REGULARITY OF MINIMIZERS UP TO DIMENSION 7 IN DOMAINS OF ...
REGULARITY OF MINIMIZERS UP TO DIMENSION 7 IN DOMAINS OF DOUBLE REVOLUTION ´ AND XAVIER ROS-OTON XAVIER CABRE
Abstract. We consider the class of semi-stable positive solutions to semilinear equations −∆u = f (u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n − m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex, we establish a priori Lp and H01 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of −∆u = λf (u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 3 for any domain Ω, in dimensions n ≤ 4 when the domain is convex, and in dimensions n ≤ 9 in the radial case. In dimensions 5 ≤ n ≤ 9 it remains an open question.
1. Introduction and results n
Let Ω ⊂ R be a smooth and bounded domain, and consider the problem −∆u = λf (u) in Ω u > 0 in Ω (1.1) u = 0 on ∂Ω,
where λ is a positive parameter and the nonlinearity f : [0, ∞) −→ R satisfies
f (τ ) = ∞. τ It is well known (see the excellent monograph [7] and references therein) that there exists an extremal parameter λ∗ ∈ (0, ∞) such that if 0 < λ < λ∗ then the problem (1.1) admits a minimal classical solution uλ , while for λ > λ∗ it has no solution, even in the weak sense. Here, minimal means smallest. Moreover, the set {uλ : 0 < λ < λ∗ } is increasing in λ, and its pointwise limit u∗ = limλ→λ∗ uλ is a weak solution of problem (1.1) with λ = λ∗ . It is called the extremal solution of (1.1). When f (u) = eu , it is well known that u ∈ L∞ (Ω) if n ≤ 9, while u∗ (x) = log |x|1 2 if n ≥ 10 and Ω = B1 . An analogous result holds for f (u) = (1 + u)p , p > 1. In the nineties H. Brezis and J.L. V´azquez [1] raised the question of determining the regularity of u∗ , depending on the dimension n, for general convex nonlinearities satisfying (1.2). The first general results were proved by G. Nedev [9, 10] – see [5] for the statement and proofs of the results of [10]. (1.2) f is C 1,γ , γ ∈ (0, 1), f is nondecreasing, f (0) > 0, and lim
τ →∞
Theorem 1.1 ([9],[10]). Let Ω be a smooth bounded domain, f be a convex function satisfying (1.2), and u∗ be the extremal solution of (1.1). Key words and phrases. Semilinear elliptic equations, regularity of stable solutions. Both authors were supported by MTM2011-27739-C04-01 (Spain) and 2009SGR345 (Catalunya). 1
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i) If n ≤ 3, then u∗ ∈ L∞ (Ω). n ii) If n ≥ 4, then u∗ ∈ Lp (Ω) for every p < n−4 . iii) Assume either that n ≤ 5 or that Ω is strictly convex. Then u∗ ∈ H01 (Ω). In 2006, the first autor and A. Capella [3] studied the radial case. Their result establishes optimal regularity results for general f . Theorem 1.2 ([3]). Let Ω = B1 be the unit ball in Rn , f be a function satisfying (1.2), and u∗ be the extremal solution of (1.1). i) If n ≤ 9, then u∗ ∈ L∞ (Ω). ii) If n ≥ 10, then u∗ ∈ Lp (Ω) for every p < pn , where (1.3)
pn = 2 +
4 √n 2+ n−1
−2
.
iii) For every dimension n, u∗ ∈ H 3 (Ω). The best known result was established in 2010 by the first author [2] and establishes the boundedness of u∗ in convex domains in dimension n = 4. Related ideas recently allowed the first author and M. Sanch´on [5] to improve the Lp estimates of Theorem 1.1 for u∗ when n ≥ 5: Theorem 1.3 ([2],[5]). Let Ω ⊂ Rn be a convex, smooth and bounded domain, f be a function satisfying (1.2), and u∗ be the extremal solution of (1.1). i) If n ≤ 4, then u∗ ∈ L∞ (Ω). 2n ii) If n ≥ 5, then u∗ ∈ Lp (Ω) for every p < n−4 = 2 + n 4−2 . 2
The boundedness of extremal solutions remains an open question in dimensions 5 ≤ n ≤ 9, even in the case of convex domains. The aim of this paper is to study the regularity of the extremal solution u∗ of (1.1) in a class of domains that we call of double revolution. The class contains domains much more general than balls, but is much simpler than general convex domains. Let n ≥ 4 and (1.4)
Rn = Rm × Rk with n = m + k, m ≥ 2 and k ≥ 2.
For each x ∈ Rn we define the variables p x21 + · · · + x2m s = q t = x2m+1 + · · · + x2n .
We say that a domain Ω ⊂ Rn is a domain of double revolution if it is invariant under rotations of the first m variables and also under rotations of the last k variables. Equivalently, Ω is of the form Ω = {x ∈ Rn : (s, t) ∈ U } where U is a domain in R2 symmetric with respect to the two coordinate axes. In fact, U = {(s, t) ∈ R2 : x = (x1 = s, x2 = 0, ..., xm = 0, xm+1 = t, ..., xn = 0) ∈ Ω} is the intersection of Ω with the (x1 , xm+1 )-plane. Note that U is smooth if and only e the intersection of U with the positive quadrant of if Ω is smooth. Let us call Ω 2 R , i.e., (1.5) e = {(s, t) ∈ R2 : s > 0, t > 0, (x1 = s, x2 = 0, ..., xm = 0, xm+1 = t, ..., xn = 0) ∈ Ω}. Ω
REGULARITY OF MINIMIZERS IN DOMAINS OF DOUBLE REVOLUTION
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Since {s = 0} and {t = 0} have zero measure in R2 , we have that Z Z v dx = cm,k v(s, t)sm−1 tk−1 dsdt Ω
e Ω
1
for every L (Ω) function v = v(x) which depends only on the radial variables s and t. Here, cm,k is a positive constant depending only on m and k. We will see that any semi-stable classical solution u of (1.1), and more generally of (1.7) below, depends only on s and t, and hence we can identify it with a function u = u(s, t) defined in (0, ∞)2 which satisfies the equation k−1 m−1 e us + ut + f (u) = 0 for s, t ∈ Ω. s t Moreover, in the case of convex domains we will also have us ≤ 0 and ut ≤ 0 and hence, u(0) = kukL∞ (see Remark 2.2). The following is our main result. We prove that, in convex domains of double revolution, the extremal solution u∗ is bounded when n ≤ 7, and belongs to H01 and certain Lp spaces when n ≥ 8. We also prove that in dimension n = 4 the convexity of the domain is not required for the boundedness of u∗ . uss + utt +
Theorem 1.4. Assume (1.4). Let Ω ⊂ Rn be a smooth and bounded domain of double revolution, f be a function satisfying (1.2), and u∗ be the extremal solution of (1.1). a) Assume either that n = 4 or that n ≤ 7 and Ω is convex. Then, u∗ ∈ L∞ (Ω). b) If n ≥ 8 and Ω is convex, then u∗ ∈ Lp (Ω) for all p < pm,k , where (1.6)
pm,k = 2 +
4 √m 2+ m−1
+
√k 2+ k−1
−2
.
c) Assume either that n ≤ 6 or that Ω is convex. Then, u∗ ∈ H01 (Ω). Remark 1.5. Let qm,k = 2+√mm−1 + 2+√kk−1 . Since 2+√xx−1 is concave in (2, ∞), we have that qn−2,2 ≤ qm,k ≤ q n2 , n2 , and therefore p n2 , n2 ≤ pm,k ≤ pn−2,2 . Thus, asymptotically as n → ∞, √ 2 2 4 √ ' p n2 , n2 ≤ pm,k ≤ pn−2,2 ' 2 + √ . 2+ n n Instead, in a general convex domain, Lp estimates are only known for p ' 2+ n8 (see Theorem 1.3 ii) above), while in the radial case one has Lp estimates for p ' 2 + √4n (see Theorem 1.2 ii). In fact, the optimal exponent (1.3) in the radial case can be obtained by setting m = n and k = 0 in (1.6), but recall that in this paper we always assume m ≥ 2 and k ≥ 2. The proofs of the results in [9, 10, 3, 2, 5] use the semi-stability of the extremal solution u∗ . In fact, one first proves estimates for any regular semi-stable solution u of −∆u = f (u) in Ω (1.7) u = 0 on ∂Ω, then one applies these estimates to the minimal solutions uλ , and finally by monotone convergence the estimates also hold for the extremal solution u∗ .
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t e Ω
Q P
s
Figure 1. A non-convex domain for which the maximum of u∗ will not be u∗ (0) Recall that a classical solution u of (1.7) is said to be semi-stable if the second variation of energy at u is nonnegative, i.e., if Z (1.8) Qu (ξ) = |∇ξ|2 − f 0 (u)ξ 2 dx ≥ 0 Ω
C01 (Ω).
for all ξ ∈ The proof of the estimates in [3, 2, 5] was inspired by the proof of Simons theorem on the nonexistence of singular minimal cones in Rn for n ≤ 7. The key idea is to take ξ = |∇u|η (or ξ = ur η in the radial case) and compute Qu (|∇u|η) in the semi-stability property satisfied by u. Then, the expression of Qu in terms of η does not depend on f , and a clever choice of the test function η leads to Lp and L∞ bounds depending on the dimension n. In this paper we will proceed in a similar way, proving first results for general semi-stable solutions of (1.7) and then applying them to uλ to deduce estimates for u∗ . We will take ξ = us η and ξ = ut η separately instead of ξ = |∇u|η, to obtain bounds for Z Z (1.9) u2s s−2α−2 dx and u2t t−2β−2 dx, √
Ω
√
Ω
where α < m − 1 and β < k − 1. When the domain Ω is convex, we will have the additional information kukL∞ = u(0), us ≤ 0, and ut ≤ 0, which combined with (1.9) will lead to L∞ and Lp estimates for u∗ . Instead, when the domain Ω is not convex the maximum of u may not be achieved at the origin – see Figure 1 for an example in which u(0) will be much smaller than kukL∞ . Thus, in nonconvex domains we can not apply the same argument. However, if the maximum is away from {s = 0} and {t = 0} (as in Figure 1) then the problem is essentially two dimensional near the maximum, since dx = cm,k sm−1 tk−1 dsdt and both s and t will be positive and bounded below. We will still have to prove some boundary estimates, for instance estimates near the points P and Q in Figure 1. But, by the same reason as before, near P the coordinate s is positive and bonded below. Thus, the problem near P will be essentially k + 1 dimensional, and k = n − m ≤ n − 2. This will allow us, if n − 1 is small enough, to use Nedev’s [9] W 2,p estimates to obtain boundary estimates.
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Our result for general semi-stable solutions of (1.7) reads as follows. It states global estimates controlled in terms of boundary estimates. Proposition 1.6. Assume (1.4). Let Ω ⊂ Rn be a smooth and bounded domain of double revolution, f be any C 1,γ function, and u be a positive bounded semi-stable solution of (1.7). Let δ be a positive real number, and define Ωδ = {x ∈ Ω : dist(x, ∂Ω) < δ}. Then, for some constant C depending only on Ω, δ, n, and also p in part b) below, one has: a) If n ≤ 7 and Ω is convex, then kukL∞ (Ω) ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) . b) If n ≥ 8 and Ω is convex, then kukLp (Ω) ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) for each p < pm,k , where pm,k is given by (1.6). c) For all n ≥ 4, kukH01 (Ω) ≤ CkukH 1 (Ωδ ) . To prove part b) of Proposition 1.6 we will need a new weighted Sobolev inequality in (R+ )2 = {(s, t) ∈ R2 : s > 0, t > 0}. It states the following. Proposition 1.7. Let a > −1 and b > −1 be real numbers, being positive at least one of them, and let D = 2 + a + b. Let u be a nonnegative Lipschitz function with compact support in R2 such that us ≤ 0 and ut ≤ 0 in (R+ )2 , with strict inequality when u > 0. Then, for each 1 ≤ q < D there exist a constant C, depending only on a, b, and q, such that !1/q∗ !1/q Z Z ∗ (1.10) sa tb |u|q dsdt ≤C sa tb |∇u|q dsdt , (R+ )2
where q ∗ =
(R+ )2
Dq D−q .
In section 4 we establish this weighted Sobolev inequality as a consequence of a new weighted isoperimetric inequality. Remark 1.8. When a and b are nonnegative integers, inequality (1.10) is a direct consequence of the classical Sobolev inequality in RD . Namely, define in RD = Ra+1 × Rb+1 the variables s and t as before, with m = a + 1 and k = b + 1. Then, for functions u defined in RD depending only on the variables s and t, write the integrals appearing in the classical Sobolev inequality in RD in terms of s and t. Since dx = ca,b sa tb dsdt, the obtained inequality is precisely the one given in Proposition 1.7. Thus, the previous proposition extends the classical Sobolev inequality to the case of non-integer exponents a and b. In another article, [4], we prove inequality (1.10) with (R+ )2 replaced by (R+ )d and with sa tb replaced by the monomial Ad 1 weight xA = xA 1 · · · xd . We also prove a related isoperimetric inequality with best constant, a weighted Morrey’s inequality, and we determine extremal functions for some of these inequalities. The paper is organized as follows. In section 2 we prove the estimates of Proposition 1.6. Section 3 deals with the regularity of the extremal solution of (1.1). Finally, in section 4 we prove the weighted Sobolev inequality of Proposition 1.7.
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2. Proof of Proposition 1.6 In this section we prove the estimates of Proposition 1.6. For this, we will need two preliminary results. Lemma 2.1 ([3, 2]). Let u be a bounded semi-stable solution of (1.7), and let c be a C 2 (Ω) function. Then, Z Z 0 2 c{∆c + f (u)c}η dx ≤ c2 |∇η|2 dx Ω
Ω
for all η ∈ Lip(Ω) with η|∂Ω = 0.
Proof. It suffices to set ξ = cη in the semi-stability condition (1.8) and then integrate by parts. The fact that we can take η ∈ Lip(Ω) can be deduced by density arguments. Remark 2.2. Note that when the domain is of double revolution, any bounded semi-stable solution u of (1.7) will depend only on the variables s and t. To prove this, define v = xi uxj − xj uxi , with i 6= j. Note that u will will depend only on s and t if and only if v ≡ 0 for each i, j ∈ {1, ..., m} and for each i, j ∈ {m + 1, ..., n}. We first see that, for such indexes i, j, v is a solution of the linearized equation of (1.7): ∆v
=
∆(xi uxj − xj uxi )
= xi ∆uxj + 2∇xi · ∇uxj − xj ∆uxi − 2∇xj · ∇uxi = xi (∆u)xj − xj (∆u)xi = −f 0 (u){xi uxj − xj uxi } = −f 0 (u)v. Moreover, since u = 0 on ∂Ω then v = 0 on ∂Ω. Thus, multiplying the equation by v and integrating by parts, we obtain Z {|∇v|2 − f 0 (u)v 2 }dx = 0. Ω
But since u is semi-stable, λ1 (∆ + f 0 (u); Ω) ≥ 0. If λ1 (∆ + f 0 (u); Ω) > 0, the preious inequality leads to v ≡ 0. If λ1 (∆ + f 0 (u); Ω) = 0, then we must have v = Kφ1 , where K is a constant and φ1 is the first eigenfunction, which we may take to be positive in Ω. But since v is the derivative of u along the vector field ∂t = xi ∂xj − xj ∂xi , and its integral curves are closed, v can not have constant sign. Thus, K = 0, that is, v ≡ 0. Hence, we have seen that any classical semi-stable solution u of (1.7) depend only on the variables s and t. Moreover, by the classical result of Gidas-Ni-Nirenberg [8], when Ω is even and convex with respect each coordinate, we have uxi > 0 when xi > 0, for i = 1, ..., n. In particular, when Ω is a convex domain of double revolution, we have that us < 0 and ut < 0 for s > 0, t > 0. In particular, kukL∞ (Ω) = u(0). On the other hand, since f ∈ C 1,γ then a bounded solution u of (1.7) satisfies u ∈ C 3,γ (Ω). In particular, u ∈ C 3 (Ω), and us , ut ∈ C 2 (Ω). Finally, note that since u is an even function of s and t, then us = 0 when s = 0 and ut = 0 when t = 0.
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We now apply Lemma 2.1 separately with c = us and with c = ut , and then we choose appropriately the test function η to get the following result. This estimate is the key ingredient in the proof of Proposition 1.6. Lemma 2.3. Assume (1.4). Let Ω ⊂ Rn be a smooth and bounded domain of double revolution, f be any C 1,γ function, and u be a positive bounded semi-stable solution of (1.7). Let α and β be such that α2 < m − 1 and β 2 < k − 1. Then, for each δ > 0 there exists a constant C, which depends only on Ω, δ, n, α, and β, such that Z 1/2 2 −2α−2 (2.1) us s + u2t t−2β−2 dx ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) , Ω
where
Ωδ = {x ∈ Ω : dist(x, ∂Ω) < δ}. Proof. We will prove only the estimate for u2s s−2α−2 ; the other term can be estimated similarly. Differentiating the equation −∆u = f (u) with respect to s, we obtain us −∆us + (m − 1) 2 = f 0 (u)us , s and hence, setting c = us ∈ C 2 (Ω) in Lemma 2.1, we have that Z Z 2 2η (m − 1) us 2 dx ≤ u2s |∇η|2 dx s Ω Ω for all η ∈ Lip(Ω), η|∂Ω = 0. Let us set in the last inequality η = η , where −α s ρ if s > 0 in Ωδ/3 η = and ρ= −α ρ if s ≤ 1 in Ω\Ωδ/2 , and ρ is a smooth function. Then, since α2 < 21 (α2 + m − 1) < m − 1, 1 2 in (Ω\Ωδ/2 ) ∩ {s > } 2 (α + m − 1)s−2α−2 ρ2 1 2 −2α−2 2 −2α (α + m − 1)s ρ + Cs in Ωδ/2 ∩ {s > } |∇η |2 ≤ 2 −2α C in Ω ∩ {s ≤ },
and therefore Z Z Z m − 1 − α2 u2s s−2α−2 ρ2 dx ≤ C u2s s−2α dx+C−2α u2s dx, 2 Ω∩{s>} Ωδ/2 ∩{s>} Ω∩{s≤} where C denote different constants depending only on the quantities appearing in the statement of the lemma. Now, since us ∈ L∞ (Ω) the√last term is bounded by Ckus kLi nf ty m−2α . Making → 0 and using that 2α < 2 m − 1 ≤ m, we deduce Z Z 2 −2α−2 2 us s ρ dx ≤ C u2s s−2α dx. Ω
Ωδ/2
Hence, since ρ ≡ 1 in Ω\Ωδ/2 , Z Z (2.2) u2s s−2α−2 dx ≤ C Ω\Ωδ/2
Ωδ/2
u2s s−2α dx ≤ C
Z
Ωδ/2
u2s s−2α−2 dx.
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From this we deduce that, for another constant C, Z Z 2 −2α−2 (2.3) us s dx ≤ C u2s s−2α−2 dx. Ω
Ωδ/2
Let ν < 1 to be chosen later. On the one hand using that us (0, t) = 0 we obtain that, if δ is small enough, |us (s, t)| ≤ Csν kus kC 0,ν in Ωδ/2 ∩ {s < δ}. Moreover, since −∆u = f (u) in Ωδ then kukC 1,ν (Ωδ/2 ) ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) , and therefore, ks−ν us kL∞ (Ωδ/2 ∩{s 0, we will have Z Z 2 −2α−2 −ν 2 us s dx ≤ ks us kL∞ (Ωδ/2 ) s−2α−2+2ν dx ≤ Cks−ν us k2L∞ (Ωδ/2 ) , Ωδ/2
Ωδ/2
and hence by (2.3) and (2.4), Z 2 u2s s−2α−2 dx ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) . Ω
Using Lemma 2.3 we can now establish Proposition 1.6. Proof of Proposition 1.6. a) We assume Ω to be convex. Recall that in this case kukL∞ = u(0); see Remark 2.2. On the one hand, making the change of variables σ = s2+α , τ = t2+β in the integral in (2.1), one has ( m sm−1 ds = cα σ 2+α −1 dσ k tk−1 dt = cβ τ 2+β −1 dτ, and thus, (2.5)
Z
Ω
k m σ 2+α −1 τ 2+β −1 (u2σ + u2τ )dσdτ ≤ C kuk2L∞ (Ωδ ) + kf (u)k2L∞ (Ωδ ) .
Here, to simplify, we are abusing notation and calling again Ω the image of the two e in (1.5) after the transformation (s, t) 7→ (σ, τ ). Therefore, dimensional√domain Ω 2 2 setting ρ = σ + τ and taking into account that in {τ < σ < 2τ } we have σ > ρ2 and τ ≥ ρ3 , we obtain Z m k (2.6) ρ 2+α + 2+β −2 (u2σ + u2τ )dσdτ ≤ C kuk2L∞ (Ωδ ) + kf (u)k2L∞ (Ωδ ) . Ω∩{τ 0 such that Ωδ ∩ Br0 (0) = ∅. Thus, s ≥ c in Ωδ ∩ {s > t} and t ≥ c in Ωδ ∩ {s < t}, and Z Z t|∇u∗ |p dsdt ≤ C, s|∇u∗ |p dsdt ≤ C. Ωδ ∩{s>t}
Ωδ ∩{s 3 then we can apply Sobolev’s inequality in dimension 3 (as explained in Remark 1.8, to obtain u∗ ∈ L∞ (Ωδ ∩{s > t}) and u∗ ∈ L∞ (Ωδ ∩{s < t}). Therefore u∗ ∈ L∞ (Ωδ ), as claimed. To prove part c) in the non-convex case, let n ≤ 6. By Proposition 1.6, it suffices to prove that u∗ ∈ H 1 (Ωδ ) for some δ > 0. Take r0 and δ such that Ωδ ∩Br0 (0) = ∅, as in part a). n . Thus, In [9] is proved that u∗ ∈ W 2,p (Ω) for p < n−2 Z Z tk−1 |D2 u∗ |p dsdt ≤ C, sm−1 |D2 u∗ |p dsdt ≤ C. Ωδ ∩{s>t}
Ωδ ∩{s t}) and u∗ ∈ H 1 (Ωδ ∩ {s < t}), and therefore, u∗ ∈ H 1 (Ωδ ).
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4. Weighted Sobolev inequality It is well known that the classical Sobolev inequality can be deduced from an isoperimetric inequality. This is done by applying first the isoperimetric inequality to the level sets of the function and then using the coarea formula. In this way one deduces the Sobolev inequality with exponent 1 on the gradient. Then, by applying H¨ older’s inequality one deduces the general Sobolev inequality. Since in our case we have us ≤ 0 and ut ≤ 0, with strict inequality when u > 0, it suffices to prove a weighted isoperimetric inequality for domains Ω ⊂ (R+ )2 satisfying the following properties: P1) If (s, t) ∈ Ω, then (s0 , t0 ) ∈ Ω for all s0 and t0 with 0 < s0 < s and 0 < t0 < t. P2) Ωt = {s > 0 : (s, t) ∈ Ω} and Ωs = {t > 0 : (s, t) ∈ Ω} are strictly decreasing in t and s, respectively. We will denote m(Ω) =
Z
sa tb dsdt
and
m(∂Ω) =
Z
sa tb dsdt.
∂Ω∩(R+ )2
Ω
Proposition 4.1. Let Ω ⊂ (R+ )2 be a bounded Lipschitz domain satisfying P1 and P2, and let a > −1 and b > −1 be real numbers, being positive at least one of them. Then, there exists a constant C, depending only on a and b such that m(Ω)
D−1 D
≤ Cm(∂Ω),
where D = a + b + 2. Proof. First, by symmetry we can suppose a > 0. Properties P 1 and P 2 ensure the existence of a non-increasing function ψ ∈ W 1,1 (R) with compact support such that Ω = {(s, t) ∈ (R+ )2 : t < ψ(s)}. Then, Z +∞ Z +∞ q m(Ω) = sa ψ b+1 ds, m(∂Ω) = sa ψ b 1 + ψ˙ 2 ds, 0
0
where ψ˙ is the derivative of ψ. λD Let λ > 0 be such that m(Ω) = a+1 . Then, we claim that ψ(s) < λ for s > λ. Assume that it is false. Then, we would have v(s0 ) ≥ λ for some s0 > λ, and hence Z s0 Z λ λD a b+1 m(Ω) ≥ s ψ ds > sa λb+1 = , a+1 0 0 a contradiction. On the other hand, since a > 0, b + 1 > 0, and ψ˙ ≤ 0, Z +∞ q m(∂Ω) = sa ψ b 1 + ψ˙ 2 ds 0 Z +∞ b+1 ˙ ≥ c sa ψ b 1 − ψ ds a 0 b+1 Z +∞ d ψ = c sa ψ b − ds ds a 0 Z +∞ 1 1 a b+1 = c s ψ + ds. ψ s 0
REGULARITY OF MINIMIZERS IN DOMAINS OF DOUBLE REVOLUTION
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Finally, taking into accout that ψ(s) < λ for s > λ, we obtain that ψ1 + 1s ≥ λ−1 for each s > 0, and Z +∞ D−1 1 1 a b+1 m(∂Ω) ≥ c s ψ + ds ≥ cλ−1 m(Ω) = cm(Ω) D , ψ s 0 as claimed.
Now we are able to prove our Sobolev inequality. Proof of Proposition 1.7. We will prove first the case q = 1. Letting χA the characteristic function of the set A, we have Z +∞ χ[u(x)>τ ] dτ, u(x) = 0
where x = (s, t). So, by Minkowski’s integral inequality ! D−1 Z Z D sa tb |u|
D D−1
+∞
≤
dsdt
(R+ )2
=
! D−1 D
sa tb χ[u(x)>τ ] dsdt
(R+ )2
0
Z
Z
+∞
m({u(x) > τ })
D−1 D
dτ
dτ.
0
Since us ≤ 0 and ut ≤ 0, with strict inequality when u > 0, then the level sets {u(x) > τ } satisfy P1 and P2, so Proposition 4.1 implies that m({u(x) > τ }) whence Z
a b
s t |u|
D D−1
! D−1 D
dsdt
(R+ )2
≤C
Z
D−1 D
≤ Cm({u(x) = τ }),
+∞
m({u(x) = τ })dτ = C
Z
sa tb |∇u|dsdt,
(R+ )2
0
where we have used the coarea formula. It remains to prove the case 1 < q < D. Take u satisfying the hypothesis of ∗ Proposition 1.7, and define v = uγ , where γ = 1q∗ . In particular, γ > 1, so that v ∈ C 1 , and we can apply the weighted Sobolev inequality with q = 1 to get ! D−1 !1/1∗ Z Z Z D ∗
sa tb |u|q dsdt
(R+
D
sa tb |v| D−1 dsdt
=
)2
(R+
sa tb |∇v|dsdt.
≤C
(R+ )2
)2
Now, |∇v| = γuγ−1 |∇u|, and by H¨older’s inequality it follows that !1/q Z !1/q0 Z Z a b a b q a b (γ−1)q 0 s t |∇v|dsdt ≤ C s t |∇u| dsdt s t |u| dsdt . (R+ )2
(R+ )2
(R+ )2
But from the definition of γ and q ∗ it follows that 1 1 1 − ∗ = 0, (γ − 1)q 0 = q ∗ , ∗ 1 q q and hence !1/q∗ !1/q Z Z ∗ a b q a b q ≤C s t |∇u| dsdt , s t |u| dsdt (R+ )2
(R+ )2
14
´ AND XAVIER ROS-OTON XAVIER CABRE
as desired.
References
[1] H. Brezis, J.L. V´ azquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Compl. Madrid 10 (1997), 443-469. [2] X. Cabr´ e, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Appl. Math. 63 (2010), 1362–1380. [3] X. Cabr´ e, A. Capella, Regularity of radial minimizers and extremal solutions of semi-linear elliptic equations, J. Funct. Anal. 238 (2006), 709-733. [4] X. Cabr´ e, X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights, preprint. [5] X. Cabr´ e, M. Sanch´ on, Geometric-type Sobolev inequatilies and applications, arXiv 1111.2801v1. [6] D. de Figueiredo, P.L. Lions, R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), 41-63. [7] L. Dupaigne, Stable Solutions to Elliptic Partial Differential Equations, CRC Press, 2011. [8] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. [9] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S´ er. I Math. 330 (2000), 997-1002. [10] G. Nedev, Extremal solutions of semilinear elliptic equations, Preprint, 2001. `cnica de Catalunya, Departament de Matema ` tica ApliICREA and Universitat Polite cada I, Diagonal 647, 08028 Barcelona, Spain E-mail address: [email protected] `cnica de Catalunya, Departament de Matema ` tica Aplicada I, DiUniversitat Polite agonal 647, 08028 Barcelona, Spain E-mail address: [email protected]
Abstract. We consider the class of semi-stable positive solutions to semilinear equations −∆u = f (u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n − m variables. We assume 2 ≤ m ≤ n − 2. When the domain is convex, we establish a priori Lp and H01 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of −∆u = λf (u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 3 for any domain Ω, in dimensions n ≤ 4 when the domain is convex, and in dimensions n ≤ 9 in the radial case. In dimensions 5 ≤ n ≤ 9 it remains an open question.
1. Introduction and results n
Let Ω ⊂ R be a smooth and bounded domain, and consider the problem −∆u = λf (u) in Ω u > 0 in Ω (1.1) u = 0 on ∂Ω,
where λ is a positive parameter and the nonlinearity f : [0, ∞) −→ R satisfies
f (τ ) = ∞. τ It is well known (see the excellent monograph [7] and references therein) that there exists an extremal parameter λ∗ ∈ (0, ∞) such that if 0 < λ < λ∗ then the problem (1.1) admits a minimal classical solution uλ , while for λ > λ∗ it has no solution, even in the weak sense. Here, minimal means smallest. Moreover, the set {uλ : 0 < λ < λ∗ } is increasing in λ, and its pointwise limit u∗ = limλ→λ∗ uλ is a weak solution of problem (1.1) with λ = λ∗ . It is called the extremal solution of (1.1). When f (u) = eu , it is well known that u ∈ L∞ (Ω) if n ≤ 9, while u∗ (x) = log |x|1 2 if n ≥ 10 and Ω = B1 . An analogous result holds for f (u) = (1 + u)p , p > 1. In the nineties H. Brezis and J.L. V´azquez [1] raised the question of determining the regularity of u∗ , depending on the dimension n, for general convex nonlinearities satisfying (1.2). The first general results were proved by G. Nedev [9, 10] – see [5] for the statement and proofs of the results of [10]. (1.2) f is C 1,γ , γ ∈ (0, 1), f is nondecreasing, f (0) > 0, and lim
τ →∞
Theorem 1.1 ([9],[10]). Let Ω be a smooth bounded domain, f be a convex function satisfying (1.2), and u∗ be the extremal solution of (1.1). Key words and phrases. Semilinear elliptic equations, regularity of stable solutions. Both authors were supported by MTM2011-27739-C04-01 (Spain) and 2009SGR345 (Catalunya). 1
´ AND XAVIER ROS-OTON XAVIER CABRE
2
i) If n ≤ 3, then u∗ ∈ L∞ (Ω). n ii) If n ≥ 4, then u∗ ∈ Lp (Ω) for every p < n−4 . iii) Assume either that n ≤ 5 or that Ω is strictly convex. Then u∗ ∈ H01 (Ω). In 2006, the first autor and A. Capella [3] studied the radial case. Their result establishes optimal regularity results for general f . Theorem 1.2 ([3]). Let Ω = B1 be the unit ball in Rn , f be a function satisfying (1.2), and u∗ be the extremal solution of (1.1). i) If n ≤ 9, then u∗ ∈ L∞ (Ω). ii) If n ≥ 10, then u∗ ∈ Lp (Ω) for every p < pn , where (1.3)
pn = 2 +
4 √n 2+ n−1
−2
.
iii) For every dimension n, u∗ ∈ H 3 (Ω). The best known result was established in 2010 by the first author [2] and establishes the boundedness of u∗ in convex domains in dimension n = 4. Related ideas recently allowed the first author and M. Sanch´on [5] to improve the Lp estimates of Theorem 1.1 for u∗ when n ≥ 5: Theorem 1.3 ([2],[5]). Let Ω ⊂ Rn be a convex, smooth and bounded domain, f be a function satisfying (1.2), and u∗ be the extremal solution of (1.1). i) If n ≤ 4, then u∗ ∈ L∞ (Ω). 2n ii) If n ≥ 5, then u∗ ∈ Lp (Ω) for every p < n−4 = 2 + n 4−2 . 2
The boundedness of extremal solutions remains an open question in dimensions 5 ≤ n ≤ 9, even in the case of convex domains. The aim of this paper is to study the regularity of the extremal solution u∗ of (1.1) in a class of domains that we call of double revolution. The class contains domains much more general than balls, but is much simpler than general convex domains. Let n ≥ 4 and (1.4)
Rn = Rm × Rk with n = m + k, m ≥ 2 and k ≥ 2.
For each x ∈ Rn we define the variables p x21 + · · · + x2m s = q t = x2m+1 + · · · + x2n .
We say that a domain Ω ⊂ Rn is a domain of double revolution if it is invariant under rotations of the first m variables and also under rotations of the last k variables. Equivalently, Ω is of the form Ω = {x ∈ Rn : (s, t) ∈ U } where U is a domain in R2 symmetric with respect to the two coordinate axes. In fact, U = {(s, t) ∈ R2 : x = (x1 = s, x2 = 0, ..., xm = 0, xm+1 = t, ..., xn = 0) ∈ Ω} is the intersection of Ω with the (x1 , xm+1 )-plane. Note that U is smooth if and only e the intersection of U with the positive quadrant of if Ω is smooth. Let us call Ω 2 R , i.e., (1.5) e = {(s, t) ∈ R2 : s > 0, t > 0, (x1 = s, x2 = 0, ..., xm = 0, xm+1 = t, ..., xn = 0) ∈ Ω}. Ω
REGULARITY OF MINIMIZERS IN DOMAINS OF DOUBLE REVOLUTION
3
Since {s = 0} and {t = 0} have zero measure in R2 , we have that Z Z v dx = cm,k v(s, t)sm−1 tk−1 dsdt Ω
e Ω
1
for every L (Ω) function v = v(x) which depends only on the radial variables s and t. Here, cm,k is a positive constant depending only on m and k. We will see that any semi-stable classical solution u of (1.1), and more generally of (1.7) below, depends only on s and t, and hence we can identify it with a function u = u(s, t) defined in (0, ∞)2 which satisfies the equation k−1 m−1 e us + ut + f (u) = 0 for s, t ∈ Ω. s t Moreover, in the case of convex domains we will also have us ≤ 0 and ut ≤ 0 and hence, u(0) = kukL∞ (see Remark 2.2). The following is our main result. We prove that, in convex domains of double revolution, the extremal solution u∗ is bounded when n ≤ 7, and belongs to H01 and certain Lp spaces when n ≥ 8. We also prove that in dimension n = 4 the convexity of the domain is not required for the boundedness of u∗ . uss + utt +
Theorem 1.4. Assume (1.4). Let Ω ⊂ Rn be a smooth and bounded domain of double revolution, f be a function satisfying (1.2), and u∗ be the extremal solution of (1.1). a) Assume either that n = 4 or that n ≤ 7 and Ω is convex. Then, u∗ ∈ L∞ (Ω). b) If n ≥ 8 and Ω is convex, then u∗ ∈ Lp (Ω) for all p < pm,k , where (1.6)
pm,k = 2 +
4 √m 2+ m−1
+
√k 2+ k−1
−2
.
c) Assume either that n ≤ 6 or that Ω is convex. Then, u∗ ∈ H01 (Ω). Remark 1.5. Let qm,k = 2+√mm−1 + 2+√kk−1 . Since 2+√xx−1 is concave in (2, ∞), we have that qn−2,2 ≤ qm,k ≤ q n2 , n2 , and therefore p n2 , n2 ≤ pm,k ≤ pn−2,2 . Thus, asymptotically as n → ∞, √ 2 2 4 √ ' p n2 , n2 ≤ pm,k ≤ pn−2,2 ' 2 + √ . 2+ n n Instead, in a general convex domain, Lp estimates are only known for p ' 2+ n8 (see Theorem 1.3 ii) above), while in the radial case one has Lp estimates for p ' 2 + √4n (see Theorem 1.2 ii). In fact, the optimal exponent (1.3) in the radial case can be obtained by setting m = n and k = 0 in (1.6), but recall that in this paper we always assume m ≥ 2 and k ≥ 2. The proofs of the results in [9, 10, 3, 2, 5] use the semi-stability of the extremal solution u∗ . In fact, one first proves estimates for any regular semi-stable solution u of −∆u = f (u) in Ω (1.7) u = 0 on ∂Ω, then one applies these estimates to the minimal solutions uλ , and finally by monotone convergence the estimates also hold for the extremal solution u∗ .
´ AND XAVIER ROS-OTON XAVIER CABRE
4
t e Ω
Q P
s
Figure 1. A non-convex domain for which the maximum of u∗ will not be u∗ (0) Recall that a classical solution u of (1.7) is said to be semi-stable if the second variation of energy at u is nonnegative, i.e., if Z (1.8) Qu (ξ) = |∇ξ|2 − f 0 (u)ξ 2 dx ≥ 0 Ω
C01 (Ω).
for all ξ ∈ The proof of the estimates in [3, 2, 5] was inspired by the proof of Simons theorem on the nonexistence of singular minimal cones in Rn for n ≤ 7. The key idea is to take ξ = |∇u|η (or ξ = ur η in the radial case) and compute Qu (|∇u|η) in the semi-stability property satisfied by u. Then, the expression of Qu in terms of η does not depend on f , and a clever choice of the test function η leads to Lp and L∞ bounds depending on the dimension n. In this paper we will proceed in a similar way, proving first results for general semi-stable solutions of (1.7) and then applying them to uλ to deduce estimates for u∗ . We will take ξ = us η and ξ = ut η separately instead of ξ = |∇u|η, to obtain bounds for Z Z (1.9) u2s s−2α−2 dx and u2t t−2β−2 dx, √
Ω
√
Ω
where α < m − 1 and β < k − 1. When the domain Ω is convex, we will have the additional information kukL∞ = u(0), us ≤ 0, and ut ≤ 0, which combined with (1.9) will lead to L∞ and Lp estimates for u∗ . Instead, when the domain Ω is not convex the maximum of u may not be achieved at the origin – see Figure 1 for an example in which u(0) will be much smaller than kukL∞ . Thus, in nonconvex domains we can not apply the same argument. However, if the maximum is away from {s = 0} and {t = 0} (as in Figure 1) then the problem is essentially two dimensional near the maximum, since dx = cm,k sm−1 tk−1 dsdt and both s and t will be positive and bounded below. We will still have to prove some boundary estimates, for instance estimates near the points P and Q in Figure 1. But, by the same reason as before, near P the coordinate s is positive and bonded below. Thus, the problem near P will be essentially k + 1 dimensional, and k = n − m ≤ n − 2. This will allow us, if n − 1 is small enough, to use Nedev’s [9] W 2,p estimates to obtain boundary estimates.
REGULARITY OF MINIMIZERS IN DOMAINS OF DOUBLE REVOLUTION
5
Our result for general semi-stable solutions of (1.7) reads as follows. It states global estimates controlled in terms of boundary estimates. Proposition 1.6. Assume (1.4). Let Ω ⊂ Rn be a smooth and bounded domain of double revolution, f be any C 1,γ function, and u be a positive bounded semi-stable solution of (1.7). Let δ be a positive real number, and define Ωδ = {x ∈ Ω : dist(x, ∂Ω) < δ}. Then, for some constant C depending only on Ω, δ, n, and also p in part b) below, one has: a) If n ≤ 7 and Ω is convex, then kukL∞ (Ω) ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) . b) If n ≥ 8 and Ω is convex, then kukLp (Ω) ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) for each p < pm,k , where pm,k is given by (1.6). c) For all n ≥ 4, kukH01 (Ω) ≤ CkukH 1 (Ωδ ) . To prove part b) of Proposition 1.6 we will need a new weighted Sobolev inequality in (R+ )2 = {(s, t) ∈ R2 : s > 0, t > 0}. It states the following. Proposition 1.7. Let a > −1 and b > −1 be real numbers, being positive at least one of them, and let D = 2 + a + b. Let u be a nonnegative Lipschitz function with compact support in R2 such that us ≤ 0 and ut ≤ 0 in (R+ )2 , with strict inequality when u > 0. Then, for each 1 ≤ q < D there exist a constant C, depending only on a, b, and q, such that !1/q∗ !1/q Z Z ∗ (1.10) sa tb |u|q dsdt ≤C sa tb |∇u|q dsdt , (R+ )2
where q ∗ =
(R+ )2
Dq D−q .
In section 4 we establish this weighted Sobolev inequality as a consequence of a new weighted isoperimetric inequality. Remark 1.8. When a and b are nonnegative integers, inequality (1.10) is a direct consequence of the classical Sobolev inequality in RD . Namely, define in RD = Ra+1 × Rb+1 the variables s and t as before, with m = a + 1 and k = b + 1. Then, for functions u defined in RD depending only on the variables s and t, write the integrals appearing in the classical Sobolev inequality in RD in terms of s and t. Since dx = ca,b sa tb dsdt, the obtained inequality is precisely the one given in Proposition 1.7. Thus, the previous proposition extends the classical Sobolev inequality to the case of non-integer exponents a and b. In another article, [4], we prove inequality (1.10) with (R+ )2 replaced by (R+ )d and with sa tb replaced by the monomial Ad 1 weight xA = xA 1 · · · xd . We also prove a related isoperimetric inequality with best constant, a weighted Morrey’s inequality, and we determine extremal functions for some of these inequalities. The paper is organized as follows. In section 2 we prove the estimates of Proposition 1.6. Section 3 deals with the regularity of the extremal solution of (1.1). Finally, in section 4 we prove the weighted Sobolev inequality of Proposition 1.7.
´ AND XAVIER ROS-OTON XAVIER CABRE
6
2. Proof of Proposition 1.6 In this section we prove the estimates of Proposition 1.6. For this, we will need two preliminary results. Lemma 2.1 ([3, 2]). Let u be a bounded semi-stable solution of (1.7), and let c be a C 2 (Ω) function. Then, Z Z 0 2 c{∆c + f (u)c}η dx ≤ c2 |∇η|2 dx Ω
Ω
for all η ∈ Lip(Ω) with η|∂Ω = 0.
Proof. It suffices to set ξ = cη in the semi-stability condition (1.8) and then integrate by parts. The fact that we can take η ∈ Lip(Ω) can be deduced by density arguments. Remark 2.2. Note that when the domain is of double revolution, any bounded semi-stable solution u of (1.7) will depend only on the variables s and t. To prove this, define v = xi uxj − xj uxi , with i 6= j. Note that u will will depend only on s and t if and only if v ≡ 0 for each i, j ∈ {1, ..., m} and for each i, j ∈ {m + 1, ..., n}. We first see that, for such indexes i, j, v is a solution of the linearized equation of (1.7): ∆v
=
∆(xi uxj − xj uxi )
= xi ∆uxj + 2∇xi · ∇uxj − xj ∆uxi − 2∇xj · ∇uxi = xi (∆u)xj − xj (∆u)xi = −f 0 (u){xi uxj − xj uxi } = −f 0 (u)v. Moreover, since u = 0 on ∂Ω then v = 0 on ∂Ω. Thus, multiplying the equation by v and integrating by parts, we obtain Z {|∇v|2 − f 0 (u)v 2 }dx = 0. Ω
But since u is semi-stable, λ1 (∆ + f 0 (u); Ω) ≥ 0. If λ1 (∆ + f 0 (u); Ω) > 0, the preious inequality leads to v ≡ 0. If λ1 (∆ + f 0 (u); Ω) = 0, then we must have v = Kφ1 , where K is a constant and φ1 is the first eigenfunction, which we may take to be positive in Ω. But since v is the derivative of u along the vector field ∂t = xi ∂xj − xj ∂xi , and its integral curves are closed, v can not have constant sign. Thus, K = 0, that is, v ≡ 0. Hence, we have seen that any classical semi-stable solution u of (1.7) depend only on the variables s and t. Moreover, by the classical result of Gidas-Ni-Nirenberg [8], when Ω is even and convex with respect each coordinate, we have uxi > 0 when xi > 0, for i = 1, ..., n. In particular, when Ω is a convex domain of double revolution, we have that us < 0 and ut < 0 for s > 0, t > 0. In particular, kukL∞ (Ω) = u(0). On the other hand, since f ∈ C 1,γ then a bounded solution u of (1.7) satisfies u ∈ C 3,γ (Ω). In particular, u ∈ C 3 (Ω), and us , ut ∈ C 2 (Ω). Finally, note that since u is an even function of s and t, then us = 0 when s = 0 and ut = 0 when t = 0.
REGULARITY OF MINIMIZERS IN DOMAINS OF DOUBLE REVOLUTION
7
We now apply Lemma 2.1 separately with c = us and with c = ut , and then we choose appropriately the test function η to get the following result. This estimate is the key ingredient in the proof of Proposition 1.6. Lemma 2.3. Assume (1.4). Let Ω ⊂ Rn be a smooth and bounded domain of double revolution, f be any C 1,γ function, and u be a positive bounded semi-stable solution of (1.7). Let α and β be such that α2 < m − 1 and β 2 < k − 1. Then, for each δ > 0 there exists a constant C, which depends only on Ω, δ, n, α, and β, such that Z 1/2 2 −2α−2 (2.1) us s + u2t t−2β−2 dx ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) , Ω
where
Ωδ = {x ∈ Ω : dist(x, ∂Ω) < δ}. Proof. We will prove only the estimate for u2s s−2α−2 ; the other term can be estimated similarly. Differentiating the equation −∆u = f (u) with respect to s, we obtain us −∆us + (m − 1) 2 = f 0 (u)us , s and hence, setting c = us ∈ C 2 (Ω) in Lemma 2.1, we have that Z Z 2 2η (m − 1) us 2 dx ≤ u2s |∇η|2 dx s Ω Ω for all η ∈ Lip(Ω), η|∂Ω = 0. Let us set in the last inequality η = η , where −α s ρ if s > 0 in Ωδ/3 η = and ρ= −α ρ if s ≤ 1 in Ω\Ωδ/2 , and ρ is a smooth function. Then, since α2 < 21 (α2 + m − 1) < m − 1, 1 2 in (Ω\Ωδ/2 ) ∩ {s > } 2 (α + m − 1)s−2α−2 ρ2 1 2 −2α−2 2 −2α (α + m − 1)s ρ + Cs in Ωδ/2 ∩ {s > } |∇η |2 ≤ 2 −2α C in Ω ∩ {s ≤ },
and therefore Z Z Z m − 1 − α2 u2s s−2α−2 ρ2 dx ≤ C u2s s−2α dx+C−2α u2s dx, 2 Ω∩{s>} Ωδ/2 ∩{s>} Ω∩{s≤} where C denote different constants depending only on the quantities appearing in the statement of the lemma. Now, since us ∈ L∞ (Ω) the√last term is bounded by Ckus kLi nf ty m−2α . Making → 0 and using that 2α < 2 m − 1 ≤ m, we deduce Z Z 2 −2α−2 2 us s ρ dx ≤ C u2s s−2α dx. Ω
Ωδ/2
Hence, since ρ ≡ 1 in Ω\Ωδ/2 , Z Z (2.2) u2s s−2α−2 dx ≤ C Ω\Ωδ/2
Ωδ/2
u2s s−2α dx ≤ C
Z
Ωδ/2
u2s s−2α−2 dx.
´ AND XAVIER ROS-OTON XAVIER CABRE
8
From this we deduce that, for another constant C, Z Z 2 −2α−2 (2.3) us s dx ≤ C u2s s−2α−2 dx. Ω
Ωδ/2
Let ν < 1 to be chosen later. On the one hand using that us (0, t) = 0 we obtain that, if δ is small enough, |us (s, t)| ≤ Csν kus kC 0,ν in Ωδ/2 ∩ {s < δ}. Moreover, since −∆u = f (u) in Ωδ then kukC 1,ν (Ωδ/2 ) ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) , and therefore, ks−ν us kL∞ (Ωδ/2 ∩{s 0, we will have Z Z 2 −2α−2 −ν 2 us s dx ≤ ks us kL∞ (Ωδ/2 ) s−2α−2+2ν dx ≤ Cks−ν us k2L∞ (Ωδ/2 ) , Ωδ/2
Ωδ/2
and hence by (2.3) and (2.4), Z 2 u2s s−2α−2 dx ≤ C kukL∞ (Ωδ ) + kf (u)kL∞ (Ωδ ) . Ω
Using Lemma 2.3 we can now establish Proposition 1.6. Proof of Proposition 1.6. a) We assume Ω to be convex. Recall that in this case kukL∞ = u(0); see Remark 2.2. On the one hand, making the change of variables σ = s2+α , τ = t2+β in the integral in (2.1), one has ( m sm−1 ds = cα σ 2+α −1 dσ k tk−1 dt = cβ τ 2+β −1 dτ, and thus, (2.5)
Z
Ω
k m σ 2+α −1 τ 2+β −1 (u2σ + u2τ )dσdτ ≤ C kuk2L∞ (Ωδ ) + kf (u)k2L∞ (Ωδ ) .
Here, to simplify, we are abusing notation and calling again Ω the image of the two e in (1.5) after the transformation (s, t) 7→ (σ, τ ). Therefore, dimensional√domain Ω 2 2 setting ρ = σ + τ and taking into account that in {τ < σ < 2τ } we have σ > ρ2 and τ ≥ ρ3 , we obtain Z m k (2.6) ρ 2+α + 2+β −2 (u2σ + u2τ )dσdτ ≤ C kuk2L∞ (Ωδ ) + kf (u)k2L∞ (Ωδ ) . Ω∩{τ 0 such that Ωδ ∩ Br0 (0) = ∅. Thus, s ≥ c in Ωδ ∩ {s > t} and t ≥ c in Ωδ ∩ {s < t}, and Z Z t|∇u∗ |p dsdt ≤ C, s|∇u∗ |p dsdt ≤ C. Ωδ ∩{s>t}
Ωδ ∩{s 3 then we can apply Sobolev’s inequality in dimension 3 (as explained in Remark 1.8, to obtain u∗ ∈ L∞ (Ωδ ∩{s > t}) and u∗ ∈ L∞ (Ωδ ∩{s < t}). Therefore u∗ ∈ L∞ (Ωδ ), as claimed. To prove part c) in the non-convex case, let n ≤ 6. By Proposition 1.6, it suffices to prove that u∗ ∈ H 1 (Ωδ ) for some δ > 0. Take r0 and δ such that Ωδ ∩Br0 (0) = ∅, as in part a). n . Thus, In [9] is proved that u∗ ∈ W 2,p (Ω) for p < n−2 Z Z tk−1 |D2 u∗ |p dsdt ≤ C, sm−1 |D2 u∗ |p dsdt ≤ C. Ωδ ∩{s>t}
Ωδ ∩{s t}) and u∗ ∈ H 1 (Ωδ ∩ {s < t}), and therefore, u∗ ∈ H 1 (Ωδ ).
´ AND XAVIER ROS-OTON XAVIER CABRE
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4. Weighted Sobolev inequality It is well known that the classical Sobolev inequality can be deduced from an isoperimetric inequality. This is done by applying first the isoperimetric inequality to the level sets of the function and then using the coarea formula. In this way one deduces the Sobolev inequality with exponent 1 on the gradient. Then, by applying H¨ older’s inequality one deduces the general Sobolev inequality. Since in our case we have us ≤ 0 and ut ≤ 0, with strict inequality when u > 0, it suffices to prove a weighted isoperimetric inequality for domains Ω ⊂ (R+ )2 satisfying the following properties: P1) If (s, t) ∈ Ω, then (s0 , t0 ) ∈ Ω for all s0 and t0 with 0 < s0 < s and 0 < t0 < t. P2) Ωt = {s > 0 : (s, t) ∈ Ω} and Ωs = {t > 0 : (s, t) ∈ Ω} are strictly decreasing in t and s, respectively. We will denote m(Ω) =
Z
sa tb dsdt
and
m(∂Ω) =
Z
sa tb dsdt.
∂Ω∩(R+ )2
Ω
Proposition 4.1. Let Ω ⊂ (R+ )2 be a bounded Lipschitz domain satisfying P1 and P2, and let a > −1 and b > −1 be real numbers, being positive at least one of them. Then, there exists a constant C, depending only on a and b such that m(Ω)
D−1 D
≤ Cm(∂Ω),
where D = a + b + 2. Proof. First, by symmetry we can suppose a > 0. Properties P 1 and P 2 ensure the existence of a non-increasing function ψ ∈ W 1,1 (R) with compact support such that Ω = {(s, t) ∈ (R+ )2 : t < ψ(s)}. Then, Z +∞ Z +∞ q m(Ω) = sa ψ b+1 ds, m(∂Ω) = sa ψ b 1 + ψ˙ 2 ds, 0
0
where ψ˙ is the derivative of ψ. λD Let λ > 0 be such that m(Ω) = a+1 . Then, we claim that ψ(s) < λ for s > λ. Assume that it is false. Then, we would have v(s0 ) ≥ λ for some s0 > λ, and hence Z s0 Z λ λD a b+1 m(Ω) ≥ s ψ ds > sa λb+1 = , a+1 0 0 a contradiction. On the other hand, since a > 0, b + 1 > 0, and ψ˙ ≤ 0, Z +∞ q m(∂Ω) = sa ψ b 1 + ψ˙ 2 ds 0 Z +∞ b+1 ˙ ≥ c sa ψ b 1 − ψ ds a 0 b+1 Z +∞ d ψ = c sa ψ b − ds ds a 0 Z +∞ 1 1 a b+1 = c s ψ + ds. ψ s 0
REGULARITY OF MINIMIZERS IN DOMAINS OF DOUBLE REVOLUTION
13
Finally, taking into accout that ψ(s) < λ for s > λ, we obtain that ψ1 + 1s ≥ λ−1 for each s > 0, and Z +∞ D−1 1 1 a b+1 m(∂Ω) ≥ c s ψ + ds ≥ cλ−1 m(Ω) = cm(Ω) D , ψ s 0 as claimed.
Now we are able to prove our Sobolev inequality. Proof of Proposition 1.7. We will prove first the case q = 1. Letting χA the characteristic function of the set A, we have Z +∞ χ[u(x)>τ ] dτ, u(x) = 0
where x = (s, t). So, by Minkowski’s integral inequality ! D−1 Z Z D sa tb |u|
D D−1
+∞
≤
dsdt
(R+ )2
=
! D−1 D
sa tb χ[u(x)>τ ] dsdt
(R+ )2
0
Z
Z
+∞
m({u(x) > τ })
D−1 D
dτ
dτ.
0
Since us ≤ 0 and ut ≤ 0, with strict inequality when u > 0, then the level sets {u(x) > τ } satisfy P1 and P2, so Proposition 4.1 implies that m({u(x) > τ }) whence Z
a b
s t |u|
D D−1
! D−1 D
dsdt
(R+ )2
≤C
Z
D−1 D
≤ Cm({u(x) = τ }),
+∞
m({u(x) = τ })dτ = C
Z
sa tb |∇u|dsdt,
(R+ )2
0
where we have used the coarea formula. It remains to prove the case 1 < q < D. Take u satisfying the hypothesis of ∗ Proposition 1.7, and define v = uγ , where γ = 1q∗ . In particular, γ > 1, so that v ∈ C 1 , and we can apply the weighted Sobolev inequality with q = 1 to get ! D−1 !1/1∗ Z Z Z D ∗
sa tb |u|q dsdt
(R+
D
sa tb |v| D−1 dsdt
=
)2
(R+
sa tb |∇v|dsdt.
≤C
(R+ )2
)2
Now, |∇v| = γuγ−1 |∇u|, and by H¨older’s inequality it follows that !1/q Z !1/q0 Z Z a b a b q a b (γ−1)q 0 s t |∇v|dsdt ≤ C s t |∇u| dsdt s t |u| dsdt . (R+ )2
(R+ )2
(R+ )2
But from the definition of γ and q ∗ it follows that 1 1 1 − ∗ = 0, (γ − 1)q 0 = q ∗ , ∗ 1 q q and hence !1/q∗ !1/q Z Z ∗ a b q a b q ≤C s t |∇u| dsdt , s t |u| dsdt (R+ )2
(R+ )2
14
´ AND XAVIER ROS-OTON XAVIER CABRE
as desired.
References
[1] H. Brezis, J.L. V´ azquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Compl. Madrid 10 (1997), 443-469. [2] X. Cabr´ e, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Appl. Math. 63 (2010), 1362–1380. [3] X. Cabr´ e, A. Capella, Regularity of radial minimizers and extremal solutions of semi-linear elliptic equations, J. Funct. Anal. 238 (2006), 709-733. [4] X. Cabr´ e, X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights, preprint. [5] X. Cabr´ e, M. Sanch´ on, Geometric-type Sobolev inequatilies and applications, arXiv 1111.2801v1. [6] D. de Figueiredo, P.L. Lions, R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), 41-63. [7] L. Dupaigne, Stable Solutions to Elliptic Partial Differential Equations, CRC Press, 2011. [8] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. [9] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S´ er. I Math. 330 (2000), 997-1002. [10] G. Nedev, Extremal solutions of semilinear elliptic equations, Preprint, 2001. `cnica de Catalunya, Departament de Matema ` tica ApliICREA and Universitat Polite cada I, Diagonal 647, 08028 Barcelona, Spain E-mail address: [email protected] `cnica de Catalunya, Departament de Matema ` tica Aplicada I, DiUniversitat Polite agonal 647, 08028 Barcelona, Spain E-mail address: [email protected]
