Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

Definition 1.1

A point x₀ ∈ 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 x₀. 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.

The dimension estimate n − 10s is optimal in the limit s ↑ 1 as there are singular weak stable solutions for s = 1. However, due to the regularity results of Ros-Oton [19] for extremal solutions, one would expect that the Hausdorff dimension of the singular set would be at most n − n^*(s), where n^*(s) > 10s is implicitly defined by the second condition in (1.4) with equality. This remains as an open question.

To prove Theorem 1.1 we follow the approach of Wang [21, 22] for the classical case. More precisely, Wang considered the energy

Then he showed that there exists ε₀ > 0 (independent of u, x₀, r) such that if 𝓕(u, x₀, r) ≤ ε₀ then u is regular in a small neighborhood of x₀. This, and the fact that e^u is in Llocp for every p < 5, thanks to Farina’s estimate [11], imply that the Hausdorff dimension of the singular set is at most n − 10.

Contrary to the above energy (1.5), in our case we shall consider the following energy involving interior and boundary quantities

where u is the Caffarelli-Silvestre extension of u in R+n+1 , i.e.,

where

and d_n,s > 0 is a normalizing constant such that ∫_ℝⁿP(X,y)dy = 1, see [1]. The function u satisfies

where κs=Γ(1−s)22s−1Γ(s). As in the classical case, the Farina type estimate for the nonlocal case also imply that if u is a stable weak solution then e^u ∈ Llocp for every p < 5, see Lemma 2.1. Due to this integrability restrictions we have the dimension estimate n − 10s in Theorem 1.1. Let us also emphasize here that the stability condition will be used only to obtain the Farina type estimate in Lemma 2.1 and to get uniform bounds in the space L_s(ℝⁿ) for a family of rescaled solutions in Lemma 2.3.

It is important to note that each term in the energy 𝓔(u, x₀, r) controls the other one in a suitable way. More precisely, due to the stability hypothesis, the second term controls the L² norm of e^u, see (2.2), whereas, 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 energy decay estimate (see Proposition 3.4 and Lemma 3.5 for precise statements), that is, if 𝓔(u, x₀, r) ≤ ε₀ for some sufficiently small ε₀ > 0 then there exists θ ∈ (0, 1) such that 𝓔(u, x₀,θr) ≤ 12 𝓔(u, x₀, r).

We remark that the energy estimate does not seem to work if we only consider one of the two terms in 𝓔(u, x₀, 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 second term in the definition of the energy. On the other hand, if we only consider the second term in the energy, then the L¹ norm of v (as defined in ℝef{def-barv}) is of the order E due to the fact that there is a sacrifice in the power of ε when we use Hölder inequality, which is not good enough (compare (3.5) and (3.7)) to prove the energy decay estimate in Proposition 3.4.

The article is organized as follows: In section 2 we list some preliminary results, including the conclusions presented in our previous work [14] and some known results that would be used in the current article. In section 3, we give the proof of Theorem 1.1.

Notations

1. represent points in R+n+1 = ℝⁿ × [0, ∞) and ℝⁿ = ∂ R+n+1 .

2. the ball centered at x with radius r in ℝⁿ, B_r := B_r(0).

3. the ball centered at x with radius r in Rn+1,Brn+1:=Brn+1(0).

4. the intersection of Brn+1 (x) and R+n+1 , i.e., Brn+1(x)∩R+n+1,Dr:=Dr(0).

5. represents the non-negative part of u, i.e., u₊ = max(u, 0).

6. a generic positive constant which may change from line to line.

Lemma 2.1

([14]). Let u ∈ Ḣ^s(Ω) be a weak stable solution to (1.1). Given a function Φ be of the form Φ(x, t) = ϕ(x)η(t) for some ϕ ∈ Cc∞ (Ω) and η = 1 in a small neighborhood of the origin, for every α ∈ (0, 2) we have

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 M^p(Ω) if f ∈ L¹(Ω), and it satisfies

The norm ∥f∥_M^p(Ω) is defined to be the infimum of constants C > 0 for which the above inequality holds.

Lemma 2.2

Let f∈Mn2s−δ(B3) for some δ > 0. We set

Then we have

Proof

We set

Then

where ρ = ∣y∣. We can derive from (2.3) and integration by parts that

Thus we finish the proof.□

For any x₀ ∈ B₁, we set

The following lemma is crucial for the proof of Lemma 3.5.

Lemma 2.3

Let u be a stable solution to (1.1) with Ω = B₁. Let u^λ be defined as in (2.4) for some ∣x₀∣ < 1 and 0<λ<(1−|x0|)1+n2s. Then

for some C > 0 independent of u.

Proof

It is easy to see that u^λ is a stable solution to (1.1) on B_R with R:=1λ(1−|x0|). Hence, by (2.1)

This would imply that

As λ < 1 we get

Therefore, changing the variable x₀ + λ x ↦ y we obtain

We conclude the lemma.□

Lemma 3.1

Let e^αu ∈ L¹(B₁). Then t^1−2se^αu ∈ Lloc1 (B₁ × [0, ∞)). Moreover,

1. there exists δ = δ(n, s) > 0 and C = C(n, s,∥u₊∥_Ls(ℝⁿ)) > 0 such that

   ∥t1−2seαu¯∥L1(D1/2)≤C∥eαu∥L1(B1)+∥eαu∥L1(B1)δ,

2. for every δ₀ > 0 small there exists r₀ = r₀(n, s,δ₀) > 0 small such that

   ∫0r0∫B1/2t1−2seαu¯dxdt≤C∥eαu∥L1(B1)+∥eαu∥L1(B1)1−δ0.

Proof

For X = (x, t) ∈ D_1/2 we have

where

for some positive constant δ depending on n and s only. The last inequality easily follows from

Therefore, by Jensen’s inequality

where the last inequality follows from Hölder inequality. Integrating the above inequality with respect to t on the interval [0,12] we obtain i).

To to prove ii), we notice that for a given δ₀ > 0 small we can choose r₀ > 0 sufficiently small such that (3.1) holds with δ = 1−δ₀ for every x ∈ B_1/2 and 0 < t ≤ r₀. Then ii) follows in a similar way.□

Up to a translation and dilation of the domain Ω, we can simply assume that B₁ ⊂ Ω. For fixed 0 < r < 1 we decompose

where

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.

Lemma 3.2

Setting v := v(x, 0) we have

for some γ ∈ (0, 1) 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. On the one hand, 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

with γ, p satisfying

we obtain

Here we choose p slightly bigger than 2 in (3.4), and use the fact that the L^q and L¹ norms of Γ are uniformly bounded in B_r if r stays bounded. The lemma follows as v(x, t) ≤ v(x).□

Lemma 3.3

Setting w = w(x₀, 0) we have for every 0 < ρ < R := (r − ∣x₀∣)

where

Proof

We prove the lemma only for x₀ = 0. From (3.3) we have that

Therefore, for 0 < ρ < r

This implies that ρ^2s−n−1 ∫_∂Dρ∖Bρ t^1−2se^wdσ is monotone increasing with respect to ρ. As a consequence, we have that ρ^2s−n−2 ∫_Dρ t^1−2se^wdxdt 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, x₀, r). For simplicity, we write 𝓔(u, x₀, r) by 𝓔(x₀, r), and consider x₀ = 0 and r = 1 in the following proposition.

Proposition 3.4

Let u be a stable solution to (1.1) with Ω = B₁ for some u ∈ L_s(ℝⁿ). Then there exists ε₀ ∈ (0, 1) and θ ∈ (0, 1) depending only on n, s and ∥u₊∥_Ls(ℝⁿ) such that if

then

Proof

It follows from (2.2) that

Then writing u = v+w, where v is given by (3.2) with r = 12 , we get from Lemma 3.2 that

We shall give estimation on 𝓔(0, r) for suitable r. Using Lemma 3.1 one can find r₁ = r₁(n, s,γ) ∈ (0, 12 ) such that for every 0 < r ≤ r₁

where C > 0 depends on n, s and ∥u₊∥_Ls(ℝⁿ). Together with (3.6), (3.7) and Hölder inequality

This in turn implies that

Moreover, by Lemma 3.3

Combining the above two estimates

In a similar way, by Lemma 3.3 we can also obtain,

Thus, for 0 < r ≤ r₁,

for some C = C(n, s,∥u₊∥_Ls(ℝⁿ)) > 0, where we used s < 1 and r₁ ≤ 12 .

Finally, we first choose θ > 0 small enough such that Cθ2s=14, and then choose ε₀ > 0 small such that Cθ4s−n−2ε0γ4=14. Then for ε ≤ ε₀ we obtain

Hence, we finish the proof.□

Though the proof of the following lemma is quite standard, thanks to Proposition 3.4 and Lemma 2.3, for completeness we give a sketch of the proof.

Lemma 3.5

Let u ∈ L_s(ℝⁿ) be a stable solution to (1.1) with Ω = B₁. Then there exists ε̂₀ > 0 depending only on n, s and ∥u₊∥_Ls(ℝⁿ) such that if

then u is continuous in B_1/6.

Proof

For x0∈B12 and 0<λ≤λ0:=2−2−n2s we set

Then by Lemma 2.3 we see that

for some C₁ > 0 independent of x0∈B12 and 0 < λ ≤ λ₀. Therefore, by Proposition 3.4, we can find ε₀ > 0 and θ ∈ (0, 1) (depending only on n, s and C₁) such that

It follows from the definition of the energy 𝓔 that

Consequently, if we choose 0<ε^0<ε0C(λ0), then from (3.8) we get

By an iteration argument one obtains

In particular, setting α:=−log⁡2log⁡θ>0 we get

which gives

Now we decompose u = u₁+u₂, with

where c(n, s) is chosen such that

By (3.9) and Lemma 2.2 we conclude that u₁ is bounded in B_1/3. While u₂ satisfies (−Δ)^s u₂ = 0 in B_1/3, and it implies that u₂ 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 ∈ Llocp (Ω) for every p ∈ [1, 5) by Lemma 2.1. 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∥_Ls(ℝⁿ) such that if

then u is continuous on B_1/12.

Proof

By Hölder inequality we get that

Hence, by Lemma 3.1

for some δ > 0. This shows that

Then by Lemma 3.5 we get that u is continuous in B_1/12 provided Cεpδ/p is sufficiently small.□

Proof

Proof of Theorem 1.1. The problem (1.1) is invariant under the rescaling

Therefore, if

for some p ∈ [1, 5), then u is continuous in B_r/12(x₀), thanks to Lemma 3.6. Thus, if x₀ ∈ 𝓢 (𝓢 is the singular set) we see that for every r > 0 small

Hence, by the well-known Besicovitch covering theorem we have that the Hausdorff dimension of 𝓢 is at most n − 2ps with p ∈ [1,5), see for instance [15, Lemma 1.3.5, Lemma 2.1.1] or [13, Proposition 9.21]. Hence, we conclude that the Hausdorff dimension of 𝓢 is at most n − 10s and it completes the proof.□