Boundary blow-up solutions to fractional elliptic equations in a measure framework

Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $R^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$ and $\omega$ be the Hausdorff measure on $\partial\Omega$. We denote by $\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}$ a measure $$\langle\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha},f\rangle=\int_{\partial\Omega}\frac{\partial^\alpha f(x)}{\partial \vec{n}_x^\alpha} d\omega(x),\quad f\in C^1(\bar\Omega),$$ where $\vec{n}_x$ is the unit outward normal vector at point $x\in\partial\Omega$. In this paper, we prove that problem $$ \begin{array}{lll} (-\Delta)^\alpha u+g(u)=k\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}\quad&{\rm in}\quad \bar\Omega,\\[2mm] \phantom{(-\Delta)^\alpha +g(u)} u=0\quad&{\rm in}\quad \Omega^c \end{array} $$ admits a unique weak solution $u_k$ under the hypotheses that $k>0$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$ and $g$ is a nondecreasing function satisfying extra conditions. We prove that the weak solution is a classical solution of $$ \begin{array}{lll} \ \ \ (-\Delta)^\alpha u+g(u)=0\quad&{\rm in}\quad \Omega,\\[2mm] \phantom{------\} \ u=0\quad&{\rm in}\quad R^N\setminus\bar\Omega,\\[2mm] \phantom{} \lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty. \end{array} $$

Comparing with the Laplacian case, a much richer structure for the solutions set appears for the non-local case. Recently, the authors in [5] obtained very different phenomena of the boundary blow-up solutions to elliptic equations involving the fractional Laplacian, precisely, (−∆) α u + |u| p−1 u = 0 in Ω, u = 0 in Ω c , (1.2) lim x∈Ω,x→∂Ω u(x) = +∞, where p > 0 and the fractional Laplacian (−∆) α with α ∈ (0, 1) is defined by here for ǫ > 0, The existence of boundary blow-up solution of (1.2) is derived by constructing appropriate super and sub-solutions and this construction involves the one dimensional truncated Laplacian of power functions given by where τ ∈ (−1, 0) and χ (0,1) is the characteristic function of the interval (0, 1). It is known that there exists a unique zero point of (1.3) in (−1, 0), denoting τ 0 (α). Then Special existence for τ = τ 0 (α). Assume that .
Then for any t > 0, there is a positive solution u of equation ( Our interest in this article is to introduce a new method to study the boundary blow-up solutions of semilinear fractional elliptic equations and answer above questions. The main idea is to find suitable type measure concentrated on the whole boundary and then by making basic estimates to prove that the corresponding weak solution solves (1.2). Our first result is stated as follows: Proposition 1.2 Let α ∈ (0, 1) and τ 0 (α) is the zero point of C(·) when C(·) given by (1.3), then τ 0 (α) = α − 1.
We observe that the critical value 1 − 2α τ 0 (α) in Proposition 1.1 turns out to be 1+α 1−α . In what follows, we would like to show the details of our new method and answer the second and third questions in the following.

A new method and main results
Let α ∈ (0, 1) and ω be the Hausdorff measure on ∂Ω. We denote by ∂ α ω ∂ n α a measure where n x is the unit inward normal vector of ∂Ω at point x and In this paper, we are concerned with the existence and uniqueness of weak solution to the semilinear fractional elliptic problem where k > 0 and g : R + → R + is continuous. In [6], the authors studied problem (1.6) replaced ∂ α ω ∂ n α by ∂ α ν ∂ n α where ν is a Radon measure concentrated on boundary measure. They proved that such a problem has a unique weak solution if g is a continuous nondecreasing function satisfying g(0) ≥ 0 and Moreover, [6] analyzed the isolated singularity of weak solution of (1.6) in the case that ν = δ x 0 with x 0 ∈ ∂Ω. Our aim in this article is to investigate how the Hausdorff measure on ∂Ω works on the weak solution of (1.6). Before starting our main theorems we make precise the notion of weak solution used in this note. Definition 1.1 We say that u is a weak solution of (1.6), if u ∈ L 1 (Ω), g(u) ∈ L 1 (Ω, ρ α dx) and where ρ(x) = dist(x, ∂Ω) and X α ⊂ C(R N ) denotes the space of functions ξ satisfying: in Ω for all ε ∈ (0, ε 0 ]. Now we are ready to state our first result for problem (1.6).
Theorem 1.1 Assume that k > 0, ρ(x) = dist(x, ∂Ω) and g is a continuous nondecreasing function satisfying g(0) ≥ 0 and Then (i) problem (1.6) admits a unique positive weak solution u k ; (ii) the mapping k → u k is increasing and there exists c 1 ≥ 1 independent of k such that (1.10) We remark that in Theorem 1.1 extends the special existence of boundary blow up solutions to fractional elliptic equation (1.10) with general nonlinearity g in integral subcritical case with the critical exponents 1+α 1−α , which is larger than N +α N −α . Specially, letting g ≡ 0, there exists infinitely many boundary blow up α−harmonic functions.
From [6] and Theorem 1.1, the Dirac mass and Hausdorff measure have different contribution to the solution of (−∆) α u + g(u) = 0 in Ω.
Our interest is to understand what singularity of the solution to where x 0 ∈ ∂Ω and δ x 0 is the Dirac mass concentrated x 0 on the boundary. Inspired by Definition 1.1, it is natural to give the definition of weak solution of (1.12) as following.

Theorem 1.3
Assume that x 0 ∈ ∂Ω, g is a continuous nondecreasing function satisfying and for some λ > 0, g(s + t) ≤ λ[g(s) + g(t)], ∀s, t > 0. (1.14) Then problem (1.6) admits a unique positive weak solution v such that Moreover, if assume additionally that g is C β locally in R with β > 0, then v is a classical solution of (1.10).
From Theorem 1.3, we find out a classical solution of (1.10) with explosive rate ρ(x) α−1 + ρ(x) α |x−x 0 | N , this answers the question 3 in the first part of the introduction. The boundary blow-up solutions of (1.10) could be searched for by making use of measure type data on boundary and the main difficulty is to do the estimate of G α [ ∂ α ω ∂ n α ] and g(G α [ ∂ α ω ∂ n α ]). Especially, it is dedicate to make the estimate of g(G α [ ∂ α ω ∂ n α ]) near the boundary when the nonlinearity g is just integral-subcritical, i.e. (1.8).
This article is organized as follows. In Section §2 we present some preliminaries to the Marcinkiewicz type estimate for G α [ ∂ α ω ∂ n α ] and present the existence and uniqueness of weak solution of (1.6) when g is bounded. Section §3, §4 are devoted to prove Theorem 1.1 and Theorem 1.2. Finally, we obtain one typical solution that blows up along the boundary with different power rate.

The Marcinkiewicz type estimate
In order to obtain the weak solution of (1.6) with integral subcritical nonlinearity, we have to introduce the Marcinkiewicz space and recall some related estimate.
Lemma 2.1 There exists c 5 ≥ 1 such that for any x ∈ Ω, Proof. Since ∂Ω is C 2 , then there exists t 0 ∈ (0, 1 2 ) such that for any x ∈ Ω t := {z ∈ Ω, ρ(x) < t} with t < t 0 , there exists a unique x ∂ ∈ ∂Ω such that and for any x ∈ E, there exists a unique x t ∈ E t satisfying (2.4). Moreover, for x ∈ C t with t ∈ (0, t 0 ), there exists a unique x ∂ ∈ ∂Ω such that Denotes by ω t a measure on C t generated by ω such that for t ∈ (0, t 0 ), By compactness we only have to prove that (2.3) holds in a neighborhood of any point x ∈ ∂Ω and without loss of generality, we may assume that is the ball centered at origin with radium t 0 in R N −1 . We choose some s 0 ∈ (0, t 0 ) small enough, there exists c 7 ≥ 1 such that for any Borel set E ⊂ B s 0 (0) ∩ ∂Ω, Therefore, where c 9 , c 10 > 0 and c 11 = +∞ 0 Therefore, for t ∈ (0, t 0 ), For x ∈ Ω \ Ω t 0 and y ∈ ∂Ω, we observe that |x − y| ≥ t 0 , then ∂Ω 1 |x−y| N−α dω(y) is bounded by some constant dependent of t 0 and the diameter of Ω, thus, (2.3) holds.
given by (2.6) and p * = 1+α 1−α . Then there exists c 17 > 0 such that Proof. For any Borel set E of Ω satisfying Then there exists c 18 > 0 such that and together with (2.3), we deduce that 1+α .
Together with 2α we derive (2.9). This completes the proof.

Existence for bounded nonlinearity
We extend Hausdorff measure ω toΩ by zero inside Ω, still denoting ω. For bounded C 2 domain, it follows [19, p 57] that ω is a Radon measure inΩ. In the approximating to weak solution of (1.6), we consider a sequence {g n } of C 1 nonnegative functions defined on R + such that g n (0) = g(0), g n ≤ g n+1 ≤ g, sup s∈R + g n (s) = n and lim n→∞ g n − g L ∞ loc (R + ) = 0.
(2.10) Proposition 2.3 Assume that {g n } n is given by (2.10). Then in Ω, admits a unique positive weak solution u k,n satisfying (i) the mapping k → u k,n is increasing, the mapping n → u k,n is decreasing (2.13) Proof.
Since ω is a Radon measure inΩ, we could apply [6, Theorem 1.1] to obtain that problem (2.11) admits a unique weak solution u k,n satisfying that (i) and u k,n is a classical solution of (−∆) α u + g n (u) = 0 in Ω, u = 0 in R N \Ω.
Proof of Theorem 1.1. To prove the existence of weak solution. Take {g n } a sequence of C 1 nondecreasing functions defined on R satisfying g n (0) = g(0) and (2.10). By Proposition 2.3, problem (2.11) admits a unique weak solution u k,n such that For any compact set K ⊂ Ω, we observe from [6, Lemma 3.2] that for some β ∈ (0, α), Therefore, up to some subsequence, there exists u k such that lim n→∞ u k,n = u k in Ω.
Regularity of u k,n and u k . Since g n is C 1 in R, then by [6, Lemma 3.2], we have u k,n C 2α+β (K) ≤ c 50 k, (3.11) for any compact set K and some β ∈ (0, α). Then u k,n is C 2α+β locally in Ω. Together with the fact that u n,k is classical solution of (2.13), we derive by Theorem 2.2 in [5] that u k is a classical solution of (2.13).
To prove (1.9.) Plugging (3.1) and (3.3) into (3.10), we obtain that (1.9). In this subsection, we consider the limit of {u k } as k → ∞, where u k is the weak solution of here p ∈ (1 + 2α, 1+α 1−α ). From Theorem 1.1, we know that k → u k is increasing and u k is a classical solution of (1.2).
In order to control the limit of {u k } as k → ∞, we have to obtain barrier function, i.e. a suitable super solution of (1.2). To this end, we consider C 2 function w p satisfying We see that w p ∈ L 1 (Ω) if 2α p−1 < 1, i.e. p > 1 + 2α.
Proof of Theorem 1.2 (i). For p ∈ (1 + 2α, 1+α 1−α ), we have that and it follows by (3.3) that Then lim x∈Ω,ρ(x)→0 wp(x) = 0 and we claim that In fact, if it fails, then there exists z 0 ∈ Ω such that Then we have (−∆) α (u k − λ 0 w p )(z 0 ) < 0, which contradicts the fact that By monotonicity of the mapping k → u k , there holds which is a classical solution of (1.2) and By applying Stability Theorem [5,Theorem 2.4], we obtain that u ∞ is a classical solution of (1.2). Finally, we claim that there exists c 51 > 0 such that for t ∈ (0, t 0 ), where σ > 0 will be chosen later, then k = σt and for x ∈ C t , we apply Lemma 3.1 with p ∈ (1 + 2α, 1+α 1−α ) that where we choose σ such that c 52 σ p−1 2 (N −α)p−α−N = 1 2 . Then for any x ∈ Ω, there exists k > 0 such that x ∈ Ω and then This ends the proof.

4.2
The limit of {u k } blows up when p ∈ (0, 1 + 2α] In this subsection, we derive the blow-up behavior of the limit of {u k } when p ∈ (0, 1 + 2α].
To this end, we have to do more estimate for u k .
The proof ends.
Proposition 5.1 Let G α [ ∂ α ν ∂ n α ] given by (5.1) and p * N = N +α N −α . Then there exists c 55 > 0 such that From Proposition 2.2 and p * = 1+α 1−α > N +α N −α , on the one hand, we have that On the other hand, for λ > 0, denote in Ω, admits a unique positive weak solution u k,j,n satisfying (i) the mappings k → u k,j,n , j → u k,j,n are increasing, the mapping n → u k,j,n is decreasing (ii) u k,n is a classical solution of (2.13).
Lemma 5.1 There exists c 58 > 1 such that