Nonlinear elliptic equations with self-adjoint integro-differential operators and measure data under sign condition on the nonlinearity

1.1 Formulation of the problem

Let E be a locally compact separable metric space and m be a full support positive Radon measure on E. Consider a self-adjoint operator ( A , D ( A ) ) on L ²(E; m) generating a Markov semigroup (T _t ), a Carathéodory function f : E × R → R satisfying the sign condition:

and Borel measure μ on E which obeys ∫ _E ρ d|μ| < ∞ for a strictly positive weight ρ : E → R (class of such measures shall be denoted by M ρ ). In the present paper, we study the existence problem for the following equation:

Throughout the paper, we assume that there exists Green’s function G for A (see Section 2.2), and T _t ρ ≤ ρ m-a.e. for any t ≥ 0 (e.g. both ρ ≡ 1 and ρ being the principal eigenfunction for −A satisfy the last requirement). By a solution to (2), we mean a Borel function u on E (finite m-a.e.) such that f ( ⋅ , u ) ∈ L ρ 1 ( E ; m ) , and

Let ( E , D ( E ) ) be the Dirichlet form generated by ( A , D ( A ) ) :

and Cap be the capacity generated by A: for open U ⊂ E,

and for arbitrary B ⊂ E, Cap(B) = inf{Cap(V): B ⊂ V, V is open}. Using this subadditive set function, we may consider a unique decomposition:

of any measure μ ∈ M ρ , where μ _c  ⊥ Cap and μ _d ≪ Cap. It is well known (see [1]) that if μ ≪ Cap, then there exists a solution to (2). On the other hand, by [2], Theorem 4.14], for any measure μ ∈ M ρ with non-trivial μ _c , one can find a function f (even non-increasing and independent of x ∈ E) satisfying the above conditions such that there is no solution to (2) with A = Δ on a bounded domain D ⊂ R d with zero Dirichlet boundary condition. This shows that the presence of the non-trivial concentrated part μ _c of measure μ completely changes the picture. Our goal is to study the existence and non-existence mechanism for (2) hidden behind the relation between operator A, right-hand side f and concentrated part μ _c .

1.2 Main results of the paper

Fix a strictly positive bounded Borel function ϱ ∈ L ¹(E; m) ∩ L ²(E; m) such that

and Rϱ is bounded (the required function always exists, see e.g. [3], Lemma 6.1]). For the existence results, we need one more assumption on f, namely that for any u ̲ , u ̄ ∈ L ϱ 1 ( E ; m ) such that f ( ⋅ , u ̲ ) , f ( ⋅ , u ̄ ) ∈ L ρ 1 ( E ; m ) we have

The above condition is satisfied e.g. provided that f is non-increasing with respect to y or there exists an increasing function g on R such that c 1 g ( y ) ≤ | f ( x , y ) | ≤ c 2 g ( y ) , x ∈ E , y ∈ R for some c ₁, c ₂ > 0. In Section 4, we extend Perron’s method of supersolutions and subsolutions and prove the following result (Theorem 1).

Notice that ϕ ≡ 1 can be taken provided that ρ ∈ L ¹(E; m).

Clearly, when μ is a good measure, then μ = μ ^*,f and so, by assertion (2) of the above theorem, for any μ ∈ G ( f ) there exists a maximal solution to (2). For brevity, and when there is no risk of confusion, we mostly omit the superscript f on μ ^*,f and u ^*,f .

In case G ≥ μ ( f ) ≠ ∅ , we may also consider the problem of the existence of the smallest good measure greater than μ. By a simple calculation we find that

where f ̃ ( x , y ) ≔ − f ( x , − y ) , x ∈ E , y ∈ R , is a solution to the latter problem. Therefore, by Theorem M.2 provided G ≥ μ ( f ) ≠ ∅ , we get the existence of μ * , f ∈ G ( f ) such that

Using the notion of reduced measures and the results of Theorems M.1 and M.2, we get the following result (we follow the idea of A.C. Ponce).

Observe that contrary to Theorem M.1, we do not demand in Theorem M.3 that the subsolution be less than or equal to the supersolution m-a.e.!

In Section 6, we prove a series of results concerning the properties of the set G ( f ) and the reduction operator μ ⟼ μ*. They exhibit that the properties of these two mathematical objects, proved before in the literature for Dirichlet Laplacian and non-increasing f, extend to the general framework considered here. The only troublesome property turns out to be the continuity of μ ⟼ μ*. The proofs of this property known in the literature make essential use of the monotonicity of f and even give Lipschitz continuity of μ ⟼ μ*. Here, due to merely sign condition imposed on f we are forced to adopt a different strategy. Therefore, the main concern of Section 6 will be continuity of the reduction operator. We also observe that certain mapping defined via the reduction operator is a continuous metric projection onto G ( f ) .

One of the illustrative results concerning the structure of the set G ( f ) , easily following from the results of Section 6, is the following equality

where A ( f ) is the class of admissible measures: it consists of measures μ ∈ M ρ such that | f ( ⋅ , R μ ) | ∈ L ρ 1 ( E ; m ) . In Section 7, we prove much stronger result. Namely, let us set B L ρ 1 ( 0 , r ) ≔ u ∈ L ρ 1 ( E ; m ) : ‖ u ‖ L ρ 1 ( E ; m ) ≤ r .

The last assertion implies that for any g, satisfying the same conditions as f, if

then μ ^*,f = μ ^*,g for positive μ ∈ M ρ (see Corollary 10). In other words, the reduction operator and the class of good measures, for f independent of the state variable, depend only on the behavior of f at infinity.

1.3 Some comments on the literature related to the problem

Concerning the existence results for (2) with μ ≪ Cap, we mention the paper by Brezis and Strauss [6], and by Konishi [7], where f is assumed to be non-increasing and independent of x ∈ E, and μ ∈ L ¹(E; m), the article by the author of the present paper and A. Rozkosz [8], where f is non-increasing, and their another paper [9], where f merely satisfies the sign condition.

As to the existence results for (2) with general measure data, above all, Bénilan and Brezis’ paper [10] should be mentioned. It was published in 2004, however it summarizes, among other things, the existence and non-existence results on the problem (2), with A = Δ and non-increasing f, achieved in the period 1975–2004 (see Appendix A in [10]).

In 2004 Brezis, Marcus and Ponce [2], [5] introduced the notion of reduced measures for (2) with A = Δ and f being non-increasing. Since then the research on equations of the form (2) has flourished once more, mostly with A being the Dirichlet Laplacian or Dirichlet (rarely regional) fractional Laplacian. We limit ourselves to mentioning [4], [11], [12], [13], [14], [15] in case of Dirichlet Laplacian or divergence form diffusion operators and [16], [17], [18], [19], [20] in case of the fractional Laplacian.

The theory of reduced measures introduced in [2], [5] for the Laplace operator has been generalized by the author of the present paper in [21] to the class of Dirichlet operators considered here but with f being non-increasing as in [2], [5]. The goal of the present paper is to analyze equation (2) under the assumption that f merely satisfies the sign condition (1).

The main results of the present paper may be divided into three groups. Theorem M.2 belongs to the first one and it is the extension to f’s satisfying sign condition (1) of the theory of reduced measure that is the main concern of [21] (also a slight strengthening it since in [21] we assumed that f(⋅, y) = 0 for y ≤ 0). The second group (Theorems M.1 and M.3) consists of problems that are only in (substantial) question for non-monotone f and has been considered before solely for diffusion/Laplace operator (see [4], [15]). Theorems M.4 and M.5 fall into the third group that consists of the results that partially strengthen the existing results for monotone f and generalize them to f satisfying sign condition (1).

2.1 Elements of potential theory

Let us remind that Cap is a set function defined by (4). We say that a property P holds quasi-everywhere (q.e. for short) on E if it holds except a set B ⊂ E such that Cap(B) = 0. An increasing sequence {F _n } of closed subsets of E is called a nest iff Cap(E\F _n ) → 0, n → ∞. A function u on E is called quasi-continuous iff for any ɛ > 0 there exists a closed set F _ɛ ⊂ E such that Cap(E\F _ɛ ) ≤ ɛ and u | F ε is continuous. By [22], Theorem 2.1.2], u is quasi-continuous if and only if there exists a nest {F _n } such that for any n ≥ 1, u | F n is continuous. An increasing sequence {F _n } of closed subsets of E is called a generalized nest iff for every compact K ⊂ E, Cap(K\F _n ) → 0, n → ∞. A Borel measure μ on E is called smooth iff it is absolutely continuous with respect to Cap, and there exists a generalized nest {F _n } such that |μ|(F _n ) < ∞, n ≥ 1. M ρ 0 stands for a subset of M ρ consisting of smooth measures. We say that a measurable function u on E is quasi-integrable iff for every ɛ > 0 there exists a closed set F _ɛ ⊂ E such that Cap(E\F _ɛ ) ≤ ɛ and 1 F ε u ∈ L 1 ( E ; m ) . We say that a measurable function u on E is locally quasi-integrable iff for every compact K ⊂ E, 1 _K u is quasi-integrable.

Since we assumed that (T _t ) is transient, there exists a strictly positive function g on E such that

Therefore, there exists the extension D e ( E ) of D ( E ) such that: D e ( E ) ⊂ L 1 ( E ; g ⋅ m ) , D ( E ) = D e ( E ) ∩ L 2 ( E ; m ) , and for any u ∈ D e ( E ) , there exists a sequence { u n } ⊂ D ( E ) that is a Cauchy sequence in the norm  ‖ ⋅ ‖ E and satisfies u _n → u in L ¹(E; g ⋅ m) and E ( u n , u n ) → E ( u , u ) (see [22], Theorem 1.5.1, Theorem 1.5.2]). Clearly, ( E , D e ( E ) ) is a Hilbert space. By [22], Theorem 2.1.7], any u ∈ D e ( E ) possesses an m-version which is quasi-continuous. In what follows for u ∈ D e ( E ) we denote by u ̃ quasi-continuous m-version of u.

2.2 Probabilistic potential theory

Let ∂ be either an isolated point added to E – provided E is compact – or a one-point compactification of E – provided E is not compact. We let E _∂ ≔ E ∪ {∂}. Throughout the paper, we adopt the convention that whenever f is a function defined on B ⊂ E, then it is automatically extended to B ∪ {∂} by putting f(∂) = 0. Let

Recall that a function ω: [0, ∞) → E _∂ is called cádlág if it is right-continuous on [0, ∞) and left-limited on (0, ∞). We endow Ω with the Skorohod topology d. Then (Ω, d) is a separable metric space (see [23], Section 12]). We also consider shift operators ( θ t ) t ≥ 0 :

and a family of projection operators (also called the canonical process) X _t : Ω → E _∂ , t ≥ 0, X _t (ω) ≔ ω(t), ω ∈ Ω. Let F t 0 ≔ σ ( X s − 1 ( B ) : s ≤ t , B ∈ B ( E ∂ ) ) . By [22], Theorem 7.2.1], there exists a family ( P x ) x ∈ E ∂ of Borel probability measures on Ω and a right-continuous filtration ( F t ) t ≥ 0 on Ω such that X ≔ ( P x ) x ∈ E ∂ , ( F t ) t ≥ 0 is a Hunt process on E _∂ associated with ( A , D ( A ) ) , i.e. for any f ∈ B b ( E ) ∩ L 2 ( E ; m ) ,

By the very definition of a Hunt process, F t 0 ⊂ F t , t ≥ 0 . The question of uniqueness of X is treated in [22], Theorem 4.2.8]. Let ζ stand for the lifetime of process X , i.e.

We let F ∞ ≔ σ ( F t : t ≥ 0 ) . For t ∈ [0, ∞], we denote by b F t a set of bounded real valued F t measurable functions. In what follows, we consider the following notation

We define for any f ∈ B + ( E ) ,

We let R ≔ R ₀. We say that a property P holds almost surely (a.s.) (resp. quasi almost surely (q.a.s.)) on Ω if it holds P _x -a.s. for any x ∈ E (resp. for q.e. x ∈ E). A Borel measurable positive function f on E is called α -excessive, where α ≥ 0, if

In case α = 0, we just say that f is excessive.

Recall that a family ( A t ) t ≥ 0 of R ∪ { + ∞ } valued functions on Ω is called an additive functional of X if there exist Λ ∈ F ∞ and N ⊂ E such that

1. θ _t (Λ) ⊂ Λ, t ≥ 0, Cap(N) = 0, P _x (Λ) = 1, x ∈ E\N,

   furthermore, for any ω ∈ Λ,

2. A _t+s (ω) = A _s (ω) + A _t (θ _s ω), s, t ≥ 0,

3. |A _t (ω)| < ∞, t ∈ [0, ζ(ω)),

4. t ⟼ A _t (ω) is cádlág on [0, ζ(ω)),

5. A _t (ω) = A _ζ(ω)(ω), t ≥ ζ(ω),

6. for any t ≥ 0, A _t is F t -measurable, and A ₀(ω) = 0.

Λ is called a defining set of (A _t ), and N is called an exceptional set of (A _t ). An additive functional (AF for short) (A _t ) of X is said to be positive if A _t (ω) ≥ 0, t ≥ 0, ω ∈ Λ. We say that an AF (A _t ) of X is continuous if t ⟼ A _t (ω) is continuous on [0, ∞) for any ω ∈ Λ. In what follows, we frequently use the notion of positive continuous additive functionals (PCAF for short) of X . We say that (A _t ) is a martingale additive functional (MAF for short) of X if it is an AF of X and an ( F t ) -martingale under measure P _x for any x ∈ E\N. Analogously, we say that (A _t ) is a local MAF of X if it is an AF of X , and it is a local ( F t ) -martingale under measure P _x for any x ∈ E\N. By [22], Theorem 5.1.4] there is a ono-to-one correspondence between PCAFs of X and positive smooth measures – so called Revuz duality. PCAF (A _t ) of X and positive smooth measure ν on E are in Revuz duality if for any positive f ∈ B ( E ) ,

By [22], Theorem 5.1.3, Theorem 5.1.4], there exists a unique PCAF of X satisfying the above identity for any f ∈ B + ( E ) . We shall denote it by A ^ν . We say that ν is strictly smooth if the exceptional set N of A ^ν is empty.

3.1 Part of A on an open set D ⊂ E

For a given self-adjoint Dirichlet operator A and an open set D ⊂ E, we may define self-adjoint Dirichlet operator A _|D as follows: let

By [22], Theorem 4.4.3], E | D is a Dirichlet form, and if E is regular, then E | D is regular too. Therefore, by [22], Theorem 1.3.1, Theorem 1.4.1], there exists a unique self-adjoint Dirichlet operator ( B , D ( B ) ) on L ²(D; m) (⊂ L ²(E; m)) such that D ( B ) ⊂ D ( E | D ) , and

We let A _|D ≔ B. The operator A _|D is called a part of A on D (or restriction of A to D). This operation on a Dirichlet operator A is often used when approaching the Dirichlet problem on D for A.

3.2 Perturbation of A by a smooth measure

Let ν be a positive smooth measure on E, and A be a regular self-adjoint Dirichlet operator on L ²(E; m). Define

By [27], Theorem IV.4.4], E ν is a symmetric Dirichlet form on L ²(E; m). Therefore, there exists a unique self-adjoint operator B on L ²(E; ν) such that D ( B ) ⊂ D ( E ν ) , and

We set A _ν ≔ B. Formally, −A _ν = −A + ν, so −A _ν may be called a perturbation of −A be the measure ν.

3.3 Resurrected (censored) operator

Let A be a regular self-adjoint Dirichlet operator on L ²(E; m) with regular symmetric Dirichlet form ( E , D ( E ) ) . By the Beurling–Deny formulae (see [22], Theorem 3.2.1]) for any u , v ∈ C c ( E ) ∩ D ( E ) ,

where E ( c ) is a symmetric form, with domain D ( E ( c ) ) = D ( E ) ∩ C c ( E ) , which satisfies the strong local property:

such that v is constant on a neighbourhood of supp[u]. J is a symmetric positive Radon measure on E × E\ diag and κ is a positive Radon measure on E. Such E ( c ) , J , κ are uniquely determined by ( E , D ( E ) ) . Let E res ( u , v ) ≔ E ( u , v ) − ∫ E u v d κ . By [28], Theorem 5.2.17], ( E res , D ( E res ) ) is a regular symmetric Dirichlet form on L ²(E; m) with D ( E res ) described in [28], (5.2.25)]. Therefore, there exists a unique self-adjoint Dirichlet operator ( B , D ( B ) ) such that D ( B ) ⊂ D ( E res ) and

We let A ^res ≔ B. Operator A ^res is very useful when trying to interpret the Dirichlet problem on D for purely jumping Dirichlet operators A (see e.g. [29]).