Semilinear equations for non-local operators: Beyond the fractional Laplacian
Introduction
Let , , be a bounded open set, a function, a signed measure on and a signed measure on . In this paper we study the semilinear problem
The operator is a second-order operator of the form where is a complete Bernstein function without drift satisfying certain weak scaling conditions. The operator can be written as a principal value integral where the singular kernel is completely determined by the function . In case , , is the fractional Laplacian and the kernel is proportional to .
The operator is a boundary trace operator first introduced in [14] in the case of the fractional Laplacian, and extended to more general non-local operators in [8] — see Section 2.6 for the precise definition.
Motivated by the recent preprint [3] we consider solutions of (1.1) in the weak dual sense, cf. Definition 3.1, and show that for bounded open sets this is equivalent to the notion of weak solution as in [1, Definition 1.3].
For the nonlinearity throughout the paper we assume the condition
(F) is continuous in the second variable and there exist a function and a continuous function such that .
Semilinear problems for the Laplacian have been studied for at least 40 years and we refer the reader to the monograph [42] for a detailed account. The study of semilinear problems for non-local operators is more recent and is mostly focused on the fractional Laplacian, see [1], [2], [4], [5], [6], [14], [18], [23], [24]. One of the important differences between the local and non-local equations is that in the non-local case the boundary blow-up solutions are possible even for linear equations. To be more precise, there exist non-negative harmonic functions for the operator that blow up at the boundary. In this paper we will restrict ourselves to the so called moderate blow-up solutions, that is those bounded by harmonic functions with respect to the operator . This restriction is a consequence of the problem (1.1) itself, namely of the boundary trace requirement on the solution. In this respect we follow [1], [14] where the boundary behavior of solutions was also imposed. Note that in [1] the theory was developed for the fractional Laplacian in a bounded open set , while [14] extends part of the theory to regular open sets. This extension was possible mainly due to potential-theoretic results from [15].
The goal of this paper is to generalize results from [1], [14] and at the same time to provide a unified approach. The first main contribution of the paper is that we replace the fractional Laplacian with a more general non-local operator. This is possible due to potential-theoretic and analytic properties of such operators developed in the last ten years. For the most recent development see [9], [10], [28], [29], [30]. Here we single out the construction of the boundary trace operator for the operator in the recent preprint [8]. The second main contribution is that we obtain some of the results from [1] (which deals with open sets) for regular open subsets of . To achieve this goal we combine methods from [1] with those of [14].
Let us now describe the content of the paper in more detail. In the next section we introduce notions relevant to the paper and recall known results. This includes the notion of the non-local operator , the underlying stochastic process and its killed version upon exiting an open set, the notion of harmonic function with respect to (or ), and the Green, Poisson and Martin kernel of an arbitrary open subset of . We explain accessible and inaccessible boundary points and its importance to the theory. The boundary trace operator is introduced in Section 2.6, cf. Definition 2.1. The section ends with several auxiliary results about continuity of Green potentials.
Section 3 is central to the paper and contains two main results on the existence of a solution to the semilinear problem (1.1) in arbitrary bounded open sets. The first result, Theorem 3.6, can be thought of as a generalization of [1, Theorem 1.5]. It assumes the existence of a subsolution and a supersolution to the problem (1.1) and gives several sufficient conditions for the existence of a solution. As in almost all existence proofs of semilinear problems, the solution is obtained by using Schauder’s fixed point theorem. As a corollary of the third part of that theorem, in Corollary 3.8 we obtain a generalization of the main result of [14]. Theorem 3.10 deals with nonpositive function and is a generalization of [1, Theorem 1.7]. The main novelty of our approach is contained in using Lemma 3.9 to approximate a nonnegative harmonic function by an increasing sequence of Green potentials. This replaces the approximation used in [1] which works only in smooth open sets.
In the last two sections we look at the semilinear problem for in bounded open sets and at some related questions. In Section 4 we first recall the notion of the renewal function whose importance comes form the fact that it gives exact decay rate of harmonic functions at the boundary. We then state known sharp two-sided estimates for the Green function, Poisson kernel, Martin kernel and the killing function in terms of the renewal function. Section 4.3 may be of independent interest — there we give the boundary behavior of the Green potential and the Poisson potential of a function of the distance to the boundary. We next provide a sufficient integral condition (in terms of the renewal function) for a function of the distance to the boundary to be in the Kato class. In Section 4.6 we invoke a powerful result from [28] to show the existence of generalized normal derivative at the boundary which is used in the equivalent formulation of the weak dual solution. We end the section with a discussion on the relationship of the boundary trace operator with the boundary operator used in [1], [2].
The last section revisits Theorem 3.10 and Corollary 3.8 in bounded sets. In case when for some function , we give a sufficient and necessary integral condition for (a version of) Theorem 3.10 to hold in terms of , and the renewal function. Building on Lemma 4.5 we next give a sufficient condition for Corollary 3.8 to hold in a bounded set. Finally, we end by establishing Theorem 5.3 that extends Corollary 3.8 for nonnegative nonlinearities . This result generalizes [1, Theorem 1.9].
TheAppendix has two parts. In the first part we provide a proof of Lemma 3.9 in a more general context. In the second part, we give quite technical proofs of Proposition 4.1, Proposition 4.2. The proof of Proposition 4.1 is modeled after the proof of [3, Theorem 3.4], while the proof of Proposition 4.2 is somewhat simpler.
We end this introduction with a few words about notation. Let be an open set. Then denotes the family of all bounded continuous real valued functions on , the family of all continuous functions vanishing at infinity (i.e. if for every there exists a compact subset such that for all ), the family of all infinitely differentiable functions with compact support, Borel measurable functions on , and bounded Borel measurable functions on . If is a measure on , then denotes integrable functions, locally integrable functions and essentially bounded functions on . In case when is the Lebesgue measure on , we simply write , and . Denote by the boundary of , if , and if . For open, denotes that the closure is contained in . For , denotes -finite signed measures on and denotes the variation of . For two positive functions and , means that the quotient stays bounded from above by a positive constant, and that the quotient stays bounded between two positive constants. Finally, unimportant constants in the paper will be denoted by small letters , , , , and their labeling starts anew in each new statement. More important constants we denote by a big letter , where e.g. means that the constant depends only on parameters and .
Section snippets
The process and the jumping kernel
Let be a pure jump Lévy process in , , with the characteristic exponent given by where is a measure on satisfying — the Lévy measure. Thus the Fourier transform of the distribution of is given by We further assume that where is a complete Bernstein function, cf. [45, Chapter 6]. This means that where is a completely
The semilinear problem in bounded open set
Let us now turn to the semilinear problem. For functions and let be a function defined by
Definition 3.1 Let be a function, and a measure concentrated on , such that on . A function is called a weak dual solution to the semilinear problem if satisfies the equality for every . If in the equation
The renewal function
We start this section by introducing a function which plays a prominent role in studying the boundary behavior in open sets.
Let be a one-dimensional subordinate Brownian motion with the characteristic exponent , . We can think of as one of the components of the process . Let be the supremum process of and let be the local time of at zero. We refer the readers to [7, Chapter VI] for details. The inverse local time is
Corollary 3.8 revisited
Recall that in Corollary 3.8 we assumed that the function satisfies (F) with nondecreasing and that and , where . We give sufficient conditions for these assumptions in case of a bounded open set. We will additionally assume that for a function and that satisfies the following doubling condition: There exists such that This implies that for all there exists such that
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