On semilinear elliptic equations with diffuse measures

We consider semilinear equation of the form -Lu=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}$$-Lu=f(x,u)+\mu $$\end{document}, where L is the operator corresponding to a transient symmetric regular Dirichlet form E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}$$\end{document}, 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 diffuse measure with respect to the capacity associated with E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}$$\end{document}, and the lower-order perturbing term f(x, u) satisfies the sign condition in u and some weak integrability condition (no growth condition on f(x, u) as a function of u is imposed). We prove the existence of a solution under mild additional assumptions on E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}$$\end{document}. We also show that the solution is unique if f is nonincreasing in u.


Introduction
Let E be a locally compact separable metric space, m be a positive Radon measure on E such that supp [m] = E, and let (E, D(E)) be a regular transient symmetric Dirichlet form on L 2 (E; m). In this paper, we consider semilinear equations of the form − Lu = f (·, u) + μ.
The study of problems of the form (1.3) with μ ∈ L 1 (D; dx) was initiated by Brezis and Strauss [9] (in fact, in [9] more general second-order elliptic differential operator is considered). In [9] it is proved that if f satisfies the sign condition and where F a (x) = sup |y|≤a |f (x, y)|, x ∈ E (1. 5) with E = D, then there exists a solution to (1.3) for μ belonging to some class which is "arbitrarily smaller" than L 1 (D; dx). If f satisfies stronger monotonicity condition: (f (x, y 1 ) − f (x, y 2 ))(y 1 − y 2 ) ≤ 0, x ∈ E, y 1 , y 2 ∈ R, (1.6) then the solution exists for any μ ∈ L 1 (D; dx) and is unique. Later, Gallouët and Morel [16] proved the existence of a solution to (1.3) for any μ ∈ L 1 (D; dx) and f satisfying (1.2), (1.5). Orsina and Ponce [28] have subsequently generalized and strengthened this result by showing that a solution to (1.3) exists for any diffuse measure μ and any f satisfying (1.2) and an integrability condition weaker than (1.5).
Equations of the form (1.1) in the case where L is a general, possibly nonlocal, operator associated with a transient regular Dirichlet form were considered by Klimsiak and Rozkosz [22,24] in case f satisfies the monotonicity condition, and by Klimsiak [20] in case f satisfies the sign condition (in fact, in [20] systems of equations with right-hand side satisfying a generalized sign condition are considered).
In [20,22,24] the proofs of the existence results rely heavily on probabilistic methods. In particular, we make an extensive use of the theory of backward stochastic differential equations and we use some results from stochastic analysis and probabilistic potential theory. In the present paper we give new, rather short analytical proofs of some of the results of [20,22]. We are motivated by the desire to make them accessible to people working in PDEs that are not familiar with probabilistic methods.
Let D e (E) denote the extended Dirichlet space of (E, D(E)). In the present paper we provide a proof of the existence of a solution u in the sense of duality (or, equivalently, renormalized solution; see Sect. 3 We also show that if u is a solution to (1.1), then T k (u) = ((−k)∨u)∧k ∈ D e (E) for every k > 0 and where μ T V is the total variation norm of μ. Furthermore, if (1.6) is satisfied, then the solution u is unique. Condition (1.7) holds true in many interesting situations. For instance, it holds if the embedding V 1 → L 2 (E; m) is compact, (1.8) where V 1 denotes the space D(E) equipped with the norm determined by the form E 1 (·, ·) := E(·, ·) + (·, ·). Another condition, which is often satisfied in practice and implies (1.7), is the so called absolute continuity condition saying that R α (x, ·) m for any α > 0 and x ∈ E, (1.9) where R α (x, ·) is the resolvent kernel associated with E. For symmetric forms considered in this paper, condition (1.9) is equivalent to the condition m for any t > 0 and x ∈ E, (1.10) where P t (x, ·) is the transition kernel associated with E.
The main idea of our proofs resembles the idea used in case of problem (1.3) (see the proof of Theorem B.4 in Brezis et al. [8] and also Ponce [31,Chapter 19]). Let V denote the extended space D e (E) equipped with the norm determined by E. We first prove the existence of a solution to (1.1) with μ ∈ M 0,b ∩ V , where M 0,b is the set of all diffuse measures on E and V is the dual of V . This step can be viewed as some modification of the result of Brezis and Browder [7] on absorption problems (1.3) with μ ∈ H −1 (D). To get the existence for general μ ∈ M 0,b , we approximate it by a suitably chosen sequence {μ n } ⊂ M 0,b ∩ V and show that solutions u n of (1.1) corresponding to the measures μ n converge to a solution of (1.1). In this second step we use some a priori estimates for u n in V and condition (1.7). In [25] it is proved that any μ ∈ M 0,b admits decomposition of the form μ = g + ν with g ∈ L 1 (E; m) and ν ∈ M 0,b ∩ V (this generalizes the corresponding result proved by Boccardo et al. [5] for the form associated with Δ). Therefore, similarly to [8], in the second step of the proof it is enough to approximate by {μ n } the measure μ = g · m. This, however, does not simplify the reasoning, so in the present paper we give a direct approximation of μ ∈ M 0,b (without recourse to [25]).
In the present paper we confine ourselves to single equation with operator corresponding to symmetric regular Dirichlet forms. For results (proved with the help of probabilistic methods) for quasi-regular, possibly nonsymmetric forms, we refer the reader to [24], and for results for systems of equations to [20]. Also note that equations with f = 0 but Δ replaced by the Schrödinger operator are treated in [29] and [31,Chapter 22].
In the paper we deal exclusively with equations with diffuse measures. (see [2,3,8]). Results on (1.3) with general bounded measure μ and f satisfying the monotonicity condition are found in [3,8,13], and for equations with f satisfying the sign condition (1.2) in [31,Chapter 21]. The Dirichlet problem for linear equations with nonlocal operators and bounded measure μ is studied in [19,26,30]. In Klimsiak [21] general equations of the form (1.1) with general bounded measure μ and f satisfying (1.6) are considered. The question whether one can extend the existence results of [31] to some nonlocal operators or extend some existence results of [21] to f satisfying (1. 2) remains open.

Preliminaries
In this paper, E is a locally compact separable metric space and m is a Radon measure such that supp[m] = E, i.e. m is a nonnegative measure on the σ-field of Borel subsets of E which is finite on compact sets and strictly positive on nonempty open sets.
In what follows (E, D(E)) is a symmetric regular Dirichlet form on L 2 (E; m). We denote by (·, ·) the usual inner product in L 2 (E; m). As usual, In the whole paper we assume that (E, D(E)) is transient. Recall that this means that there exists a bounded strictly m-a.e. positive g ∈ L 1 (E; m) such that For an equivalent formulation, see [15,Section 1.5]. The extended Dirichlet space associated with (E, D(E)) (see [15,Section 1.5] for the definition) will be denoted by (E, D e (E)). Note that D e (E) with the inner product E is a Hilbert space (see [15,Theorem 1.5.3]). In the sequel this space will be denoted by V . We denote by V the dual space of V . The duality pairing between V and V will be denoted by ·, · . In the paper we define 0-order quasi notions with respect to E (capacity Cap (0) , exceptional sets, nests, quasi-continuity) as in [ Then, as usual, for an arbitrary A ⊂ E, we set We say that A ⊂ E is exceptional if Cap (0) (A) = 0, and we say that a property of points in E holds quasi-everywhere (q.e. in abbreviation) if it holds outside some exceptional subset of E. For a measure μ on E and a function u : E → R, we use the notation whenever the integral is well defined. For a signed Borel measure μ, we denote by μ + and μ − its positive and negative parts, and by |μ| the total variation measure, i.e. |μ| = μ + + μ − . We denote by In particular, if μ ∈ M 0,b ∩V , then the function Gμ is well defined and belongs to V . Let B + (E) (resp. B b (E)) denote the set of all positive (resp. bounded) real Borel functions on E, and let (G α ) α>0 denote the strongly continuous resolvent on L 2 (E; m) associated with (E, D(E)). Recall that αG α is Markovian for each Since G α is positivity preserving, we can extend it to any positive f ∈ B + (E) by where μ n = 1 Fn · μ and {F n } is a generalized nest such that μ n ∈ M 0,b ∩ V , n ≥ 1. Note that Gμ is defined uniquely up to m-equivalence.
In the paper, for a function u on E, we denote byũ its quasi-continuous m-version (whenever it exists). We will freely use, without explicit mention, the following fact: if u 1 ≤ u 2 m-a.e. and u 1 , u 2 have quasi-continuous m-versions, Let β > 0. In the proof of the lemma below we will need the symmetric form E (β) defined by Since βG β is a symmetric linear operator on L 2 (E; m), by [15,Lemma 1.4.1] there exists a unique nonnegative symmetric Radon measure σ on the product space E × E such that for any Borel functions u, v ∈ L 1 (E; m), with respect to m. Then 0 ≤ s β ≤ 1 m-a.e., and by a direct computation one can check that for a Borel u ∈ L 2 (E; m) one can rewrite and From this we conclude that for any β > 0, Letting β → ∞ and using [15, Theorem 1.5.2(ii)] we obtain the desired inequality.
For k ≥ 0 and u : E → R, we write It is clear that u n u m-a.e. By the 0-order version of [15, (2.1.10)], for all ε, δ > 0 we have Sinceũ n+1 −ũ n ∈ D e (E), it follows from the above inequality and (2.10) that Taking ε = 2 −n and letting δ 0 we get  (2.11) it is clear that {F n } is a nest andũ defined q.e. as u = lim n→∞ũn is quasi-continuous. Of course,ũ is an m-version of u. and Proof. By Lemma 2.2, Gμ + , Gμ − are finite m-a.e., so Gμ is well defined m-a.e. Let {F n } be a generalized nest for μ such that μ n = 1 Fn μ ∈ V , n ≥ 1. Set u n = Gμ n , u = Gμ. Then u n ∈ D e (E) and by (2.2) and (2.3), By the Banach-Saks theorem, there is a subsequence (n l ) such that the Cesàro Proof. By Lemma 2.3, T k ( Gμ) ∈ D e (E) and (2.12) holds true. By this and the 0-order version of [15, (2.1.10)], for all ε, δ > 0 we have which implies the desired inequality. Proof. We can and do assume that μ n ≤ 2 −2n , n ≥ 1. Then by Lemma 2.4, This proves the lemma because by the definition of F , Gμ n → 0 q.e. on F .
which proves the lemma.

14)
where Gη is a quasi-continuous m-version of Gη.
Proof. We first assume that μ ∈ M 0,b ∩ V and η ∈ L 1 (E; m) ∩ V . Then Since E is symmetric, this implies (2.14). Now assume that μ ∈ M 0,b , η ∈ L 1 (E; m) and Gη is bounded. Let {F n } be a generalized nest such that By what has already been proved, Letting n → ∞ we get (2.14).

Existence and uniqueness of solutions
Throughout this section, we assume that μ ∈ M 0,b and f : E × R → R is a Carathéodory function, i.e. f (·, y) is measurable on E for each fixed y ∈ R, and f (x, ·) is continuous on R for each fixed x ∈ E.
Following [22] we adopt the following definition.

Remark 3.5.
In [23] the following definition of a solution of (1.1) is introduced: and for every k ∈ N and every bounded v ∈ D e (E), Note that in case of local operators, this is essentially [ Proof. Let {F n } be a generalized nest such that 1 Fn (|f (·, u)|·m+|μ|) ∈ M 0,b ∩ V . For n ≥ 1 we set f n = 1 Fn f (·, u), μ n = 1 Fn · μ and u n = G(f n · m + μ n ). Then u n ∈ D e (E). For a > 0, k ∈ N we set Since ψ := (1/k)ψ a,k satisfies the assumptions of Lemma 2.1, ψ a,k (u n ) ∈ D e (E) and E(u n , ψ a,k (u n )) ≥ 0. Letũ n be a quasi-continuous m-version of u n . Then ψ a,k (ũ n ) is a quasicontinuous m-version of ψ a,k (u n ). By Proposition 3.3, E(u n , ψ a,k (u n )) = f n · m, ψ a,k (u n ) + μ n , ψ a,k (ũ n ) . Hence By Proposition 3.3(v),ũ n →ũ q.e. Therefore letting n → ∞ in (3.4) and using the dominated convergence theorem we obtain

By (1.2) and the definition of
Letting k → ∞ in the above inequality yields part (i) of the lemma. To get (ii), we observe that v = u 1 − u 2 is a solution to the problem ). Since f satisfies (1.6), g satisfies (1.2). Therefore the desired inequality follows from part (i).
Note that from Lemma 3.6(i) with a = 0 the following absorption estimate follows: Proof. By Proposition 3.3(ii) and Lemma 2.3, T k (u) ∈ D e (E) and which when combined with (3.5) yields (3.6).  Proof. We divide the proof into two steps.
Step 1 We first assume that Under the hypothesis (1.5) the operator A n : V → V defined as A n (u) = −Lu − f n (u) is pseudomonotone (see, e.g., [32, Section 2.1] for the definition). Indeed, it is clear that A n maps bounded sets of V into bounded sets of V . Next, suppose that u k → u weakly in V . Then for any v ∈ V , Furthermore, by (1.7), we can assume that u k → u m-a.e. Consequently, we can assume that f n (·, u k ) → f n (·, u) in V . Therefore, for any v ∈ V , Accordingly, A n is pseudomonotone. Since by (1.2), for u ∈ V we have A n u, u = E(u, u) − (f n (·, u), u) ≥ E(u, u), the operator A n is also coercive. Therefore A n is surjective by standard result in the theory of pseudomonotone mappings (see, e.g., [32, Theorem 2.6]). Thus, there exists a weak solution u n ∈ D e (E) of the equation i.e. for any v ∈ D e (E), Taking u n as a test function in (3.8) we get By (1.7) and (3.10) there is u ∈ D e (E) and a subsequence (still denoted by n) such that u n → u m-a.e. and weakly in V . Then, by the definition of f n , f n (·, u n ) → f (·, u) m-a.e. By (3.10), for any Borel subset B of E and a > 0 we have From the above inequality and (1.5) we conclude that the sequence {f n (·, u n )} is equi-integrable and tight. Hence f n (·, u n ) → f (·, u) in L 1 (E; m) by Vitali's convergence theorem (see, e.g., [14,Theorem 2.24]). Therefore letting n → ∞ in (3.8) we see that for any bounded v ∈ D e (E). Let η ∈ L 1 (E; m) be such that G|η| ∞ < ∞, and let {F n } be a generalized nest such that η n = 1 Fn η ∈ V . Then Gη n is bounded and Gη n ∈ V . Therefore taking v = Gη n as a test function in (3.11) we get By Lemma 2.5, Gη n → Gη as n → ∞. Therefore letting n → ∞ in the above equation we obtain (3.1). Thus u is a solution of (1.1).
Step 2 We now show how to dispense with the assumption that μ + , μ − ∈ M 0,b ∩ V . By the 0-order version of [ Step 1, there exists a solution u n ∈ D e (E) of the equation (3.12) In particular, for any η ∈ L 1 (E; m) such that G|η| is bounded. By Lemma 3.6, for any Borel subset B of E and a > 0 we have (3.14) By Proposition 3.8, whereas by the 0-order version of [15, (2.1.10)] and (3.19), If k > a, then {|u n | > a} = {|T k (u n )| > a}, so for any k > a, Hence From this and (3.14) it follows that if m(B) ≤ γ, then B |f (x, u n (x))| m(dx) ≤ ε. Furthermore, by (1.5) and σ-finitness of m, there exists a Borel set E 0 ⊂ E such that m(E 0 ) < ∞ and E\E0 F a (x) m(dx) < ε/2. Therefore taking B = E \ E 0 in (3.14) we get E\E0 |f (x, u n (x))| m(dx) ≤ ε. This shows that the sequence {f (·, u n )} is equiintegrable and tight. On the other hand, by (1.7) and (3.15), for each k > 0 the sequence {T k (u n )} n is, up to a subsequence, convergent m-a.e., so using the diagonal argument, one can find a subsequence, still denoted by (n), such that {u n } converges m-a.e. to some u. Hence f (·, u n ) → f (·, u) m-a.e. Consequently, by Vitali's theorem, f (·, u n ) → f (·, u) in L 1 (E; m). Let k > 0, and let g be a strictly positive function such that Gg ∞ < ∞ and g ∈ L 1 (E; m). Taking η = g1 {un≥k} and η = −g1 {un≤−k} as test functions in (3.13) we obtain (3.17) We already know that the sequence {f (·, u n )} is equi-integrable and tight.
Since u n is a solution of (3.12), for any η ∈ L 1 (E; m) such that G|η| ∞ < ∞ and any k ≥ 0 we have  (3.18) where g k = kg 1+kg . By what has already been proved, letting n → ∞ in (3.18) yields Since μ(E \ F ) = 0 by (2.6), letting k → ∞ and using Lemma 2.5 shows that u is a solution to (1.1). can be shortened. Indeed, by (1.6) and Lemma 3.6(ii), Since By the above, {f (·, u n )} is convergent in L 1 (E; m). The rest of the proof runs as the proof of Theorem 3.10 (see the reasoning following the statement that {f (·, u n )} is equi-integrable).
If (1.8) is satisfied, then the following Poincaré-type inequality holds true: there exists c > 0 such that This condition is sometimes expressed by saying that E is Green-bounded (see, e.g., [6,10] Then (E c , D(E)) is a regular symmetric Dirichlet form on L 2 (E; m). Obviously,

Proposition 3.13.
Assume that E is Green-bounded. If u is a solution to (1.1), then u ∈ L 1 (E; m).
(ii) (Neumann boundary conditions) Assume additionally that ∂D is Lipschitz. Consider the form E defined by (4.4), but with domain H 1 (D). Then (E, H 1 (D)) is a regular symmetric Dirichlet form on L 2 (D; dx) with D = D ∪ ∂D (see [15,Example 4.5.3]), and clearly so is (E λ , H 1 (D)) with λ ≥ 0. Moreover, if λ > 0, then (E λ , H 1 (D)) is transient because D is Green-bounded (see Lemma 3.12). The generator L λ of (E λ , H 1 (D)) is equal to L − λ, where L is the generator of (E, H 1 (D)). By Rellich's theorem, H 1 (D) → L 2 (D; dx) is compact, so the results of the paper apply to Eq. (1.1) with L replaced by the operator L − λ defined above. A solution u to such equation can be viewed as a solution to the Neumann problem where n denotes the unit outward normal to ∂D.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.