Partial regularity of stable solutions to the fractional Gel'fand-Liouville equation

We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-\Delta)^su=e^u\quad\mathrm{in}\quad \Omega\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$


INTRODUCTION
Let Ω be an open subset of R n . We consider the following fractional Gel'fand equation where s ∈ (0, 1), and u satisfies e u ∈ L 1 loc (Ω) and u ∈ L s (R n ). Here the space L s (R n ) is defined by L s (R n ) := u ∈ L 1 loc : u L s (R n ) < ∞ , u L s (R n ) := R n |u(x)| 1 + |x| n+2s dx < ∞.
The non-local operator (−∆) s is defined by (−∆) s ϕ(x) = c n,s P.V. |Γ(−s)| being a normalizing constant. We are interested in the classical question of regularity of solutions to (1.1). More precisely, we aim at studying the partial regularity of stable weak solutions of (1.1). By a weak solution we mean that u satisfies (1.1) in the sense of distribution, that is R n u(−∆) s φdx = Ω e u φdx, ∀φ ∈ C ∞ c (Ω).
For equation (1.1) in the whole space R n , the study on the classification of stable solutions and finite Morse index solutions has attracted a lot of attentions in recent decades. In the classical case, that is s = 1, Farina [8] and Dancer -Farina [3] established non-existence of stable solutions to (1.1) for 2 ≤ n ≤ 9 and non-existence of finite Morse index solutions to (1.1) for 3 ≤ n ≤ 9. While for the fractional case, Duong -Nguyen [6] proved that Eq. (1.1) has no regular stable solution for n < 10s. Later, the authors of this paper applied monotonicity formula to give an optimal condition on n, s such that Eq. (1.1) has no stable solutions in whole space, see [9]. In addition, the authors also showed that weak stable solutions are smooth, provided n < 10s.
A counterpart issue for equation (1.1) in bounded domain is to analyze the extremal solution. Let us first recall the definition of extremal solutions. Given a bounded smooth domain Ω ⊂ R n , consider the following problem where f : R → R be a C 1 function which is convex and non-negative. There exists a constant λ * > 0 such that for each λ ∈ [0, λ * ) there exists a unique solution u λ (x), which is a classical stable solution, see [7, chapter 3]. The solution u λ * := lim λ↑λ * u λ is called the extremal solution. When f (u) is the exponential nonlinearity, it is proved that if dimension n ≤ 9, the extremal solution is always smooth in the classical setting, see [7], while in the fractional case, Ros-Oton -Serra [12,13] proved that the extremal solutions are regular for n < 10s. For more general nonlinearity f (u), very recently, Cabré et al. [2] showed that the extremal solutions are regular when n ≤ 9 and f (u) is positive, non-decreasing, convex and superlinear at ∞. The restriction on the dimension n < 10 is sharp in the sense that there are singular weak stable solutions for n ≥ 10. Thus, a natural question is to understand the dimension of singular sets when n ≥ 10 (n ≥ 10s for the fractional case). This problem has been widely studied for the Lane-Emden equation, that is the nonlinearity is given by f (u) = u p , in various settings, e.g., estimating the singular set of stable solutions, finite Morse index solutions and stationary solutions, see e.g. [5,7,10,11,14]. While for the exponential nonlinearity, in the classical case, the partial regularity result has been studied by Wang [15,16] and he proved the Hausdorff dimension of the singular set for weak stable solutions is at most n − 10. In dimension n = 3, Da Lio [4] showed that the dimension of singular set for stationary solution can be at most 1.

Definition 1.1.
A point x 0 ∈ Ω is said to be a singular point for a solution u to (1.1) if u is not bounded in any small neighborhood of x 0 . The singular set S is the collection of all singular points.
It follows from the above definition that the singular set S is closed in Ω. Moreover, by standard regularity theory one gets that u is regular in Ω \ S.
Our main result is the following: Theorem 1.1. Let u be a stable weak solution of (1.1). Then the Hausdorff dimension of the singular set S is at most n − 10s.
Theorem 1.1 is a nonlocal version of Wang's [15,16] result. In order to prove small energy regularity results (see Proposition 3.4 ), we shall consider the following energy where P(X, y) = d n,s t 2s |(x − y, t)| n+2s , and d n,s > 0 is a normalizing constant such that R n P(x, y) = 1, see [1]. The function u satisfies It is important to note that each term in the energy E (u, x 0 , r) controls the other one in a suitable way. More precisely, due to the stability hypothesis, the second term controls the L 2 norm of e u , see (2.2), where as, by Jensen's inequality, the second term is controlled by the first one, see Lemma 3.1. This interplay turned out to be very crucial in proving the energy decay estimate, see Proposition 3.4.
We remark that the energy decay estimate does not seem to work if we only consider one of the two terms in E (u, x 0 , r). For instance, on one hand, if we only consider the first term (as in the local case), then we lack a Harnack type inequality for non-negative fractional sub-harmonic functions. However, in the fractional case, we do have a Harnack type inequality involving the extension function, see Lemma 3.3. This suggests to consider the send term in the definition of the energy. On the other hand, if we only consider the second term in the energy, then the L 1 norm of v (as defined in 3.2) is of the order √ E, which is not good enough to prove the energy decay estimate.
The article is organized as follows: In section 2 we list some preliminary results, including the conclusions presented in our previous work [9] and some known results that would be used in the current article. In section 3, we give the proof of Theorem 1.1. Notations: represents the non-negative part of u, i.e., u + = max(u, 0). C a generic positive constant which may change from line to line.

PRELIMINARY RESULTS
In this section we present several results that will be used in next section. First, we extend the stability condition in the fractional setting. Notice that u is welldefined as u ∈ L s (R n ). Moreover, t . We recall from [9] that the stability condition (1.2) can be generalized to the extended function u. Precisely, if u is stable in Ω then (Ω) and η = 1 in a small neighborhood of the origin, we have for every α ∈ (0, 2) Though the following regularity result on Morrey's space is well-known, we give a proof for convenience. We recall that a function f is in the Morrey's space The norm f M p (Ω) is defined to be the infimum of constants C > 0 for which the above inequality holds. .
Proof. We set where ρ = |x − y|. We can derive from (2. Thus we finish the proof.
For any x 0 ∈ B 1 , we set

4)
The following lemma is crucial in the proof of small energy regularity estimate.
Proof. It is easy to see that u λ is a stable solution to (1.1) on B R with R : This would imply that As λ < 1 we get u λ + (x) < u + (x 0 + λx). Therefore, changing the variable x 0 + λx → y we obtain u + (y)dy + Cλ 2s |y|>2 u + (y) |y| n+2s dy ≤ C u + L s (R n ) .
We conclude the lemma.

PROOF OF THEOREM 1.1
In this section, we shall present the ε-regularity result and the proof of Theorem 1.1. Before we start the iteration process for the ε-regularity, we need a more refinement result of [9, Lemma 3.3]. Lemma 3.1. Let e αu ∈ L 1 (B 1 ). Then t 1−2s e αu ∈ L 1 loc (B 1 × [0, ∞)). Moreover, i) there exists δ = δ(n, s) > 0 and C = C(n, s, u + L s (R n ) ) > 0 such that . ii) for every δ 0 > 0 small there exists r 0 = r 0 (n, s, δ 0 ) > 0 small such that for some positive constant δ depending on n and s only. Therefore, by Jensen's inequality max e αu(y) , e αδu(y) where the last inequality follows from Hölder inequality. Integrating the above inequality with respect to t on the interval [0, 1 2 ] we obtain i). To to prove ii), we notice that for a given δ 0 > 0 small we can choose r 0 > 0 sufficiently small such that (3.1) holds with δ = 1 − δ 0 for every x ∈ B 1/2 and 0 < t ≤ r 0 . Then ii) follows in a similar way.
We can simply assume that B 1 ⊂ Ω. For fixed 0 < r < 1 we decompose where C(n, s) > 0 is a dimensional constant such that Then w satisfies Notice that w is continuous up to the boundary B r . Now we prove some elementary properties of the functions v and w.
for some γ > 0 and C > 0 independent of u.
Proof. The function v can be written as a convolution, and in fact, where χ A denotes the characteristic function of the set A. In particular, by Young's inequality, we obtain where p, q verify the following conditions On the other hand, it is easy to see that Therefore, together with the interpolation inequality Here we choose p slightly bigger than 2 in (3.4), and use the fact that the L q and L 1 norms of Γ are uniformly bounded in B r if r stays bounded. The lemma follows as v(x, t) ≤ v(x).
Proof. We prove the lemma only for This implies that ρ 2s−n−1 ∂D ρ \B ρ t 1−2s e w dσ is monotone increasing with respect to ρ. As a consequence, we have that ρ 2s−n−2 D ρ t 1−2s e w dxdt is increasing in ρ.
Hence, using that w < u and w is continuous up to the boundary B r , where the former conclusion follows from the fact that v > 0, we get It completes the proof.
We now prove the following energy decay estimate for E (u, x 0 , r). For simplicity, we set E (u, x 0 , r) by E (x 0 , r) and first consider x 0 = 0 in the following proposition Proposition 3.4. Let u be a stable solution to (1.1) with Ω = B 1 for some u ∈ L s (R n ). Then there exists ε 0 > 0 and θ ∈ (0, 1) depending only on n, s and u + L s (R n ) such that if Proof. It follows from (2.2) that (3.7) Then using Lemma 3.1 one can find r 1 = r 1 (n, s, γ) > 0 such that for every 0 < r ≤ r 1 where C > 0 depends on n, s and u + L s (R n ) . Together with (3.6), (3.7) and Hölder inequality This in turn implies that Moreover, by Lemma 3.3 r 4s−n−2 D r ∩{v≤1} t 1−2s e u dxdt ≤ Cr 4s−n−2 D r t 1−2s e w dxdt ≤ Cr 2s ε.
In a similar way we can also obtain r 2s−n B r e u dx ≤ C(r 2s ε + r 2s−n ε 1+ γ 4 ).
It is not difficult to see that Together with the above proposition and Lemma 2.3, one can easily get the following lemma Lemma 3.5. Let u be a stable solution to (1.1) with Ω = B 1 for some u ∈ L s (R n ). Then there exists ε 0 > 0 and θ ∈ (0, 1) depending only on n, s and u + L s (R n ) such that if By an iteration argument one can show that for some constant C > 0 depending only on θ, where α := − log 2 log θ > 0. Particularly, e u(y) dy ≤ Cr n−2s+α , ∀x ∈ B 1 2 and 0 < r ≤ 1 2 . (3.8) Then we decompose u = u 1 + u 2 , with u 1 (x) = c(n, s) B 1/2 1 |x − y| n−2s e u(y) dy, ∀x ∈ B 1 2 , where c(n, s) is chosen such that c(n, s)(−∆) s 1 |x − y| n−2s = δ(x − y).
By (3.8) and Lemma 2.2 we conclude that u 1 (x) is regular in B 1/6 . While u 2 satisfies (−∆) s u 2 = 0 in B 1/2 and it implies that u 2 is smooth in B 1/6 . As a consequence, we get that u is continuous in B 1/6 .
We recall that if u is stable in Ω then e u ∈ L p loc (Ω) for every p ∈ [1, 5). Consequently, the small energy regularity results can be stated as follows: Lemma 3.6. For every 1 ≤ p < 5 there exists ε p > 0 depending only on n, s and u L s (R n ) such that if B 1 e pu dx ≤ ε p , then u is continuous on B 1/6 . Proof. By Hölder inequality we get that e pu dx ≤ ε p , for some p ∈ [1, 5), then u is continuous in B r/6 (x), thanks to Lemma 3.6. Thus, if x ∈ S (S is the singular set) we see that for every r > 0 r 2ps−n B r (x) e pu dx > δ.
Thus by the well-known Besicovitch covering lemma we have that the Hausdorff dimension of S is at most n − 2ps with p ∈ [1, 5). Hence, we conclude that the Hausdorff dimension of S is at most n − 10s and it completes the proof.