={\mathcal {G}}_\rho \), where the closure is taken in the space \(({\mathcal {M}}_\rho ,\Vert \cdot \Vert _{\rho })\).

### Proof

*E*such that \(R\psi \le \rho ,\, m\)-a.e. Let us observe that

*u*of (4.1). Let \(\theta _n=(1-\frac{1}{n})\) and let \(\{F_n\}\) be a nest such that \(c(n):=\Vert R(\mathbf {1}_{F_n} f(\cdot ,u))\Vert _\infty <\infty \) (such a nest exists, because \(f(\cdot ,u)\in \mathcal {M}_\rho \subset {\mathbb M}\)). Let \(\mu _n=-\theta _n A u-\mathbf {1}_{F_n} f(\cdot ,u)\). By (A),

## 6 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 \({\mathbb R}^d\).

### Theorem 6.1

Let \(\mu \in {\mathbb M}\) and *u* be a solution of (3.1). If \(u\ge 0\) then \(\mu _c\ge 0\).

### Proof

*u*. By the definition of a solution of (3.1), for every \(\alpha \ge 0\),

### Proposition 6.2

*u*be a solution of (3.1) and let \(\varphi \) be a positive convex Lipschitz continuous function on \({\mathbb R}\) such that \(\varphi (0)=0\). Then \(A\varphi (u)\in {\mathbb M}\). Moreover,

### Proof

*u*. By the definition of a probabilistic solution of (3.1),

*M*. By the Itô-Meyer formula,

*A*, where \(\varphi '\) is the left derivative of \(\varphi \). Let \(A^p\) denote the dual predictable projection of

*A*(one can find a version of \(A^p\) which is independent of

*x*; see [9]). Since \(A^p\) is predictable, it is continuous, because the filtration \(({\mathcal {F}}_t)\) is quasi-left continuous. Therefore there exists a positive smooth measure \(\nu \) such that \(A^p=A^\nu \). For q.e. \(x\in E\) we have

*w*(

*X*) is a supermartingale. Therefore

*w*is an excessive function. On the other hand,

The following version of Kato’s inequality was proved by H. Brezis and A.C. Ponce [6] (see also H. Brezis, M. Marcus and A.C. Ponce [5]) in case *A* is the Laplace operator on a bounded domain in \({\mathbb R}^d\)).

### Theorem 6.3

*u*be a solution of (3.1). Then \(Au^+\in {\mathbb M}\) and

### Proof

### Remark 6.4

Applying in the proof of Theorem 6.1 the Itô-Meyer formula with right derivative of the function \(u\mapsto u^+\) we obtain (6.5) with \(\mathbf {1}_{\{u>0\}}\) replaced by \(\mathbf {1}_{\{u\ge 0\}}\). As a result, we get (6.2) with \(\mathbf {1}_{\{u>0\}}\) replaced by \(\mathbf {1}_{\{u\ge 0\}}\).

## 7 Equations with polynomial nonlinearity

*f*satisfying the condition

*p*.

### Proposition 7.1

If \(\mu \in V'_p \cap \mathbb {M}^+\) then \(\mu \) is a good measure relative to the function \(f(u)=-|u|^{p'}\).

### Proof

*u*be a solution of the equation

### Lemma 7.2

### Proof

Let \(B=\{u\ge \lambda \}\). Then \(\lambda ^{-1}u\ge \mathbf {1}_B\), so the required inequality follows immediately from the definition of \(\text{ Cap }_{A,p}\). \(\square \)

### Lemma 7.3

Let \(\mu \in {\mathcal {M}}_b^+\). If \(\mu \le c\cdot \)Cap\(_{A,p}\) for some \(c\ge 0\), then \(\mu \in V_{p}'\).

### Proof

### Lemma 7.4

*E*such that

### 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. \(\square \)

As a corollary to Lemma 7.4 we get the following proposition.

### Proposition 7.5

A measure \(\mu \in \mathbb {M}^+\) satisfies \(\mu \ll \text{ Cap }_{A,p}\) if and only if there exists an increasing sequence \(\{E_n\}\) of Borel subsets of *E* such that \(\mathbf {1}_{E_n}\cdot \mu \in V'_p\cap \mathbb {M}^+\) for \(n\in {\mathbb N}\) and \(\mu (E\setminus \bigcup _{n\ge 1}E_n)=0\).

### Theorem 7.6

Assume (7.1). If \(\mu \in \mathbb {M}\) and \(\mu ^+\ll \text{ Cap }_{A,p'}\) then \(\mu \in {\mathcal {G}}\).

### Proof

By Theorem 5.11 we may assume that \(\mu \ge 0\). By Lemma 4.10 there exists a strictly positive bounded excessive function \(\rho \) such that \(\mu \in {\mathcal {M}}_\rho ^+\), and by Proposition 7.5 there exists a sequence \(\{\mu _n\}\subset V'_{p'}\cap \mathbb {M}^+\) such that \(\lim _{n\rightarrow \infty }\Vert \mu _n-\mu \Vert _\rho =0\). Therefore it is enough to show that \(\mu _n\in {\mathcal {G}}\). But this follows from Proposition 7.1. \(\square \)

### Corollary 7.7

Assume that \(\mu \in {\mathbb M}\) and an let \( f(x,u)=-u^p\), \(x\in E\), \(u\ge 0\). Then \(\mu \in {\mathcal {G}}\) if and only if \(\mu ^+\ll \text{ Cap }_{A,p'}\).

### Proof

*u*be a solution of (4.1) with \(\mu \) replaced by \(\mu ^+\). Then \(u\in L^p(E;m)\) by Proposition 4.8. Therefore

### Corollary 7.8

### Proof

It suffices to repeat step by step the proof of [5, Theorem 16]. \(\square \)

### Remark 7.9

*D*with zero boundary condition (see Remark 4.13) and for a compact \(K\subset D\) the capacity \(\text{ Cap }^D_{\alpha ,p}(K)\) is defined by (1.5).

## Notes

### Acknowledgments

Research supported by National Science Centre Grant No. 2012-07-D-ST1-02107.

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