Variational Problems with Critical Sobolev Growth and Positive Dirichlet Data

Luis A. Caffarelli^{∗} Institute for Advanced Study Princeton University Princeton, NJ 08540

Joel Spruck^{†} Department of Mathematics University of Massachusetts Amherst, MA 01003

1 Introduction

In this paper, we consider the Dirichlet problem for the conformally invariant model problem of critical Sobolev growth:

\begin{aligned}−Δu & = & λu\frac{n+2}{n−2} & in & Ω \\ u & = & ϕ & on & ∂Ω \\ u & ≥ & 0 \end{aligned} (1.1)

Problem (1.1) is formally the Euler-Lagrange equations for the variational problem ∫_{Ω} |∇u|^{2} → min (1.2) for u in the admissible class

∗Research supported in part by NSF

†Research supported in part by NSF and DOE

A = \{u ∈ H^{1}(Ω): u − h ∈ H_{0}^{1}(Ω), ∫_{Ω} u\frac{2n}{n − 2} = γ\}

where h is the harmonic extension of ϕ ≥ 0.

It is well known that for ϕ ≡ 0, the existence of nontrivial solutions is often a very subtle question. For example, if Ω is starshaped there is no solution of (1.1) while if Ω has “nontrivial topology” then there do exist solutions [B].

The purpose of this paper is to show that for positive C^{1+β} data ϕ, Problem (1.2) is well posed for any C^{2} domain Ω. More precisely, we have the following

Theorem 1.1 Let ϕ ∈ C^{1+β}(∂Ω) ≥ 0 be positive somewhere, ∂Ω ∈ C^{2}. Assume that γ > ∫_{Ω} h\frac{2n}{n−2} where h is the harmonic extension of ϕ. Then there is a positive minimizer u ∈ C^{2}(Ω)∩C^{1+β}(\overline{Ω}) of Problem (1.2) satisfying (1.1) for a positive constant λ > 0.

More precise estimates on the multiplier λ are given in Theorem 2.1. More generally, we can consider the variational problem

∫_{Ω} |∇u|^{2} → min (1.3)

for u in the admissible class

A = {u ∈ H^{1}(Ω): u − h ∈ H_{0}^{1}(Ω), ∫_{Ω} G(u) = γ}

with (formal) Euler-Lagrange equations

\begin{aligned}−Δu & = & λg(u) & in & Ω \\ u & = & ϕ & on & ∂Ω \end{aligned} (1.4)

Then we have the following Theorem 1.2 Let ϕ ∈ C^{1+β}(∂Ω) ≥ 0 be positive somewhere, ∂Ω ∈ C^{2} and γ > ∫_{Ω} G(h) where h is the harmonic extension of ϕ.

Assume that the nonlinearity g(t) is a locally Lipschitz function satisfying

(i) g(t) is non-decreasing, g(0) = 0
(1.5)  (ii) t^{−} \frac{n + 2}{n − 2} g(t) is non-increasing
(iii) g(t) ≥ κt, for t ≥ t_{0} > 0, where κ > λ_{1}(Ω)

Then there is a positive minimizer u ∈ C^{2}(Ω)∩C^{1+β}(\overline{Ω}) of Problem (1.4) satisfying (1.4) for a positive constant λ > 0. Here λ_{1}(Ω) is the first Dirichlet eigenvalue of −Δ in Ω.

Some estimates for λ are given in Proposition 5.1.

We should remark that the existence of minimal or “small norm” solutions of (1.1) is completely standard and the main point of our work is the nonexistence of the blow-up of minimizing sequences for the variational problem (1.2). This fact is based on strong apriori estimates that we derive in Sections 3 and 4. Here, the positivity and smoothness of ϕ is essential. These estimates rely on the “asymptotic symmetry” method that we developed in our paper [C-G-S]. For the convenience of the reader we have outlined the main steps in this method in Section 4.

One consequence of these strong estimates is the existence of a continuum S of solutions (u, λ) of (1.1) containing (h, 0) which is unbounded (but {(u, λ) ∈ S: λ ≥ ε} is bounded) and such that λ decreases to zero as γ = ∫_{Ω} u^{2n/n−2} → ∞.

An outline of the paper is as follows. Section 2 contains preliminary existence and multiplier estimates for the subcritical approximations to problem (1.2). In Section 3, we prove a preliminary apriori supnorm estimate for solutions of (1.4). These estimates allow us to use the method of [C-G-S] to prove uniform C^{1+β} estimates. This is carried out in Section 4. Finally, in Section 5 we prove Theorem 1.1 and then go to prove Theorem 1.2.

2 Multiplier Estimates for the Approximating Problems

In this section we will consider subcritical approximations to the variational problem (1.2) and estimate the multipliers associated to solutions.

To fix the ideas, let ϕ ∈ C^{1+β}(∂Ω) ≥ 0, ϕ ≢ 0 and consider the variational problem ∫_{Ω} |∇u|^{2} → min (2.1)

for u in the admissible class

A_{γ} = {u ∈ H^{1}(Ω): u − ϕ ∈ H_{0}^{1}(Ω), ∫ |u|^{α+1} = γ}

where 1 < α < \frac{n+2}{n−2}, and γ > ∫_{Ω} h^{α+1}, for h the harmonic extension of ϕ. The main result of this section may be stated as follows.

Theorem 2.1 There is a positive solution u ∈ C^{2}(Ω) ∩ C^{1+β}(Ω̄) to problem (2.1) satisfying

−Δu = λu^{α} in Ω
u = ϕ on ∂Ω

where λ > 0 is a positive constant. In terms of θ =: γ − ∫_{Ω} h^{α+1}, λ satisfies the asymptotic estimates

(i) c_{1}θ ≤ λ ≤ c_{2}θ for θ > 0 small
(ii) c_{3}θ^{( \frac{1−α}{α+1} )} ≤ λ ≤ c_{4}θ^{( \frac{1−α}{α+1} )} for θ > 0 large

for absolute constants c_{1}, c_{2}, c_{3}, c_{4}.

Proof. The existence of a classical solution u to problem 2.1 is well-known. That is, u satisfies

−Δu = λ|u|^{α−1}u in Ω

u = ϕ on ∂Ω (2.2)

∫_{Ω} |u|^{α+1} = γ > ∫_{Ω} h^{α+1}.

We claim that λ > 0 and so u > h in Ω. For using u − h as a test function in (2.2), we obtain

∫_{Ω} |∇u|^{2} − ∫_{Ω} ∇u · ∇h = λ ∫_{Ω}(u − h)|u|^{α−1}u = λ(γ − ∫_{Ω} h|u|^{α−1}u) (2.3)

By Hölder,

∫_{Ω} h|u|^{α−1}u ≤ (∫_{Ω} h^{α+1})^{\frac{1}{α+1}} (∫_{Ω} |u|^{α+1})^{\frac{α}{α+1}} ≤ γ (2.4)

Now observe that

∫_{Ω} ∇u · ∇h = ∫_{Ω} |∇h|^{2}

and so ∫_{Ω} |∇u|^{2} − ∫_{Ω} ∇u · ∇h = ∫_{Ω} |∇(u − h)|^{2} > 0 (2.5)

It follows from (2.3)-(2.5) that λ > 0 and thus u > h by the maximum principle.

We can obtain a sharper estimate for λ from below as follows. By convexity,

u^{α+1} − h^{α+1} ≤ (α + 1)u^{α}(u − h).

Therefore,

∫_{Ω}(u^{α+1}−h^{α+1}) ≤ (α+1) ∫_{Ω} u^{α}(u−h) ≤ (α+1)(∫_{Ω} u^{α+1})^{α/α+1}(∫_{Ω}(u−h)^{α+1})^{\frac{1}{α+1}}

and so, γ − ∫_{Ω} h^{α+1} ≤ (α + 1)γ^{\frac{α}{α+1}} (∫_{Ω}(u − h)^{α+1})^{\frac{1}{α+1}} (2.6)

Using Sobolev’s inequality, (2.6) implies

(γ − ∫_{Ω} h^{α+1})^{2} ≤ (α + 1)^{2}γ^{\frac{2α}{α+1}} c^{2} ∫_{Ω} |∇(u − h)|^{2}
= (α + 1)^{2}γ^{\frac{2α}{α+1}} c^{2}λ(γ − ∫_{Ω} hu^{α})

≤ (α + 1)^{2}γ^{\frac{2α}{α+1}} c^{2}λ(γ − ∫_{Ω} h^{α+1})

Therefore,

γ − ∫_{Ω} h^{α+1} ≤ c^{2}(α + 1)^{2}γ^{\frac{2α}{α+1}} λ

or λ ≥ (γ − ∫_{Ω} h^{α+1})/c^{2}(α + 1)^{2}γ^{\frac{2α}{α+1}} (2.7)

From our earlier discussion,

∫_{Ω} |∇(u − h)|^{2} = λ(γ − ∫_{Ω} u^{α}h) ≥ \frac{λ}{α + 1}(γ − ∫_{Ω} h^{α+1})

or λ ≤ (α + 1) ∫_{Ω} |∇(u − h)|^{2}/(γ − ∫_{Ω} h^{α+1}) (2.8)

We can use (2.8) to estimate λ from above by estimating

∫_{Ω} |∇(u − h)|^{2} = ∫_{Ω} |∇u|^{2} − ∫_{Ω} |∇h|^{2}.

For example, let η ≥ 0 be a fixed nontrivial C_{0}^{1}(Ω̄) function and set v = h+εη. Then there is a unique ε > 0 so that ∫_{Ω} v^{α+1} = γ. This choice of ε makes vεA_{γ} and so

∫_{Ω} |∇u|^{2} ≤ ∫_{Ω} |∇v|^{2} = ∫_{Ω} |∇h|^{2} + ε^{2} ∫_{Ω} |∇η|^{2} (2.9)

Again by convexity, we can estimate ε by

γ = ∫_{Ω}(h + εη)^{α+1} ≥ ∫_{Ω} h^{α+1} + (α + 1)ε ∫_{Ω} h^{α}η

so that ε ≤ C(γ − ∫_{Ω} h^{α+1}) (2.10)

The combination (2.9) (2.10) gives

∫_{Ω} |∇(u − h)|^{2} ≤ C(γ − ∫_{Ω} h^{α+1})^{2} (2.11)

Inserting (2.11) into (2.8) gives the upper bound

λ ≤ C(γ − ∫_{Ω} h^{α+1}) (2.12)

The estimates (2.7) and (2.12) give the required asymptotics for λ as stated in part i) of Theorem 2.1.

To get the correct asymptotics for λ as γ → ∞, we let v be a minimizer to the problem

∫ |∇v|^{2} → min = μ, vεH_{0}^{1}(Ω), ∫_{Ω} |v|^{α+1} = 1

and set u = (1 − ε)γ^{\frac{1}{α+1}} v + h. As before, ε is chosen so that ∫_{Ω} u^{α+1} = γ. Then

∫_{Ω} |∇u|^{2} = (1 − ε)^{2}γ^{\frac{2}{α+1}} μ + ∫_{Ω} |∇h|^{2}

so from (2.8),

λ ≤ (α + 1)(1 − ε)^{2}γ^{\frac{2}{α+1}} /(γ − ∫_{Ω} h^{α+1}) ≤ Cγ^{−( \frac{α−1}{α+1} )} (2.13)

for γ large. The estimate (2.13) together with (2.7) completes the proof of the estimates stated in part ii) of Theorem 2.1. □

3 Apriori Estimates near the Boundary

In this section, we will derive the following apriori sup norm estimate near the boundary.

Theorem 3.1 Suppose Ω is a domain with ∂Ω ∈ C^{2} and ϕ ∈ C^{1+β}(∂Ω) ≥ 0, ∂ϕ ≢ 0. Let u ∈ C^{2}(Ω) ∩ C^{1+β}(Ω̄) be a solution to the problem

\begin{aligned}−Δu & = & g(u) & in & Ω \\ u & = & ϕ & on & ∂Ω \end{aligned} (3.1)

where g(t) is a locally Lipschitz function satisfying

(i) g(t) is non-decreasing, g(0) = 0
(3.2)  (ii) t^{−( \frac{n+2}{n−2} )} g(t) is non-increasing
(iii) g(t) ≥ Kt, for t ≥ t_{0} > 0, where K > λ_{1}(Ω).

Then u(x) ≤ C in a neighborhood of ∂Ω, 0 < d(x) < d_{0} (d(x) = distance (x, ∂Ω)) where C, d_{o} depend only upon ϕ, Ω and (K − λ_{1}(Ω))^{−1}.

We first estimate u in L_{loc}^{1}(Ω) in a standard way [F-L-N].

Lemma 3.2 Let u satisfy (3.1), (3.2). Then ∫_{Ω} uv_{1} ≤ C, where v_{1} ≥ 0 is the first Dirichlet eigenfunction of Ω.

Proof. Let h be the harmonic extension of ϕ and let v_{1} ≥ 0 be the first (normalized) Dirichlet eigenfunction of Ω. Then

∫_{Ω} v_{1}Δ(h − u) − (h − u)Δv_{1} = 0, and so
∫_{Ω} v_{1}g(u) + λ_{1}(h − u)v_{1} = 0

Using 3.2 (iii), this gives

∫_{Ω} v_{1}u ≤ \frac{λ_{1}}{K − λ_{1}} ∫_{\{u<t_{0}\}} v_{1}u ≤ \frac{λ_{1}t_{0}}{K − λ_{1}} |Ω|^{\frac{l}{2}} = C.

Our aim is to use the method of moving planes and inversion to show that the Kelvin transform of u is monotone. For boundary date ϕ ≡ 0, this was done in [?] for n = 2 and in [?] for n > 2.

Let us fix the notation as follows. For a point P ∈ ∂Ω, we will choose a ball B = B_{R} of fixed radius R externally tangent to ∂Ω at P. For convenience we will assume that B = B_{R}(0) and that P = Re_{n}. We perform an inversion in B

y = Ix = \frac{R^{2}x}{|x|^{2}} (3.3)

Let κ denote a principal curvature at X ∈ ∂D with respect to the interior normal ν(X) at X. Then [?, ?], the principal curvature κ̃ of I(∂Ω) at I(X) (with respect to the interior normal) is given by

κ̃ = κ\frac{|X|^{2}}{R^{2}} + \frac{2}{R^{2}} X · ν

In particular, we may choose δ_{0} ∈ (0, 1) so small that if X ∈ ∂Ω ∩ B_{δ0R}(P), X · ν ≥ \frac{1}{2}, so that I(∂Ω ∩ B_{δ0R}(P)) is strictly convex for R sufficiently small, say κ̃ ≥ \frac{1}{2R^{2}} .

For any function ψ in Ω, the Kelvin transform of ψ is defined by

(Kψ)(y) = \frac{R^{n−2}}{|y|^{n−2}} ψ(x) = \frac{R^{n−2}}{|y|^{n−2}} ψ(\frac{R^{2}y}{|y|^{2}} ), x ∈ Ω (3.4)

and satisfies

− Δ(Kψ) = \frac{R^{n+2}}{|y|^{n+2}} Δψ (\frac{R^{2}y}{|y|^{2}} ) (3.5)

□

Lemma 3.3 Let h be the harmonic extension of ϕ in Ω. Then for δ_{0} and R small enough, Kh is monotone on I(∂Ω ∩ B_{δ_{0}}(P)), i.e. ∇(Kh) · τ > 0 for τ in a cone of directions near −e_{n}.

Proof. Using (3.4), we compute

(∇Kh)_{i}(y) = −\frac{n − 2}{|y|^{n}}R^{n−2}y_{i}h(x) + \frac{R^{n}}{|y|^{n}} (δ_{ij} − \frac{2y_{i}y_{j}}{|y|^{2}} )h_{j}(x)

This gives

∇(kh) · τ = \frac{R^{n−2}}{|y|^{n}} (−(n − 2)(y · τ)h(x) + O(R^{2})O(|∇h|)) = (3.6)

\frac{R^{n−2}}{|y|^{n}} ((n − 2)Rτ · −e_{n} + O(δ_{0}R) + O(R^{2})O(|∇h|))

Since h ∈ C^{1+β}(Ω̄), (3.6) implies that

∇(kh) · τ(y) > 0 if τ · −e_{n} > O(δ_{0}) + O(R).
□

Corollary 3.4 Ku is monotone on I(∂Ω) ∩ B_{δ_{0}R}(P)) for τ in the cone of directions near −e_{n} given in lemma 3.3.

Proof. Since u ≥ h in Ω, u = h on ∂Ω, we have that Ku ≥ Kh in I(Ω) and Ku = Kh on I(∂Ω). Therefore,

∇(Ku) · τ ≥ ∇(Kh) · τ.
□

We can now prove

Proposition 3.5 Let u satisfy the hypotheses of Theorem 3.1. Then ∇(Ku)· τ > 0 ∈ KΩ ∩ B_{δ_{0}R/4}(P), for y in a cone of directions of aperture angle α(τ, −e_{n}) ≤ δ_{0}/8R, for δ_{0}, R sufficiently small.

Proof. We may assume that K(∂Ω) ∩ B_{δ_{0}R}(P) is a graph in any of the directions τ. For any allowable τ, we denote by y_{λ} the reflection of y in the plane T_{λ} = {y · τ = λ}, and set v = Ku. Since K(∂Ω) ∩ B_{δ_{0}R}(P) is strictly convex, there is a unique value λ_{0}, such that the plane T_{λ_{0}} is tangent to K(∂Ω) (at a point near P) and then for λ > λ_{0} cuts off an open cap Σ_{λ} from KΩ. For y ∈ Σ_{λ} we define v^{λ}(y) = v(y_{λ}).

By our choice of δ_{0}, R and the cone aperture α, we may increase λ to a first value λ_{1} so that the cap Σ_{λ_{1}} contains a \frac{δ_{0}^{2}}{16} neighborhood of K(∂Ω)∩B_{δ_{0}R/4}(P) and also is contained in B_{δ_{0}R}(P). This implies that v ≢ v^{λ} for λ_{0} < λ < λ_{1}. For otherwise, ∇v · τ = 0 on T_{λ} ∩ K(∂Ω), contradicting Corollary 3.4. Using Corollary 3.4 we see that v(y) < v^{λ}(y) for λ > λ_{0}, λ close to λ_{0}. According to [?] [?] the inequalities v < v^{λ}, ∇v · τ > 0 must continue to hold for λ_{0} < λ ≤ λ_{1}, since we have ruled out the possibility v ≡ v^{λ}. □

Proof of Theorem 3.1. Let Q be any point in K(Ω)∩B_{δ_{0}R/8}(P) at distance cδ_{0}^{2} from k(∂Ω), where c is a small constant independent of δ_{0}, R chosen so that the cone Γ_{Q} of aperture α and height cδ_{0}^{2} with vertex at Q is contained in a δ_{0}^{2}/16 neighborhood of K(∂Ω) ∩ B_{δ_{0}R/4}(P). Let v = Ku. According to Lemma 3.2 we can estimate ∫_{Γ_{Q}} v ≤ C (3.7)

for a constant C uniformly controlled. Then using Proposition 3.5, v ≥ v(Q) in Γ_{Q}, and so from (3.7) we obtain

v(Q) ≤ \frac{c}{|Γ_{Q}|} = C_{1} (3.8)

In particular, using Proposition 3.5 once more, we can conclude that

v(y) ≤ C_{1}

for y in a cδ_{0}^{2} neighborhood of K(∂Ω) ∩ B_{δ_{0}R/16}. Finally, using the compactness of ∂Ω, and that v = Ku we conclude that ε_{0} ≤ u(x) ≤ c_{2} in a uniform neighborhood 0 < d(x) < d_{0} of ∂Ω. This completes the proof of Theorem 3.1. □

4 Global Estimates via the Asymptotic Symmetry Method

The apriori near the boundary estimates of the previous section allow us to apply the “asymptotic symmetry method” that we developed in [C-G-S] to prove interior Lipschitz estimates. In this section we will outline the main steps in the proof of the following crucial result. Theorem 4.1 Suppose Ω is a domain with ∂Ω ∈ C^{2} and ϕ ∈ C^{1+β}(∂Ω), ϕ ≢ 0. Let u ∈ C^{2}(Ω) ∩ C^{1+β}(Ω̄) be a solution to the problem

− Δu = g(u) in Ω u = ϕ on ∂Ω (4.1)

where g(t) is a locally Lipschitz function satisfying the conditions (3.2). Then |u(x) − u(z)| ≤ K |x − z| for x, z ∈ Ω where K depends only on ϕ, Ω and (K − λ_{1}(Ω))^{−1}.

Combining Theorems 3.1 and 4.1 and using standard elliptic regularity theorem, we obtain the following global apriori estimate.

Corollary 4.2 Let Ω, ϕ, u satisfy the hypothesis of Theorem 4.1. Then ||u||_{C^{1+β}(Ω)} ≤ C, where C depends only on ϕ, Ω and (K − λ_{1}, (Ω))^{−1}.

We first observe that from Theorem 3.1, it follows in a standard way that

||u||_{C^{1+β}(Ω_{0})} ≤ C (4.2)

where Ω_{0} = \{x ∈ Ω: 0 < d(x) < \frac{3}{4}d_{o}\}, and C depends only upon ϕ, Ω and (K − λ_{1}(Ω))^{−1}. Therefore, in proving Theorem 4.1, it suffices to consider points x, z with d(x), d(z) ≥ \frac{d_{o}}{2} . Since u > h > 0 in Ω, and u is under control in Ω_{0}, we may always reduce Theorem 4.1 to the case ϕ ≥ ε_{0} (i.e. u ≥ ε_{0} in \overline{Ω}).

Fix x_{0} ∈ Ω such that d(x_{0}) ≥ \frac{d_{o}}{2} and set R = \frac{d_{0}}{8} . Let B = B_{R}(x_{0}). For convenience, we assume x_{0} = 0. We perform an inversion in B, y = Ix and set v = Ku as in Section 3, formulas (3.3)-(3.5).

We denote by Γ_{1}, … Γ_{N} the connected components of ∂Ω with Γ_{N} the finite boundary component of R^{n}−Ω. Let Γ̃_{i} = I(Γ_{i}), i = 1, …, N. Then v is well-defined in the unbounded domain Ω̃ = IΩ. Note that Ω̃ = R^{n}\ ∪_{i=1}^{N} H_{i} where the H_{i} are “holes”, that is the H_{i} are disjoint connected domains bounded by Γ̃_{i}, and x_{0} ∈ Γ̃_{N}.

Note that in Ω̃, v satisfies

− Δv = \frac{R^{n+2}}{|y|^{n+2}} g(\frac{|y|^{n−2}}{R^{n−2}} v(y)) ≡ f(y, v) (4.3)

and because of (4.2),

||v||_{C^{1+β}(IΩ_{o})} ≤ C_{1}, 0 < δ_{0} ≤ v ≤ \frac{1}{δ_{0}} in IΩ_{0} (4.4)

where C_{1}, δ_{0} are under control.

Step 1. Extension Lemma. There exists σ = σ(δ_{0}, C_{1}, f) such that for any open set A ⊂ ∪_{i=1}^{N}H_{i} with |A| < σ, we can extend v to a Lipschitz function v̄ in R^{n} satisfying

(4.5)  (i) v̄ ≥ δ_{0}/2 in R^{n}
(ii) −Δv̄ ≥ f(y, v̄) in [\overline{Ω̃ ∪ A}]^{0}

Our aim is to show that v is asymptotically radial at infinity in the sense that

v(y) = (inf v)_{∂B_{|y|}}(1 + 0(\frac{1}{|y|}) as |y| → ∞ (4.6)

Step 2. Reflection Theorem. Assume there exists a set A′ ⊂ {(y′, 0): |y′| < R\}, |A′| < \frac{σ}{2} and a positive number M > R such that if y = (y′, y_{n}) with y′ ∉ A′ and y_{n} ≥ M then v(y) ≤ \frac{δ_{0}}{4} . Let v̄ be the extension of v to R^{n} given by the extension lemma (Step 1) corresponding to A = {y = (y′, y_{n}): |y| < R, y′ ∈ A′} Then v̄(y) ≤ v̄(y_{λ}) for y_{n} > λ ≥ M (4.7)

(Here y = (y′, y_{n}) and y_{λ} = (y′, 2λ − y_{n}) is the reflection of y in the plane y_{n} = λ)

A direction (which for convenience we have taken to be e_{n}) for which the hypothesis of the Reflection Theorem is satisfied is said to be an admissible direction. The essential meaning of admissibility is that v decays to zero uniformly, on rays parallel to the reflection direction, except possibly for a certain exceptional set of rays which hit B_{R}. The set A is the intersection of the exceptional rays with B_{R} and is required to have small measure.

The next step is to show that for M large enough the set of admissible directions covers almost all of the unit sphere Σ_{1}. The precise statement is as follows.

Step 3. Let v ≥ 0 be superharmonic in R^{n}\B_{R} and satisfy

∫_{R^{n}\B_{R}} \frac{v^{p}}{|x|^{β}} < c_{0}, for some p ≥ 1, β < n. (4.8)

Then the set of admissible directions for a given M = 2^{k} and σ = 2^{−εk} = M^{−ε} has measure |Σ_{1}| −_{c} M^{−δ} for ε = \frac{n−β}{2} , δ = \frac{n−β}{4} and c depending only on c_{0} and n.

Note that because of Lemma 3.2, ∫_{B_{2R}(x_{0})} u ≤ C and since v = Ku is given by (3.4), v satisfies (4.8) with p = 1 and β = n − 2 for suitable c_{0}.

The final step asserts that when the Reflection Theorem holds for a sufficiently large set of directions then asymptotic symmetry (4.6) holds.

Step 4. Let v ≥ 0 be superharmonic on R^{n} −B_{R} with the property that for some M > 0 and τ ∈ A (the admissible set) ⊂ Σ_{1},

v(y) ≤ v(y_{λ}) if y · τ ≥ λ > M

Then

v(y) = inf _{|z|=|y|}v(z) (1 + 0(\frac{1}{|y|)}) for |y| → ∞ (4.9)

where all error estimates are uniformly controlled.

The detailed proofs of Steps 1-4 are given in [?]. From Step 4, Theorem 4.1 follows easily. For from (4.9) it follows that

(4.10) u(x) = inf _{|x|=r} u(1 + 0(r)) for |x| = r as r → 0.

Formula (4.10) implies, by a simple chaining argument that u is bounded. For we can connect any x, d(x) ≥ \frac{d_{0}}{2} to a point x̄, d(x̄) < d_{0} in a finite number of steps k (with k under control) so that x_{1} = x̄, x_{k} = x and

u(x_{i+1}) ≤ c u(x_{i}), i = 1, …, k

Thus, u(x) = u(x_{k}) ≤ c^{k−1} u(x̄) ≤ C by Theorem 3.1.

Now that u is bounded, choosing the center point x_{0} and radius r to be x_{0} = \frac{x+z}{2} and r = |x − z|/2, (4.10) gives

|u(x) − u(z)| ≤ C |x − z|

for |x − z| sufficiently small. This completes the proof of Theorem 4.1.

5 The Existence of Smooth Minimizers

In this section we will use the apriori estimates of Section 4 to pass from approximating minimizers to the limit case.

We proceed directly to the Proof of Theorem 1.1. Let M = inf ∫_{Ω} |∇u|^{2} for u in the admissible class

A = \{u − h ∈ H_{0}^{1}(Ω): ∫_{Ω} |u|\frac{2n}{n − 2} = γ > ∫_{Ω} h\frac{2n}{n − 2}\}

Given any ε_{k} > 0 there exists η ∈ A such that

∫_{Ω} |∇η|^{2} ≤ M + ε_{k}

By Lebesgues’ theorem,

∫_{Ω} |η|^{α_{j}+1} → ∫_{Ω} |η|\frac{2n}{n − 2} = γ > ∫_{Ω} h\frac{2n}{n − 2}

for any sequence α_{j} ↗ \frac{n+2}{n−2} . Let us fix such a sequence and assume that for j large

γ_{j} =: ∫_{Ω} |η|^{α_{j}+1} > ∫_{Ω} h^{α_{j}+1}

Using the results of Sections 2–4, there exists a positive minimizer u_{jk} ∈ C^{2}(Ω) ∩ C^{1+β}(\overline{Ω}) in the admissible class A_{γj} to the subcritical variational problem (2.1) with α = α_{j}, j large. Moreover (from Corollary 4.2),

∥u_{jk}∥_{C^{1+α}(Ω)} ≤ C (5.1)

where C is a uniform constant. Since u_{jk} is a minimizer, ∫_{Ω} |∇u_{jk}|^{2} ≤ ∫_{Ω} |∇η|^{2} ≤ M + ε_{k}.

Choosing a diagonal subsequence, u_{j_{i}j_{i}} converges to a limit minimizer u ∈ C^{2}(Ω) ∩ C^{1+β}(\overline{Ω}) satisfying

−Δu = λu\frac{n+2}{n−2} in Ω
u = ϕ  on ∂Ω
u > 0  in Ω

where λ > 0 satisfies the estimates of theorem 2.1. □

We now consider a more general variational problem

∫ |∇u|^{2} → min (5.2)

u ∈ A_{γ} = {u ∈ H^{1}(Ω): u − h ∈ H_{0}^{1}(Ω), ∫_{Ω} G(u) = γ > ∫_{Ω} G(h)}

Here G(t) is the primitive of g(t), with g(t) satisfying the hypotheses of Theorem 1.2 and extended to g ≡ 0 for t ≤ 0. As before, ϕ ∈ C^{1+β}(∂Ω) ≥ 0 is positive somewhere and ∂Ω ∈ C^{2}.

We will assume, for the moment, that g(t) satisfies the subcritical growth condition

g(t) ≤ At^{α} α ∈ [1, \frac{n + 2}{n − 2}) , t > T (5.3)

for some A, T, α.

We note that given an arbitrary g(t) satisfying the conditions (1.5), we can construct subcritical approximations g_{j}(t) satisfying the same conditions by choosing t_{j} > t_{0} and setting

g_{j}(t) = \{ \begin{matrix}g(t) & t ≤ t_{j} \\ \frac{g(t_{j})}{t_{j}}t & t > t_{j} \end{matrix} (5.4)

Preliminary to proving Theorem 1.2, we have the following

Proposition 5.1 Let g(t) satisfy the hypotheses of Theorem 1.2 and the subcriticality condition (5.4). Then there exists a positive minimizer u ∈ C^{2}(Ω) ∩ C^{1+β}(\overline{Ω}) of Problem (5.2) satisfying

\begin{aligned}−Δu & = & λg(u) & in & Ω \\ u & = & ϕ & on & ∂Ω \end{aligned}

where 0 < λ < C(γ − ∫_{Ω} G(h)). If \frac{g(t)}{t} → ∞ as t → ∞, λ → 0 as t → ∞.

Proof. The existence of a classical solution to Problem (5.2) is well-known. That is, u satisfies

\begin{aligned}−Δu & = & λg(u) & in & Ω \\ u & = & ϕ & on & ∂Ω \end{aligned} (5.5)

We claim that λ > 0 and so u > h in Ω by the maximum principle. For using u − h as a test function in (5.5).

∫_{Ω} |∇u|^{2} − ∫_{Ω} ∇u · ∇h = λ ∫_{Ω}(u − h)g(u) (5.6)

Since G is convex, we have

0 < γ − ∫_{Ω} G(h) = ∫_{Ω}(G(u) − G(h)) ≤ ∫_{Ω}(u − h)g(u) (5.7)

Observing that ∫_{Ω} ∇u · ∇h = ∫_{Ω} |∇h|^{2}, we conclude from (5.6) (5.7) that λ > 0, as claimed. We also obtain the estimate

0 < λ ≤ (∫_{Ω} |∇u|^{2} − ∫_{Ω} |∇h|^{2}) / (γ − ∫_{Ω} G(h)) (5.8)

Using (5.8) we can estimate λ from above. For example, let η ≥ 0 be a fixed nontrivial C_{0}^{1}(\overline{Ω}) function and set v = h + εη. There is a unique ε > 0 so that v ∈ A_{γ}. Then

∫_{Ω} |∇u|^{2} ≤ ∫_{Ω} |∇v|^{2} = ∫_{Ω} |∇h|^{2} + ε^{2} ∫_{Ω} |∇η|^{2} (5.9)

Again by the convexity of G,

γ = ∫_{Ω} G(h + εη) ≥ ∫_{Ω} G(h) + ε ∫_{Ω} ηg(h)

so that ε ≤ C(γ − ∫_{Ω} G(h)) (5.10)

The combination (5.8)–(5.10) gives

0 < λ ≤ C(γ − ∫_{Ω} G(h)) (5.11)

To get a better upper bound as γ → ∞ let v = h+tη for a fixed nontrivial η ≥ 0 η ∈ C_{0}^{1}(Ω). Choosing t so that v ∈ A_{γ}, we have

∫_{Ω} |∇u|^{2} ≤ ∫_{Ω} |∇v|^{2} = ∫_{Ω} |∇h|^{2} + t^{2} ∫_{Ω} |∇η|^{2}

Therefore, from (5.8)

λ ≤ C \frac{t^{2}}{γ − ∫ G(h)} (5.12)

By convexity, we can estimate

γ − ∫_{Ω} G(h) ≥ t ∫_{Ω} ηg(tη) (5.13)

If we assume that \frac{g(t)}{t} → ∞ as t → ∞, we can conclude from (5.12), (5.13) that λ → 0 as γ → ∞ (5.14)

□

We are now in a position to give the Proof of Theorem 1.2:. Choose a sequence t_{j} ↗ +∞ with t_{j} > sup_{∂Ω} h. As in the proof of Theorem 1.1, given any ε_{k} > 0, there exists

η ∈ A = {u − h ∈ H_{0}^{1}(Ω): ∫_{Ω} G(u) = γ > ∫_{Ω} G(h)}

such that

∫_{Ω} |∇η|^{2} ≤ M + ε_{k}

Here, M = inf_{u∈A} ∫ |∇u|^{2}. By Lebesgues’ theorem,

∫_{Ω} G_{j}(η) → ∫_{Ω} G(η) = γ > ∫_{Ω} G(h)

where Gj^{′}|t| = g_{j}(t) and g_{j}(t) is given by (5.4). Using Proposition 5.1 and Corollary 4.2, there exists a positive minimizer u_{jk} ∈ C^{2}(Ω)∩C^{1+β}(\overline{Ω}) in the admissible class A_{γ_{j}}, where

γ_{j} =: ∫_{Ω} G_{j}(η) > ∫_{Ω} G_{j}(h) = ∫_{Ω} G(h).

Since u_{jk} is a minimizer,

∫_{Ω} |∇u_{jk}|^{2} ≤ ∫_{Ω} |∇η|^{2} ≤ M + ε_{k}

Choosing a diagonal subsequence, u_{j_{i}j_{i}} converges to a limit minimizer u ∈ C^{2}(Ω) ∩ C^{1+β}(\overline{Ω}) satisfying

−Δu = λg(u) in Ω
u = ϕ  on ∂Ω
u > 0

where λ ≥ 0. Since ∫_{Ω} G(u) = γ > ∫_{Ω} G(h), 0 < λ < C(γ − ∫_{Ω} G(h)) by the arguments of Proposition 5.1. If in addition \frac{g(t)}{t} → ∞ then λ → 0 as γ → ∞. The proof of Theorem 1.2 is complete. □