Semilinear equations for non-local operators: Beyond the fractional Laplacian

Abstract

We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. We also give results in case of bounded $C^{1,1}$ open sets.

Introduction

Let $D\subset {\mathbb{R}}^d$, $d\ge 2$, be a bounded open set, $f:D\times \mathbb{R}\to \mathbb{R}$ a function, $\lambda$ a signed measure on $D^c={\mathbb{R}}^d\setminus D$ and $\mu$ a signed measure on $\partial D$. In this paper we study the semilinear problem

$$\begin{array}{cccc}\hfill -Lu\left(x\right)& \hfill =\hfill & f(x,u\left(x\right))\hfill & \text{in }D\hfill \\ \hfill u& \hfill =\hfill & \lambda \hfill & \text{in }{D}^c\hfill \\ \hfill W_{D}u& \hfill =\hfill & \mu \hfill & \text{on }\partial D.\hfill \end{array}$$

The operator $L$ is a second-order operator of the form $L=-\varphi (-\Delta )$ where $\varphi :(0,\infty )\to (0,\infty )$ is a complete Bernstein function without drift satisfying certain weak scaling conditions. The operator $L$ can be written as a principal value integral

$$Lu\left(x\right)=\text{P.V.}{\int }_{{\mathbb{R}}^d}(u\left(y\right)-u\left(x\right))j\left(|y-x|\right)dy,$$

where the singular kernel $j$ is completely determined by the function $\varphi$. In case $\varphi \left(t\right)=t^{\alpha ∕2}$, $\alpha \in (0,2)$, $-L$ is the fractional Laplacian ${(-\Delta )}^{\alpha ∕2}$ and the kernel $j\left(|y-x|\right)$ is proportional to ${|y-x|}^{-d-\alpha }$.

The operator $W_D$ 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 $C^{1,1}$ open sets this is equivalent to the notion of weak $L^1$ solution as in [1, Definition 1.3].

For the nonlinearity $f$ throughout the paper we assume the condition

(F) $f:D\times \mathbb{R}\to \mathbb{R}$ is continuous in the second variable and there exist a function $\rho :D\to [0,\infty )$ and a continuous function $\Lambda :[0,\infty )\to [0,\infty )$ such that $\left|f(x,t)\right|\le \rho \left(x\right)\Lambda \left(\left|t\right|\right)$.

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 $L$ 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 $L$. 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 $C^{1,1}$ open set $D$, 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 $L$ in the recent preprint [8]. The second main contribution is that we obtain some of the results from [1] (which deals with $C^{1,1}$ open sets) for regular open subsets of ${\mathbb{R}}^d$. 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 $L$, the underlying stochastic process $X={\left(X_t\right)}_{t\ge 0}$ and its killed version upon exiting an open set, the notion of harmonic function with respect to $X$ (or $L$), and the Green, Poisson and Martin kernel of an arbitrary open subset of ${\mathbb{R}}^d$. 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 $f$ 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 $L$ in bounded $C^{1,1}$ 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 $W_D$ with the boundary operator used in [1], [2].

The last section revisits Theorem 3.10 and Corollary 3.8 in bounded $C^{1,1}$ sets. In case when $f(x,t)=W\left({\delta }_D\left(x\right)\right)\Lambda \left(t\right)$ for some function $W$, we give a sufficient and necessary integral condition for (a version of) Theorem 3.10 to hold in terms of $W$, $\Lambda$ and the renewal function. Building on Lemma 4.5 we next give a sufficient condition for Corollary 3.8 to hold in a bounded $C^{1,1}$ set. Finally, we end by establishing Theorem 5.3 that extends Corollary 3.8 for nonnegative nonlinearities $f$. 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 $D\subset {\mathbb{R}}^d$ be an open set. Then $C_b\left(D\right)$ denotes the family of all bounded continuous real valued functions on $D$, $C_0\left(D\right)$ the family of all continuous functions vanishing at infinity (i.e. $f\in C_0\left(D\right)$ if for every $ϵ>0$ there exists a compact subset $K\subset D$ such that $\left|f\left(x\right)\right|<ϵ$ for all $x\in D\setminus K$), $C_c^{\infty }\left(D\right)$ the family of all infinitely differentiable functions with compact support, $\mathcal{B}\left(D\right)$ Borel measurable functions on $D$, and ${\mathcal{B}}_b\left(D\right)$ bounded Borel measurable functions on $D$. If $\mu$ is a measure on $D$, then $L^1(D,\mu )$ denotes integrable functions, $L_{loc}^1(D,\mu )$ locally integrable functions and $L^{\infty }(D,\mu )$ essentially bounded functions on $D$. In case when $\mu$ is the Lebesgue measure on $D$, we simply write $L^1\left(D\right)$, $L_{loc}^1\left(D\right)$ and $L^{\infty }\left(D\right)$. Denote by $\partial D$ the boundary of $D$, ${\delta }_D\left(x\right)=\mathrm{dist}(x,\partial D)$ if $x\in D$, and ${\delta }_{D^c}\left(z\right)=\mathrm{dist}(z,\partial D)$ if $z\in {\overline{D}}^c$. For $U\subset D$ open, $U\subset \subset D$ denotes that the closure $\overline{U}$ is contained in $D$. For $A\subset {\mathbb{R}}^d$, $\mathcal{M}\left(A\right)$ denotes $\sigma$-finite signed measures on $A$ and $\left|\lambda \right|$ denotes the variation of $\lambda \in \mathcal{M}\left(A\right)$. For two positive functions $f$ and $g$, $f⪯g$ means that the quotient $f∕g$ stays bounded from above by a positive constant, and $f\asymp g$ that the quotient $f∕g$ stays bounded between two positive constants. Finally, unimportant constants in the paper will be denoted by small letters $c$, $c_1$, $c_2$, $\dots$, and their labeling starts anew in each new statement. More important constants we denote by a big letter $C$, where e.g. $C(a,b)$ means that the constant $C$ depends only on parameters $a$ and $b$.