ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 3, December 1996 S 1079-6762(96)00014-5

ON THE EXISTENCE OF POSITIVE SOLUTIONS OF YAMABE-TYPE EQUATIONS ON THE HEISENBERG GROUP

L. BRANDOLINI, M. RIGOLI, AND A. G. SETTI

(Communicated by Richard Schoen)

Abstract. We study nonexistence, existence and uniqueness of positive solutions of the equation Δ_{H^{n}}u+a(x)u−b(x)u^{σ} = 0 with σ > 1 on the Heisenberg group H^{n}. Our results hold, with essentially no changes, also for the Euclidean version of the above equation. Even in this case they appear to be new.

Introduction

Let H^{n} be the Heisenberg group of real dimension 2n + 1, i.e. the nilpotent Lie group which as a manifold is the product

H^{n} = ℂ^{n} × ℝ

and whose group structure is given by

(z, t) ∘ (z′, t′) = (z + z′, t + t′ + 2 Im(z, z′)) ,
(z, t), (z′, t′) ∈ H^{n},

where ( , ) denotes the usual Hermitian product on ℂ^{n}.

A (real) basis for the Lie algebra of left-invariant vector fields on H^{n} is given by

X_{j} = 2 Re \frac{∂}{∂z_{j}} + 2 Im z_{j} \frac{∂}{∂t}, Y_{j} = 2 Im \frac{∂}{∂z_{j}} − 2 Re z_{j} \frac{∂}{∂t}, \frac{∂}{∂t},

for j = 1, 2, …, n. The above basis satisfies Heisenberg’s canonical commutation relations for position and momentum

[X_{j}, Y_{k}] = −4δ_{j k} \frac{∂}{∂t},

all other commutators being 0. It follows that the vector fields X_{j}, Y_{k} satisfy Hörmander’s condition, and the real part of the Kohn-Spencer Laplacian, defined by

Δ_{H^{n}} = ∑_{j=1}^{n} (X_{j}^{2} + Y_{j}^{2}) , (1)

is hypoelliptic by Hörmander’s theorem ([5]).

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors March 8, 1996.

1991 Mathematics Subject Classification. Primary 35H05; Secondary 35J70.

Key words and phrases. Heisenberg group, hypoelliptic equations, CR-Yamabe problem.

In H^{n} one has a natural origin 0 = (0, 0) and a distinguished distance function from 0 defined by

ρ(x) = ρ(z, t) = (|z|^{4} + t^{2})^{1/4} ,

which is homogeneous of degree one with respect to the Heisenberg dilations (z, t) → (δz, δ^{2}t). The distance between two points x, x′ ∈ H^{n} is then given by d(x, x′) = ρ(x^{−1}x′).

We also define the density function with respect to 0 by

ψ(x) = ψ(z, t) = \frac{|z|^{2}}{ρ(z, t)^{2}} , for x ≠ 0,

and note that 0 ≤ ψ(x) ≤ 1. If u is a “radial function”, that is, u(z, t) = f (ρ(z, t)) for f : [0, +∞) → ℝ of class C^{2}, then

Δ_{H^{n}}u = ψ \{f′′(ρ) + \frac{2n + 1}{ρ}f′(ρ)\} .

In this paper we consider the equation

Δ_{H^{n}}u + a(x)u − b(x)|u|^{σ−1}u = 0, (2)

with σ > 1 constant, and determine conditions on the coefficients a(x), b(x) in order to guarantee the existence (resp., nonexistence) of positive solutions on H^{n}.

Our problem is motivated by the following geometric fact. The vector fields Z_{j} = X_{j} + iY_{j} span a subbundle T_{1,0} of the complexified tangent bundle of H^{n}, and give rise to its canonical CR structure with contact form θ, which is determined modulo the transformation

θ̃ = u^{2/n}θ (3)

for 0 < u ∈ C^{∞}(H^{n}). The choice of θ specifies a pseudo-Hermitian structure on H^{n}, and

θ_{o} = dt + ∑_{j=1}^{n} (iz^{j}dz̄^{j} − iz̄^{j}dz^{j})

defines the canonical one.

A contact form on a CR manifold M induces a scalar curvature (the Webster scalar curvature) R_{θ}, which under the change (3) of contact forms transforms according to the equation

\frac{2n + 2}{n}Δ_{θ}u + R_{θ}u = Rθ̃u^{(n+2)/n},

where Δ_{θ} is the hypoelliptic Laplacian of the pseudo-Hermitian manifold (M, θ).

The Webster scalar curvature of the canonical pseudo-Hermitian structure θ_{o} on H^{n} is identically zero, and Δ_{θ_{o}} is the operator defined in (1). Therefore, when a(x) ≡ 0, equation (2) arises as the transformation law for the Webster scalar curvature of (H^{n}, θ_{o}) under the conformal change θ̃ = u^{2/n}θ_{o} of contact forms. In this sense, (2) is the CR analogue of the Yamabe equation on Euclidean space that has been extensively investigated over the past fifteen years (see, for instance, [3], [4], [9]).

The CR-Yamabe equation has been studied in a series of papers by D. Jerison and J. Lee ([6], [7] and [8]). In particular, they consider the problem of conformally changing the contact form to one having constant Webster curvature in the compact setting. As in the Riemannian case, their analysis is based on variational techniques, which, however, are not straightforwardly extendable to the noncompact case.

We stress that the presence of the linear term in (2) introduces additional difficulties and gives rise to new interesting phenomena. Most notably, our results suggest the existence of a critical behaviour for the coefficient a(x). On the one hand, the linear term essentially does not affect existence and nonexistence of positive solutions if a(x) decays faster than quadratically in the distance (see Theorem 5 and Theorem 2, respectively). On the other hand, if a(x) decays more slowly than quadratically, Theorem 3 guarantees the existence of positive solutions under minimal conditions on b(x).

The presence of the linear term is also reflected in the uniqueness result contained in Theorem 7. Without it the uniqueness would follow by a direct application of the maximum principle.

In our analysis of equation (2) we use a variety of different techniques. The nonexistence results are for the most part based on applications of maximum principle techniques and geometric ideas to ordinary differential equations, which yield results that might be of independent interest.

The key ingredient in the proof of our existence results is the method of superand subsolutions, which is the standard approach to existence problems when lack of compactness of the ambient space makes a variational treatment unnatural. Again, geometric ideas play a crucial role in the construction of super- and subsolutions.

Our results are based on the assumption that the coefficients a(x) and b(x) in (2) satisfy bounds of the form

ψ(x) a_{1} (ρ(x)) ≤ a(x), respectively a(x) ≤ ψ(x) a_{2} (ρ(x)) ,
ψ(x) b_{1} (ρ(x)) ≤ b(x), respectively b(x) ≤ ψ(x) b_{2} (ρ(x)) ,

(4)

for suitable, nonnegative, continuous functions a_{i} and b_{i} on [0, +∞). In some sense, the presence of the factor ψ(x) in (4) reflects the anisotropic nature of the Heisenberg group. It allows us to “radialise” the problem and apply ordinary differential equation methods.

Since 0 ≤ ψ(x) ≤ 1, its occurrence as a factor in the lower bounds in (4) is not restrictive. On the other hand, the presence of ψ in the upper bounds is a genuine restriction, and since ψ(x) vanishes for x = (z, 0), it forces the corresponding coefficient to vanish along the t-axis. However, an analysis of our results shows that the restriction so introduced is necessary in our nonexistence and uniqueness theorems (Theorems 1, 2, and 7, respectively) at least if one is willing to assume that the bounds on the coefficients are expressed in terms of the distance function.

Moreover, the presence of the factor ψ in the existence theorems allows us to obtain solutions whose behaviour at infinity can be controlled by radial functions (see Theorems 3 and 5).

We also remark that all our results hold, with essentially no changes, for the Euclidean version of equation (2). It suffices to set ψ ≡ 1 throughout, and replace Δ_{H^{n}} with the standard Laplace operator of ℝ^{m}. Even in this case our results appear to be new. (For related results in the setting of complete noncompact Riemannian manifolds, however, see [10] and [1].)

Next sections contain the statements of our main results. All the proofs will be included in the forthcoming paper [2].

1. Nonexistence results

Theorem 1. Let σ > 1, and let a, b ∈ C^{0}(H^{n}) satisfy

\{ \begin{matrix}a(x) ≤ ψ(x)a_{2} (ρ(x)) , \\ b(x) ≥ ψ(x)b_{1} (ρ(x)) \end{matrix} on H^{n}, (5)

with a_{2}, b_{1} ∈ C^{0}([0, +∞)). Assume that for some constant A ≤ n,

a_{2}(t) ≤ \frac{A^{2}}{t^{2}} ,

that b_{1}(t) ≥ 0 on [0, +∞), and that for some integer k,

lim inf_{t→+∞} b_{1}(t)\frac{(log t)^{σ+1} log(log t) ⋯ log^{(k)}(t)}{t^{n(σ−1)−2}} > 0 if A = n,
lim inf_{t→+∞} b_{1}(t)\frac{(log t) log(log t) ⋯ log^{(k)}(t)}{t^{(n−\sqrt{n^{2}−A^{2}})(σ−1)−2}} > 0  if A < n,

where log^{(k)} denotes the k th composition power of log. Then the equation Δ_{H^{n}}u + a(x)u − b(x)u^{σ} = 0 (6) has no positive solution on H^{n}. Theorem 2. Assume that a, b ∈ C^{0}(H^{n}) satisfy (5) in the statement of Theorem 1. Suppose that a_{2} satisfies

a_{2}(t) ≤ min \{\frac{n^{2}}{t^{2}} , \frac{A^{2}}{t^{2+ϵ}} \} ,

for some constants A and ϵ > 0, and that there exist an integer k and a constant C > 0 such that

b_{1}(t) ≥ 0 on [0, +∞),
lim inf_{t→+∞} t^{2} log t log(log t) ⋯ log^{(k)}(t) b_{1}(t) ≥ C > 0.

Then (6) has no positive solution on H^{n}.

2. Existence results

Theorem 3. Let a, b ∈ C^{∞}(H^{n}), μ ≤ 2, 1 < σ ≤ \frac{n+2}{n},

A_{μ} > \{ \begin{matrix}0 & if & μ < 2, \\ \frac{2n}{σ − 1} & if & μ = 2, \end{matrix} ,

and let γ ∈ ℝ. Assume that a and b satisfy

ψ(x)a_{1}(ρ(x)) ≤ a(x) ≤ ψ(x)a_{2}(ρ(x)) on H^{n}

and

ψ(x)b_{1}(ρ(x)) ≤ b(x) ≤ ψ(x)b_{2}(ρ(x)) on H^{n},

respectively, for suitable a_{1}, a_{2}, b_{1}, b_{2} ∈ C^{0}([0, +∞)) with a_{1}(t) = A_{μ}t^{−μ} for t ≫ 1,

i) b_{1}(t) ≥ 0 on [0, +∞) and b_{1}(t) > 0 in [t_{0}, +∞),
ii) b_{2}(t) ≤ c_{1}t^{− \frac{2}{n} γ−μ} for t ≫ 1,

with c_{1} > 0 and t_{0} such that a_{2}(t) ≤ \frac{n^{2}}{t^{2}} on (0, t_{0}]. Then there exists a positive solution u of (6) on H^{n} satisfying the further requirement

u(x) ≥ c_{2}ρ(x)^{\frac{2γ}{n(σ−1)}}

for some constant c_{2} > 0 and ρ(x) sufficiently large.

Definition 4. We say that a solution U of equation (6) on H^{n} is maximal if for any other solution u on H^{n} we have

u(x) ≤ U(x) for x ∈ H^{n}.

Theorem 5. Let a, b ∈ C^{0}(H^{n}). Suppose there exist a_{2}, b_{1}, b_{2} ∈ C^{0}([0, +∞)) satisfying

a_{2}(t) ≤ \frac{n^{2}}{t^{2}}

and

i) b_{1}(t) ≥ 0 on [0, +∞) and b_{1}(t) > 0 for t ≫ 1,
ii) tb_{2}(t) ∈ L^{1}(+∞),

such that

0 ≤ a(x) ≤ ψ(x)a_{2}(ρ(x)) on H^{n}

and

ψ(x)b_{1}(ρ(x)) ≤ b(x) ≤ ψ(x)b_{2}(ρ(x)) on H^{n}.

Then, there exists a unique, positive, maximal solution U of (6) on H^{n} satisfying

lim_{ρ(x)→+∞} U(x) = +∞.

Furthermore, if for some constant c > 0

b_{1}(t) ≥ ct^{−k}, for k > 2 and t ≫ 1,

then

U(x) ≤ Cρ(x)^{\frac{k−2}{σ−1}}

for some constant C > 0 and ρ(x) ≫ 1.

Similarly, if for some constant c > 0

b_{2}(t) ≤ ct^{−h}, for h > 2 and t ≫ 1,

then

U(x) ≥ Cρ(x)^{\frac{h−2}{σ−1}}

for some constant C > 0 and ρ(x) ≫ 1.

In particular, in the case where

C_{1}ψ(x) [1 + ρ(x)]^{−k} ≤ b(x) ≤ C_{2}ψ(x) [1 + ρ(x)]^{−k}

for some constants C_{1}, C_{2} > 0, we find that

U(x) ≍ ρ(x)^{\frac{k−2}{σ−1}} as ρ(x) → +∞.

Theorem 6. Let a, b ∈ C^{0}(H^{n}). Suppose there exist a_{2}, b_{1} ∈ C^{0}([0, +∞)) satisfying

a_{2}(t) ≤ \frac{n^{2}}{t^{2}} ,

b_{1}(t) ≥ 0 on [0, +∞) and b_{1}(t) > 0 for t ≫ 1,

such that

0 ≤ a(x) ≤ ψ(x)a_{2}(ρ(x)) on H^{n}

and

ψ(x)b_{1}(ρ(x)) ≤ b(x) on H^{n}.

If there exists a positive subsolution u of (6), then there exists a unique, positive, maximal solution U of (6) on H^{n} satisfying

lim_{ρ(x)→+∞} U(x) = +∞.

3. A uniqueness result

Theorem 7. Let a(x), b(x) ∈ C^{0}(H^{n}) satisfy b(x) ≥ 0 on H^{n}, and

a(x) ≤ ψ(x)a_{2} (ρ(x)) on H^{n},

where a_{2} ∈ C^{0}([0, +∞) satisfies

a_{2}(t) ≤ \frac{A^{2}}{t^{2}} on (0, +∞), (7)

with A ≤ n. Let u, v ∈ C^{2}(H^{n}) be positive solutions of equation (6). If

(8) (u − v)(x) = \{ \begin{matrix}o (ρ(x)^{−n} log(ρ(x))) & for A = n, \\ o (ρ(x)^{−n+\sqrt{n^{2}−A^{2}}}) & for A < n, \end{matrix} as ρ(x) → +∞,

then u ≡ v on H^{n}.

Moreover, if instead of (7) we assume that

a_{2}(t) ≤ min \{\frac{n^{2}}{t^{2}} , \frac{A^{2}}{t^{2+ϵ}} \}

for some positive constants A and ϵ, then the same conclusion holds with (8) replaced by

(u − v)(x) = o (1) as ρ(x) → +∞.

References

1. B. Bianchini and M. Rigoli, Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifold, preprint.

2. L. Brandolini, M. Rigoli, and A. G. Setti, Positive solutions of Yamabe-type equations on the Heisenberg group, preprint.

3. K. S. Cheng and J. T. Lin, On the elliptic equations Δu = K(x)u^{σ} and Δu = K(x)e^{2u}, Trans. Amer. Math. Soc. 304 (1987), 639–668. MR 88j:35054

4. K. S. Cheng and W. M. Ni, On the structure of the conformal scalar curvature equation on ℝ^{n}, Indiana Univ. Math. J. 41 (1992), 261–278. MR 93g:35040

5. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1968), 147– 171. MR 36:5526

6. D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Cont. Math. 27 (1984), 57–63. MR 85i:58122

7. , The Yamabe problem on CR manifolds, J. Diff. Geom. 25 (1987), 167–197. MR 88i:58162 8. , Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe

problem, J. Amer. Math. Soc. 1 (1988), 1–12. MR 89b:53063

9. W. M. Ni, On the elliptic equation Δu + K(x)u^{(n+2)/(n−2)} = 0, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493–529. MR 84e:35049

10. A. Ratto, M. Rigoli, and L. Véron, Scalar curvature and conformal deformation of noncompact Riemannian manifolds, Math. Z. (to appear).

Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy

E-mail address: brandolini@vmimat.mat.unimi.it

Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy

E-mail address: rigoli@vmimat.mat.unimi.it

Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy

E-mail address: setti@vmimat.mat.unimi.it