Asymptotic estimates and non-existence results for
critical problems with Hardy term involving
Grushin-type operators
Annunziata LoiudiceDipartimento di Matematica Universit`a degli Studi di Bari Via Orabona, 4 - 70125 Bari (Italy)
annunziata.loiudice@uniba.it
Abstract
We provide the asymptotic behavior of solutions, at the singularity and at infinity, for a class of subelliptic Dirichlet problems with Hardy perturbation and critical non-linearity of the type −Lαu − µψ
2
d2u = K(z)|u|2
∗−2
u in Ω, where Lα= ∆x+ |x|2α∆y,
α > 0 is the so-called Grushin operator, Ω is an open subset of RN, 0 ∈ Ω, d is
the gauge norm naturally associated with Lα, ψ := |∇αd|, where ∇α is the Grushin
gradient, K ∈ L∞ and 0 ≤ µ < µ, where µ is the best Hardy constant for L α.
Fur-thermore, we establish some Pohozaev-type non-existence results. 2010 Mathematics Subject Classification: 35J70, 35J75, 35B40.
Keywords: subelliptic critical problem; Grushin operator; Hardy potential; asymptotic behavior; Pohozaev-type identity; non-existence results.
1
Introduction
The purpose of this paper is to investigate regularity, asymptotic behavior and non-existence results for a class of semilinear equations with Hardy term involving the so-called Grushin operator, defined on RN = Rm
x × Rny by
Lα = ∆x+ |x|2α∆y, α > 0.
We recall that this operator is elliptic for x 6= 0 and degenerates on the manifold {0} × Rn. When α is a nonnegative integer, this operator falls into the class of
H¨ormander-type operators; in the general case, it belongs to the wide class of subelliptic operators studied by Franchi and Lanconelli in [11], [12], [13]. Moreover, it constitutes the symplest prototype of the so-called ∆λ-operators introduced by Kogoj and Lanconelli in [20] and it
Let Xi = ∂x∂ i, i = 1, . . . , m, Xm+j = |x| α ∂ ∂yj, j = 1, . . . , n (1.1) and ∇α = (X1, . . . , XN).
For a given open subset Ω ⊂ RN, we shall indicate by D1,2
α (Ω) the completion of C0∞(Ω)
with respect to the norm
kuk := ˆ Ω |∇αu|2dz. Let moreover Q = m + (α + 1)n
denote the homogeneous dimension naturally attached to the operator Lα (see Section 2).
Our aim is to study qualitative properties of solutions in D1,2α (Ω) to the following class
of semilinear subelliptic equations with Hardy term and critical nonlinearity − Lαu − µψ 2 d2u = K(z)|u|2 ∗−2 u in Ω, u = 0 on ∂Ω (1.2) where Ω ⊂ RN = Rm×Rnis an arbitrary open set, 0 ∈ Ω, d is the natural gauge associated to Lα, defined, for z = (x, y) ∈ Rm× Rn, by
d(z) = (|x|2(α+1)+ (α + 1)2|y|2)2(α+1)1 ,
ψ := |∇αd|, K is a bounded coefficient, 2∗ = Q−22Q denotes the critical Sobolev exponent
in this context and 0 ≤ µ < µ, where µ =
Q−2
2 2
is the best constant in the Hardy inequality for the Grushin gradient
ˆ Ω |∇αu|2dz ≥ µ ˆ Ω ψ2 |u|2 d(z)2 dz, ∀u ∈ C ∞ 0 (Ω). (1.3)
Precisely, we establish the exact asymptotic behavior at the singularity and at infinity, in the case of unbounded domains, of finite energy solutions to problem (1.2), extending the asymptotic results obtained for the case µ = 0 by the author in [25]. Moreover, by means of Pohozaev-type arguments, we provide non existence results for problem (1.2) on bounded starshaped domains with respect to the Grushin geometry and necessary conditions for the existence of entire solutions on the whole RN.
We recall that the Hardy-type inequality (1.3) involving the Grushin gradient, which is the main functional tool in the variational formulation of problem (1.2), was proved by Garofalo in [14]; we also quote [4], [21] for related results and generalizations. In particular, in [4], among other results, it is proved that, when 0 ∈ Ω, the constant µ = (Q − 2)2/4 is the best constant in (1.3) and it is never attained; in [21] the validity of the above inequality is extended to the general class of the so-called ∆λ-operators. Some Hardy inequalities with remainder terms can also be found in [41].
Note that the Dirichlet problem involving the operator Lα has been deeply studied by
many authors under different perspectives (see e.g. [14, 20, 23, 33, 39, 40, 30, 38, 5, 34, 2, 22] and the references therein). In particular, Monti and Morbidelli in [33] investigate uniqueness and symmetry of solutions for the unperturbed critical problem involving the Grushin operator and introduce a suitable Kelvin-type transform in this context, which will be a useful tool in our asymptotic analysis; in [20] Kogoj and Lanconelli provide regularity, existence and non-existence results of Pohozaev-type for the Dirichlet problem
−Lαu = f (u) for general nonlinearities f , in the larger framework of ∆λ-operators.
In this paper we focus on the critical problem (1.2). The first part of the paper is devoted to study the singularity of solutions at 0. The result is established for a general class of nonlinearities possessing at most critical growth. In what follows, Br will indicate
the ball with center at 0 and radius r with respect to the homogeneous norm d. Theorem 1.1. Let Ω ⊂ RN be an arbitrary open set, 0 ∈ Ω. If u ∈ D1,2
α (Ω) is a weak
solution to problem
− Lαu − µψ
2
d2u = f (z, u) in Ω, (1.4)
where f : Ω × R → R is a Carath´eodory function such that
|f (z, t)| ≤ C(|t| + |t|2∗−1), ∀t ∈ R, (1.5)
there exist positive constants C and r such that |u(z)| ≤ C
d(z)√µ−√µ−µ, ∀z ∈ Br\ {0}. (1.6)
If, moreover, u is positive and f (z, t) ≥ 0 for t > 0, there exist positive constants C0
and r0 such that
u(z) ≥ C
0
d(z)√µ−√µ−µ, ∀z ∈ Br0\ {0}. (1.7)
The above asymptotic estimates in the Euclidean canonical setting go back to [17], [8], [3], and they have been an interesting tool in the study of many related problems. Recently, this kind of estimates has been extended by the author in [28] to the subelliptic framework of sub-Laplacians on Carnot groups. Following the Euclidean scheme, the estimate from above is achieved by means of Moser-type estimates, the estimate from below relies on a suitable comparison argument. We observe that the above general theorem also provides an estimate of the singularity of the L2-normalized eigenfunctions of the operator
−Lα− µψ
2
d2 with Dirichlet boundary conditions, whose existence can be easily proved as in the Heisenberg case following [31].
Now, when f is a purely critical nonlinearity and Ω is a neighborhood of ∞, by exploiting the conformal invariance of the equation, from the behavior at 0 we can deduce the exact asymptotic decay of solutions at infinity by means of the Kelvin-type transform introduced in [33]. The complete description of the asymptotics in this case is contained in the following theorem.
Theorem 1.2. Let Ω ⊂ RN be a neighborhood of ∞, 0 ∈ Ω and let K ∈ L∞ be
nonnega-tive. If u ∈ Dα1,2(Ω) is a positive solution to
−Lαu − µψ 2 d2u = K(z)u2 ∗−1 in Ω, (1.8) then u satisfies u(z) ∼ 1 d(z)√µ−√µ−µ, as d(z) → 0, (1.9) u(z) ∼ 1 d(z)√µ+√µ−µ, as d(z) → ∞. (1.10)
We remark that in the ordinary Laplacian case, when K ≡ 1 and Ω = RN all positive
solutions to (1.8) are known to have the explicit form
U(x) = [4(µ − µ)N (N − 2)] N −2 4 |x|γ0/√µ + |x|γ/√µN −22 , > 0,
where γ0 = √µ −√µ − µ and γ = √µ +√µ − µ (see Terracini [37], Jannelli [18]). So,
our estimates in the Grushin case reflect the behavior of the functions U in the Euclidean
framework. We recall that the above result in the case µ = 0 was obtained by the author in [25], where it was proved that the finite energy solutions of (1.8) in the unperturbed case are bounded at 0 and decay at infinity like d2−Q, that is the fundamental solution with pole at 0 of the involved operator (see also [26], [28], [29] and the references therein for analogous decay results concerning sublaplacians on Carnot groups).
The second part of the paper is devoted to some non-existence results for positive solutions to problem (1.2), which are obtained by means of Pohozaev-type arguments. A first non-existence result concerns bounded starshaped domains with respect to the Grushin geometry and it extends to the present singular case analogous results proved by Kogoj and Lanconelli in [20] for the case µ = 0 and autonomous nonlinearities in the larger framework of ∆λ-operators. We first recall the definition of δλ-starshaped domains, where δλdenotes the family of anisotropic dilations naturally attached to the homogeneity
of the operator Lα, i.e.
δλ(x, y) = (λx, λα+1y), λ > 0. (1.11)
In what follows, Z will indicate the infinitesimal generator of the dilations δλ, whose expression is recalled in formula (2.4) below.
Definition 1.3. Let Ω ⊂ RN be a connected open set with C1-boundary, containing 0 at
its interior. We say that Ω is δλ-starshaped with respect to the origin if hZ, νi(z) ≥ 0 ∀z ∈ ∂Ω,
where ν denotes the outer unit normal to the boundary of Ω.
Our nonexistence result on bounded starshaped domains is the following (see Section 2 for the definition of the functional space Λ2 appearing in the statement of the theorem):
Theorem 1.4. Let Ω ⊂ RN be a connected bounded domain with C1-boundary, δ
λ
-starshaped about the origin. Assume that K ∈ C1(Ω) and ZK ≤ 0 in Ω. Then, the
problem −Lαu − µψ 2 d2u = K(z)u2 ∗−1 in Ω, u = 0 on ∂Ω (1.12)
has no nonnegative nontrivial solutions u ∈ Dα1,2(Ω) ∩ Λ2(Ω \ {0}).
The result applies, in particular, if K is constant or, more generally, homogeneous of degree zero with respect to the dilations δλ, in which case ZK ≡ 0 in Ω.
Finally, concerning entire solutions of the problem in the whole RN, we prove the
following result.
Theorem 1.5. Assume that K ∈ L∞(RN) ∩ C1(RN), ZK ∈ L∞(RN) and let u ∈
D1,2α (RN) be a positive solution to −Lαu − µψ2 d2u = K(z)u 2∗−1 in RN. (1.13) Then ˆ RN ZK · u2∗dz = 0. (1.14) The above condition implies that there are no positive solutions if the radial derivative
ZK of the coefficient K does not change sign in RN and K is not δλ-homogeneous of degree
0. We remark that the necessary condition (1.14) for the existence of positive solutions to the critical problem (1.13) is the Grushin counterpart of the Euclidean condition
ˆ RN
h∇K(x), xi u2∗dx = 0
which appears in Smets [36, Prop. 2.2] and Felli-Schneider [10, Coroll. 2.3].
The paper is organized as follows. In Section 2 we introduce the functional setting of Grushin-type operators and their main properties; in Section 3 we study the behavior at the singularity of solutions to our problem, as stated in Theorem 1.1; in Section 4, we focus on the critical equation and we derive the asymptotic behavior of solutions at infinity on exterior domains on RN from the behavior at 0 by means of the Kelvin transform for the Grushin gradient; finally, in Section 5, we provide some weighted Pohozaev-type identities related to the involved operator and, consequently, we derive the non-existence results stated in Theorem 1.4 and Theorem 1.5.
2
Preliminaries and notation
Let us begin by recalling that the operator Lα = ∆x+ |x|2α∆
y is a degenerate elliptic
operator, which is homogeneous of degree two with respect to the family of anisotropic dilations δλ defined in (1.11). The number
naturally attached to the dilations δλ, plays the rˆole of a global space dimension in the
analysis of the operator Lα and it is usually called the homogeneous dimension of the space RN = Rm× Rn with respect to δλ.
Denoted by ∇α = (X1, . . . , XN) the Grushin gradient, where the Xj’s are defined in
(1.1), for any h = (h1, . . . , hN) ∈ C1(RN, RN) we shall indicate by
divα(h) = N
X
i=1
Xihi
the divergence induced by the Grushin vector fields. With this notation, Lα = divα∇α.
In what follows, it will be convenient to consider Lαas a divergence form operator on RN.
To this aim, consider the N × N matrix
Aα= IRm 0 0 |x|2αI Rn (2.1) Then Lα = div(Aα∇)
where div and ∇ are the usual Euclidean operators taken with respect to the variable
z ∈ RN. Observe that, for a function u ∈ C1(RN), it holds
|∇αu|2 = hAα∇u, ∇ui = |∇xu|2+ |x|2α|∇yu|2.
For z = (x, y) ∈ Rm× Rn, let
d(z) = d(x, y) = (|x|2(α+1)+ (α + 1)2|y|2)2(α+1)1 .
The function d(z) is homogeneous of degree one with respect to the anisotropic dilations (1.11). Moreover, there exists a suitable constant C > 0 depending on α and Q, such that
Γ(z) = C
d(z)Q−2
is the fundamental solution of −Lα with pole at the origin (see [14, Prop. 2.1]).
For R > 0, we shall denote by Bd(0, R), or simply BR, the ball with center at 0 and
radius R with respect to the homogeneous norm d, i.e.
BR= {z ∈ RN| d(z) < R}.
We recall that, if we let ψ := |∇αd|, then ψ has the following explicit form
ψ(z) = |x|α d(z)α.
Such weight function naturally appears in the Hardy-type inequality for Lα recalled in
(1.3). Moreover, if f = f (d) ∈ C2(R+) is a radial function with respect to d, the following formula holds Lα(f (d)) = ψ2 f00(d) +Q − 1 d f 0(d) . (2.2)
(see e.g. [14], formula (2.19)).
We recall that, if u is a function on RN, we say that u is homogeneous of degree k ∈ R
with respect to the dilations (1.11) if for any λ > 0
u ◦ δλ = λku. (2.3)
In Section 5, we shall consider the following smooth vector field on RN
Z = m X i=1 xi∂x∂ i + (α + 1) n X j=1 yj∂y∂ j (2.4)
which is related to Grushin geometry since it is the infinitesimal generator of the dilations
δλ, so that u satisfies (2.3) if and only if Zu = ku. In particular, since d is homogeneous
of degree one with respect to δλ, then
Zd = d. (2.5)
Letting 2∗= Q−22Q , the following global Sobolev-type inequality holds ˆ RN |u|2∗dz ≤ C ˆ RN |∇αu|2dz, ∀ u ∈ C0∞(RN)
where C is a positive constant depending on Q. This inequality can be derived as a consequence of the general embedding results by Franchi and Lanconelli in [13], as explici-tly verified by the author in [24], where also Sobolev inequalities with remainder terms for the Grushin gradient have been obtained. We quote [20] for the general Sobolev embedding inequality for the ∆λ-operators; we also refer to [1], [32] for further related results.
In our treatment, we shall need some weighted versions of the above Sobolev inequality which we here recall. If 0 ≤ s ≤ 2 and a > 2−Q2 , there exists a positive constant C depending on s, a, Q such that
ˆ RNψ s |dau|2 ∗(s) ds dz ! 2 2∗(s) ≤ C ˆ RN|d a∇ αu|2 dz, ∀u ∈ C0∞(RN) (2.6)
where 2∗(s) = 2(Q−s)Q−2 . The above inequality in the case a = 0 is a Hardy-Sobolev type inequality and it can be easily derived by combining Hardy and Sobolev inequality for the Grushin gradient; a direct proof can be found in Dou and Niu [6]. The present weighted version can be easily obtained reasoning as in the Appendix of [28], where the analogous inequality for the horizontal gradient on Stratified Lie groups is obtained.
We conclude by introducing the functional spaces we shall use throughout the paper. If Ω is an open subset of RN, we shall indicate by Λ2(Ω \ {0}) the space of functions u ∈
C(Ω \ {0}) such that Xju, Xj2u ∈ C(Ω \ {0}), for j = 1, . . . , N . The space D1,2α (Ω, d−2βdz),
or simply Dα,β1,2(Ω), with β < Q−22 , will denote the completion of C∞
0 (Ω) with respect to the weighted norm
kukD1,2
α,β(Ω):= ˆ
Ω
d−2β|∇αu|2dz.
Moreover, we shall indicate by Sα,β1,2(Ω) the space of functions u ∈ L2∗
(Ω, d−2∗β
dz) such that ∇αu ∈ L2(Ω, d−2βdz) endowed with the norm
kukS1,2 α,β(Ω):= ˆ Ω d−2∗β|u|2∗dz + ˆ Ω d−2β|∇αu|2dz.
In view of (2.6) for s = 0 and a = −β, the two norms are equivalent on C∞
0 (Ω).
3
Asymptotic behavior at the origin
In order to state the behavior of solutions to problem (1.4) at the origin, we first need to study their Lp-regularity. The following Brezis-Kato type result holds.
Lemma 3.1. Let Ω ⊂ RN be a bounded neighborhood of 0 and 0 ≤ µ < µ. Assume that V ∈ LQ/2(Ω) and g ∈ Lq(Ω), q ≥ 2. If u ∈ D1,2
α (Ω) is a weak solution of
−Lαu − µψ
2
d2u − V u + νu = g in Ω, (3.1)
where ν is such that the linear operator on the left-hand side is positive, then u ∈ Lp(Ω), ∀p < plim, where plim = 2∗min
q 2, √ µ √ µ −√µ − µ .
Proof. The proof follows the Euclidean outline in Smets [36] and it is based on the validity
of Hardy and Sobolev inequality for the Grushin gradient (see also [28] for the analogous proof for sub-Laplacians on Stratified Lie groups). We omit the details.
By Lemma 3.1, the following regularity follows for the solutions to our problem. Proposition 3.2. Let Ω be an arbitrary open subset of RN, 0 ∈ Ω. Let u ∈ Dα1,2(Ω) be a
solution of problem (1.4) under the assumption (1.5). Then u ∈ Lploc(Ω), ∀ p < plim = 2∗
√ µ √
µ −√µ − µ. (3.2) Proof. By the regularity theory developed in [12], [13], we can infer that u ∈ Λ2(Ω \ {0}). Moreover, let R > 0 be such that BR ⊂⊂ Ω and let η ∈ C0∞(BR) be a cut-off function
such that η ≡ 1 on BR/2, 0 ≤ η ≤ 1. Then, the function w := ηu satisfies the equation
−Lαw − µψ2
d2w − V w + νw = g, w ∈ D 1,2
where V := f (ξ, u)/u ∈ LQ/2(B
R), since u ∈ L2
∗
, and g := −2∇αu · ∇αη − uLαη + νw.
From ∇αη ≡ 0 on BR/2(0) and u ∈ Λ2(B
R\ {0}), it follows that the Lp regularity of g is
given by that of w. Therefore, starting from g ∈ L2∗
and arguing recursively, we improve the regularity of w by means of Lemma 3.1, until we reach the limitation imposed by µ.
The proof is therefore complete.
Remark 3.3. Note that the limit exponent for the local Lp-regularity appearing in formula
(3.2) corresponds to the exponent plim = √µ−Q√µ−µ, which is exactly the sharp Lp-weak
summability exponent of the function d−(√µ−√µ−µ)appearing in the upper bound estimate
(1.6). So, by means of (1.6), we shall obtain that, in fact, u ∈ L Q √
µ−√µ−µ,∞
loc (Ω).
In order to study the behavior of solutions at the singularity, we shall use a convenient transformation, following the Euclidean scheme. If u ∈ D1,2α (Ω) is a solution of (1.4), we
define
v := dβu, where β =pµ −pµ − µ. (3.3) By Hardy inequality (1.3), it is easy to verify that v ∈ D1,2α (Ω, d−2βdz). Moreover, using
formula (2.2), we can see by direct inspection that v satisfies
−divα
d−2β∇αv
= d−βf (z, d−βv), z ∈ Ω \ {0}. (3.4) Indeed, by the equation (1.4) we get
−(Lαd−β)v − d−βLαv − 2h∇αd−β, ∇αvi − µψ
2
d2d−βv = f (z, d−βv) (3.5) that is, taking into account (2.2),
−d−βLαv − 2h∇αd−β, ∇αvi − ψ2 β2− β(Q − 2) + µd−β−2= f (z, d−βv)
which reduces to
−d−βLαv − 2h∇αd−β, ∇αvi = f (z, d−βv) (3.6) since β2− β(Q − 2) + µ = 0 iff β =√µ ±√µ − µ. Then, multiplying equation (3.6) by
d−β, we obtain the equation for v in (3.4).
We remark that the advantage of working with equation (3.4), instead of equation (1.4), is that, under the growth assumption (1.5) on f , the finite energy solutions of (3.4) turn out to be bounded, as we will prove in what follows.
Proof of Theorem 1.1 Let u ∈ Dα1,2(Ω) be a solution of (1.4) and define v as in (3.3).
Adapting the proof in [17], let ρ > 0 be such that Bρ⊂⊂ Ω. For s, t > 1, let φ := η2vv2(s−1)t ∈ Dα1,2
Ω, d−2βdz
where vt = min {|v|, t}, η ∈ C0∞(Bρ), 0 ≤ η ≤ 1, η ≡ 1 in Br, with 0 < r < ρ and
|∇αη| ≤ 4
ρ−r. By using φ as a test function in (3.4), we get
ˆ Ω d−2β 2ηvvt2(s−1)h∇αη, ∇αvi + η2vt2(s−1)|∇αv|2+ 2 (s − 1) η2v2(s−1)t |∇αvt|2 dz = ˆ Ω d−βf z, dβv η2vvt2(s−1)dz. (3.7)
For any ε > 0 small, it holds ˆ Ω 2d−2βηvv2(s−1)t h∇αη, ∇αvi dz ≤ ε ˆ Ω d−2βη2vt2(s−1)|∇αv|2 dz + C(ε) ˆ Ω d−2β|v|2vt2(s−1)|∇αη|2 dz, (3.8)
Then, by (3.7) and (3.8), by choosing ε sufficiently small and using the growth assumptions (1.5) on f , we get ˆ Ω d−2β η2vt2(s−1)|∇αv|2+ 2 (s − 1) η2v2(s−1)t |∇αvt|2 dz ≤ C ˆ Ω d−2β η2+ |∇αη|2 |v|2v2(s−1)t dz (3.9) + C ˆ Ω d−2∗βη2|v|2∗vt2(s−1)dz.
Now, consider the following weighted Sobolev inequality for the Grushin gradient ˆ Ω d−2∗β|w|2∗ dz 2 2∗ ≤ C ˆ Ω d−2β|∇αw|2 dz, ∀w ∈ D1,2α Ω, d−2βdz (3.10) which follows from the general weighted Hardy-Sobolev inequality recalled in Section 2, formula (2.6). Taking w = ηvvts−1 in (3.10) and using (3.9), we obtain
ˆ Ω d−2∗β|ηvvts−1|2∗dz 2 2∗ ≤ C ˆ Ω d−2β ∇α ηvvts−1 2 dz ≤ C ˆ Ω d−2β |∇αη|2|v|2vt2(s−1)+ η2vt2(s−1)|∇αv|2+ (s − 1)2η2vt2(s−1)|∇αvt|2 dz ≤ Cs ˆ Ω d−2β η2+ |∇αη|2 |v|2vt2(s−1)dz + Cs ˆ Ω d−2∗βη2|v|2∗vt2(s−1)dz. (3.11) Since u ∈ Lploc(Ω) for any p < 2Q
Q−2−2(√µ−µ) (see Proposition 3.2), we choose
Q
2 < q <
Q(Q − 2)
so that (2∗− 2)q < 2Q Q − 2 − 2 (√µ − µ) and 2 < 2q q − 1 < 2 ∗.
By this choice and reasoning as in [17], the last integral in (3.11) can be estimated as follows, for any ε > 0
ˆ Ω d−2∗βη2|v|2∗vt2(s−1)dz = ˆ Ω d−2β|u|2∗−2 ηvvts−1 2 dz (3.12) ≤ kuk2L∗(2∗−2)q−2 (supp η)kdβηvvs−1t k2 Lq−12q (Ω) ≤ Cε2 ˆ Ω d−2∗β ηvvs−1t 2∗ dz 2 2∗ + Cε−2q−Q2Q ˆ Ω d−2β|ηvvs−1t |2dz. So, by (3.11) and (3.12), we have
ˆ Ω d−2∗β ηvvs−1t 2∗ dz 2 2∗ ≤ Csε2 ˆ Ω d−2∗β ηvvts−1 2∗ dz 2 2∗ + Cs ˆ Ω d−2β η2+ |∇αη|2 |v|2vt2(s−1)dz + Csε−2q−Q2Q ˆ Ω d−2β ηvvts−1 2 dz. (3.13) Taking ε = √1 2Cs in (3.13), we obtain ˆ Ω d−2∗β ηvvts−1 2∗ dz 2 2∗ ≤ Csα ˆ Ω d−2β η2+ |∇αη|2 |v|2v2(s−1)t dz, (3.14) where α = 2q−Q2q > 0. Note that
ˆ Ω d−2∗βη2∗|v|2vt2∗s−2dz ≤ ˆ Ω d−2∗β ηvvts−1 2∗ dz. (3.15) Hence, from (3.15) and (3.14) we have
ˆ Ω d−2∗βη2∗|v|2vt2∗s−2dz 2 2∗ ≤ Csα ˆ Ω d−2β η2+ |∇αη|2 |v|2vt2(s−1)dz ≤ Csα ˆ Ω d−2∗β η2+ |∇αη|2 |v|2v2(s−1)t dz. By the definition of η, we then obtain
ˆ Br d−2∗β|v|2v2t∗s−2dz 2 2∗ ≤ Cs α (ρ − r)2 ˆ Bρ d−2∗β|v|2vt2s−2dz. (3.16)
Choosing s∗ such that
Q Q − 2 < s
∗< Q
Q − 2 − 2√µ − µ,
we consider the sequence sj = s∗ 2
∗ 2
j
, j = 0, 1, 2, . . .. Let ρ0 be sufficiently small that
B2ρ0 ⊂⊂ Ω and let rj = ρ0(1 + ρ
j
0), j = 0, 1, 2, . . .. Taking ρ = rj, r = rj+1 and s = sj in (3.16), by means of Moser’s iteration technique we finally obtain
ˆ Brj+1 d−2∗β|v|2v2sj+1t −2dz ! 1 2sj+1 ≤ C (1 − ρ0)ρ0 Pj k=02sk1 ρ− Pj k=02skk 0 j Y k=0 s α 2sk k ˆ Br0 d−2∗β|v|2vt2s∗−2dz ! 1 2s∗ , (3.17)
where the integral in the right-hand side of (3.17) if finite, since ˆ Br0 d−2∗β|v|2vt2s∗−2dz ≤ r0(2s∗−2∗)β ˆ Br0 |u|2s∗dz < ∞. So, letting j → ∞ in (3.17), we get that
kvtkL∞(B
ρ0) ≤ C
with C independent of t. Therefore, letting t → +∞, we obtain that v is bounded on Bρ0, that is equivalent to the upper bound (1.6).
To prove the lower bound (1.7), let u be a positive solution of problem (1.4) and suppose that f (z, t) ≥ 0 for t > 0. As before, let v be defined as in (3.3). We follow the proof in Cao-Han [3, Theorem 1.1]. Let 0 < ρ1 < ρ2 such that Bρ2 ⊂⊂ Ω and set
ϕ(ρ) := mind(z)=ρv(z), ρ1 < ρ < ρ2. Consider the comparison function
g(z) := Ad(z)−2√µ−µ+ B,
where the constants A and B are chosen so that g(z) = ϕ(ρi) for d(z) = ρi, for i = 1, 2.
The explicit expression of A and B turns out to be
A = ϕ(ρ2) − ϕ(ρ1) ρ−22 √µ−µ− ρ−21 √µ−µ, B = ϕ(ρ1)ρ−2 √ µ−µ 2 − ϕ(ρ2)ρ−2 √ µ−µ 1 ρ−22 √µ−µ− ρ−21 √µ−µ .
By direct calculation, it is easy to verify that divα
d−2β∇αg
= 0, ∀z 6= 0.
So, taking into account (3.4) and the sign hypothesis on f , it follows that
−divα
d−2β∇α(v − g)
On the other hand, by the definition of g, we know that v ≥ g on ∂(Bρ2\ Bρ1). Therefore, by the weak maximum principle, we obtain that
v ≥ g in Bρ2 \ Bρ1. (3.18) So, taking into account the explicit form of g, from (3.18) we get
v(z) ≥ ϕ(ρ2) − ϕ(ρ1) ρ−22 √µ−µ− ρ−21 √µ−µ d(z) −2√µ−µ+ϕ(ρ1)ρ−2 √ µ−µ 2 − ϕ(ρ2)ρ−2 √ µ−µ 1 ρ−22 √µ−µ− ρ−21 √µ−µ = ρ −2√µ−µ 2 − d(z)−2 √ µ−µ ρ−2 √ µ−µ 2 − ρ−2 √ µ−µ 1 ϕ(ρ1) +d(z)−2 √ µ−µ− ρ−2√µ−µ 1 ρ−2 √ µ−µ 2 − ρ−2 √ µ−µ 1 ϕ(ρ2) ≥ d(z)2 √ µ−µ− ρ2√µ−µ 1 d(z)2√µ−µ 1 − ρ2 √ µ−µ 1 ρ−2 √ µ−µ 2 ϕ(ρ2), (3.19) for every z ∈ Bρ2\ Bρ1.
Finally, letting ρ1 → 0 in (3.19), we obtain that v(z) ≥ ϕ(ρ2) = mind(η)=ρ2v(η) > 0
for all z ∈ Bρ2 \ {0}, that is (1.7).
4
Asymptotic behavior at infinity
This section is devoted to study the asymptotic behavior at infinity of solutions to problem (1.2) on exterior domains of RN.
Due to the conformal invariance of the critical equation (1.2), the rate of decay of solutions at infinity can be directly deduced from their asymptotic behavior at 0. In the Euclidean case this is achieved by means of the classical Kelvin transform on RN (see
e.g. [8]). In our framework, we shall use the Kelvin-type transform constructed on the geometry of Grushin vector fields, introduced by Monti and Morbidelli in [33].
In order to prove the asymptotic decay result, we briefly recall the definition and the main properties of such Kelvin transform. Using the dilations (1.11), a spherical inversion
σ constructed on the Grushin geometry is defined on RN by
σ(z) = δd(z)−2(z), z 6= 0.
A direct calculation (see [33, Lemma 2]) shows that the Jacobian of σ satisfies
|Jσ(z)| = d(z)−2Q = Γ(z)
2Q
Q−2, ∀z 6= 0. (4.1) The inversion σ is a conformal map for the Grushin gradient in the following sense
|∇α(u ◦ σ)(z)|2 = |Jσ(z)|2/Q|(∇αu)(σ(z))|2 ∀u ∈ C1(RN), z 6= 0. (4.2)
By means of σ, the following Kelvin-type transform can be defined. Given a function
u : RN → R, we define u∗ : RN \ {0} → R as the function
The function u∗ has, a priori, a singularity at z = 0. This singularity, however, turns out
to be removable, as proved in [33].
Using the conformal property (4.2), it can be proved that the Kelvin transform u∗ preserves the critical equation with Hardy perturbation (1.2), as we will verify below as a consequence of Theorem 4.1.
In what follows, we shall denote by Ω∗ the image of a generic domain Ω ⊂ RN under the inversion σ. Note that, if Ω is a neighborhood of ∞, i.e. there exists a ball BR such
that BRC ⊂ Ω, then Ω∗ is a punctured neighborhood of 0, i.e. Ω∗= D \ {0}, where D is
an open set, 0 ∈ D.
For µ ∈ [0, µ[, consider the following scalar product on Dα1,2(Ω):
hu, viµ:= ˆ Ω h∇αu, ∇αvi dz − µ ˆ Ω ψ2uv d2 dz 1/2 . (4.4)
Observe that the norm induced by h·, ·i´ µ is equivalent to the Dirichlet norm kuk =
Ω|∇αu|2dz 1/2
, due to Hardy inequality (1.3). The following property holds.
Theorem 4.1. The Kelvin transform defined in (4.3) is an isometry between D1,2α (Ω) and
D1,2α (Ω∗) with respect to the scalar product (4.4).
Proof. Let u, v ∈ Dα1,2(Ω) and let u∗, v∗ their Kelvin transform. As proved in [33, Theorem
2.5], it holds that ˆ Ω h∇αu, ∇αvi dz = ˆ Ω∗ h∇αu∗, ∇αv∗i dz.
Concerning the Hardy term, we have that ˆ Ω∗ ψ2u∗v∗ d2 dz = ˆ Ω∗ ψ2d2(2−Q)u(σ(z))v(σ(z)) d2 dz = ˆ Ω ψ2uv d2 dz,
where we have used that d(σ(z)) = d(z)−1 for any z 6= 0 and that, since ψ is homogeneous
of degree 0 with respect to the dilations δλ, then ψ(σ(z)) = ψ(z), for all z ∈ RN \ {0}.
The proof is therefore complete.
Lemma 4.2. If u ∈ D1,2α (Ω) is a solution to −Lαu − µ ψ2 d2u = f (z, u) in Ω, (4.5) then u∗∈Do1,2 α (Ω∗) satisfies −Lαu∗− µψ 2 d2u∗ = 1 dQ+2f (σ(z), dQ−2u∗) in Ω∗. (4.6)
Proof. Let u ∈ D1,2α (Ω) be a solution to (4.5). By Theorem 4.1 we know that u∗ ∈ Dα1,2(Ω∗).
Let ϕ ∈ C∞
0 (Ω∗); then, we can write ϕ = φ∗, for some φ ∈ C0∞(Ω). Applying Theorem 4.1 and property (4.1) we have
ˆ Ω∗ h∇αu∗, ∇αϕi dz − µ ˆ Ω∗ ψ2u∗ϕ d2 dz = ˆ Ω∗h∇αu ∗, ∇ αφ∗i dz − µ ˆ Ω∗ψ 2u∗φ∗ d2 dz = ˆ Ω h∇αu, ∇αφi dz − µ ˆ Ω ψ2uφ d2 dz = ˆ Ω f (z, u)φ dz = ˆ Ω∗ f (σ(z), u(σ(z))) φ(σ(z)) |Jσ(z)|dz = ˆ Ω∗ d−2−Qf (σ(z), dQ−2u∗) φ∗dz = ˆ Ω∗ d−2−Qf (σ(z), dQ−2u∗) ϕ dz.
By the arbitrariness of ϕ ∈ C0∞(Ω∗), the thesis follows.
Proof of Theorem 1.2 Let u be a positive solution of equation (1.8). The behavior at the origin (1.9) directly follows from Theorem 1.1. To prove the decay estimate at infinity (1.10), observe that, by Lemma 4.2, if u satisfies the critical equation (1.8) in a neighborhood of infinity, then u∗ satisfies the same critical equation in a neighborhood of
the origin, where K(z) is substituted by K(σ(z)), i.e.
−Lαu∗− µψ2 d2u ∗ = K(σ(z))(u∗)2∗−1 in Ω∗. So, by Theorem 1.1, u∗(z) ∼ d(z)−√µ+√µ−µ for d(z) small.
Hence, taking into account that u(z) = d(z)2−Qu∗(σ(z)), the decay estimate at infinity
(1.10) follows.
5
Some integral identities and non-existence results
In this section, by means of Pohozaev-type arguments [35], we prove some non-existence results for positive solutions to problem (1.2). We recall that Pohozaev-type identities in the subelliptic setting have been extensively studied, starting from the seminal paper by Garofalo and Lanconelli [15] and they present several additional difficulties with respect to the Euclidean framework. We shall refer, in particular, to [20], [27] for their relation to the present results.
In order to treat our critical problem with Hardy perturbation (1.2), as in the previ-ous steps, it will be convenient to use the transformation v = dβu, β = √µ −√µ − µ,
introduced in (3.3), being u a weak solution of (1.2). So, taking into account (3.4), the original problem (1.2) turns into the equivalent one
−divα d−2β∇αv = K(z)|v|2 ∗−2 v d2∗β in Ω. (5.1)
As previously observed, the advantage of working with equation (5.1) is that its solutions are bounded at 0, as proved in Section 3. For a refined analysis of the regularity at 0 of solutions to equations like (5.1) in the non-degenerate case α = 0 we quote [9], where it is proved that weak solutions of such problems are, in fact, H¨older continuous at 0.
The following Pohozaev-type identity for solutions to equation (5.1) holds. The Euclid-ean counterpart is given by Theorem 2.1 in Felli-Schneider [10]; the identity in the Grushin non-singular case β = 0 with autonomous nonlinearities is contained in [20].
Theorem 5.1. Let Ω ⊂ RN be a bounded domain with C1-boundary, 0 ∈ Ω, K ∈ C1(Ω)
and v ∈ Sα,β1,2(Ω) ∩ Λ2(Ω \ {0}) be a solution of equation (5.1). Then, the following identity
holds 1 2∗ ˆ Ω ZK |v|2 ∗ d2∗β dz − 1 2∗ ˆ ∂Ω K(z)|v|2 ∗ d2∗βhZ, νi dσ =Q − 2 − 2β 2 ˆ ∂Ω d−2βvh∇αv, ναi dσ − 12 ˆ ∂Ω d−2β|∇αv|2hZ, νi dσ + ˆ ∂Ω d−2βh∇αv, ναiZv dσ (5.2)
where Z is the infinitesimal generator of the dilations δλ defined in (2.4), ν = (ν1, . . . , νN)
is the outward unit normal to ∂Ω and να = (ν1, . . . , νm, |x|ανm+1, . . . , |x|ανN).
Proof. Let us first prove (5.2) under the assumption v ∈ C2(Ω \ {0}). Due to the lack of regularity at the origin, in order to prove identity (5.2) we shall consider approximating domains Ω \ Brn, for an appropriate sequence of radii rn→ 0, as in [10], [27]. To this aim,
observe that, from Federer’s coarea formula (see [7]), if BR= Bd(0, R) is a d-ball centered
at 0 contained in Ω, then ˆ R 0 ds ˆ ∂Bs K(z)|v|2 ∗ d2∗β + |∇αv|2 d2β 1 |∇d|dσ = ˆ BR K(z)|v|2 ∗ d2∗β + |∇αv|2 d2β dz. (5.3) Now, since v ∈ Sα,β1,2(Ω) (see the definition in Section 2), the integral in the right-hand side of (5.3) is finite. This implies that there exists a sequence rn→ 0 such that
rn ˆ ∂Brn K(z)|v|2 ∗ d2∗β + |∇αv|2 d2β 1 |∇d|dσ −→ 0, as n → ∞. (5.4)
Let Ωrn := Ω \ Brn. Multiplying equation (5.1) by Zv and integrating over Ωrn we get ˆ Ωrn −divα d−2β∇αv Zv dz = ˆ Ωrn K(z)|v|2 ∗−2 v d2∗β Zv dz. (5.5)
Concerning the left-hand side of (5.5), we claim that the following Rellich-type identity holds, which represents a weighted version of the identity in [14, Theorem 2.2] or its generalization to ∆λ-operators in [20, Theorem 2.1]:
ˆ Ωrn divα d−2β∇αv Zv dz = Q − 2 − 2β 2 ˆ Ωrn d−2β|∇αv|2dz −1 2 ˆ ∂Ωrn d−2β|∇αv|2hZ, νi dσ + ˆ ∂Ωrn d−2βh∇αv, ναiZv dσ. (5.6)
To prove it, we use the general integral identity recalled in [14, formula (2.40)], whose proof can be found in Garofalo-Lanconelli [15]. Let D ⊂ RN be a bounded piecewise
C1-domain, let C = (c
ij) be a N × N symmetric matrix with C1-entries and consider
the operator div(C∇) in RN; let X be a smooth vector field in RN with components
X1, . . . , XN. Moreover, denote by v,i the partial derivative ∂v/∂zi and let cij,k denote
∂cij/∂zk. The following identity holds for v ∈ C2(D)
ˆ D div(C∇v)Xv dz = 1 2 ˆ D divX hC∇v, ∇vi dz − ˆ D ∂Xi/∂zj(C∇v)jv,idz +1 2 ˆ D Xicjk,iv,kv,jdz −12 ˆ ∂D hC∇v, ∇vihX, νi dσ + ˆ ∂D hC∇v, νiXv dσ. (5.7)
Note that the operator Pα,β:= divα d−2β∇ α
which appears in the left-hand side of (5.6) can be expressed in divergence form as
Pα,β= div(d−2βAα∇)
where “div” is the Euclidean divergence on RN and A
α is the matrix defined in (2.1).
Now, following [14, proof of Theorem 2.2], we define the approximating matrices
Aεα = IRm 0 0 (|x|2+ ε2)αI Rn
Observe that the matrix d−2βAε
α has smooth entries in domains not containing the origin,
so we can apply (5.7) on D = Ωrn with C = d−2βAεα and X = Z. By the definition of Z
(see (2.4)), we get that
(∂Zi/∂zj) = diag[1, . . . , 1 | {z } m , α + 1, . . . , α + 1 | {z } n ], so that ∂Zi/∂zj(d−2βAεα∇v)jv,i= d−2β |∇xv|2+ (α + 1)(|x|2+ ε2)α|∇yv|2 = d−2β hAεα∇v, ∇vi + α(|x|2+ ε2)α|∇yv|2 . (5.8)
Moreover,
Zi(d−2βAεα)jk,iv,kv,j= −2βd−2β−1hZ, ∇di(Aεα)jkv,kv,j+ 2d−2βα|x|2(|x|2+ ε2)α−1|∇yv|2
= −2βd−2βhAεα∇v, ∇vi + 2d−2βα|x|2(|x|2+ ε2)α−1|∇yv|2
(5.9) where we have used (2.5). Now, inserting (5.8) and (5.9) into (5.7) and taking into account that divZ = Q, we get
ˆ Ωrn div d−2βAεα∇v Zv dz = Q − 2 − 2β 2 ˆ Ωrn d−2βhAεα∇v, ∇vi dz − α ˆ Ωrn d−2β ε2 |x|2+ ε2 |x|2+ ε2 α |∇yv|2dz −1 2 ˆ ∂Ωrn d−2βhAεα∇v, ∇vihZ, νi dσ + ˆ ∂Ωrn d−2βhAεα∇v, νiZv dσ. (5.10)
Hence, letting ε → 0 in (5.10) and taking into account that hAα∇v, ∇vi = h∇αv, ∇αvi
and hAα∇v, νi = h∇αv, ναi, the claim (5.6) follows.
Concerning the right-hand side of (5.5), if we denote for simplicity f (z, v) := K(z)|v|d2∗−22∗β v and F (z, v) :=´0vf (z, t) dt, it holds ˆ Ωrn f (z, v)Zv dz = ˆ Ωrn Z(F (z, v)) dz − ˆ Ωrn hZ, ∇zF (z, v)i dz = − ˆ Ωrn divZ F (z, v) dz + ˆ ∂Ωrn F (z, v)hZ, νi dσ − ˆ Ωrn hZ, ∇zF (z, v)i dz. (5.11)
Now, taking into account that F (z, v) = 1
2∗K(z)|v|
2∗
d2∗β and that divZ = Q, from (5.11) we get ˆ Ωrn K(z)|v|2 ∗−2 v d2∗β Zv dz = β − Q 2∗ ˆ Ωrn K(z)|v|2 ∗ d2∗β dz + 1 2∗ ˆ ∂Ωrn K(z)|v|2 ∗ d2∗β hZ, νi dσ − 1 2∗ ˆ Ωrn ZK|v|2 ∗ d2∗β dz, (5.12) where we have used that hZ, ∇di = d, as observed in (2.5). So, inserting (5.6) and (5.12)
into (5.5), we obtain Q 2∗ − β ˆ Ωrn K(z)|v|2 ∗ d2∗β dz + 1 2∗ ˆ Ωrn ZK |v|2 ∗ d2∗β dz − 1 2∗ ˆ ∂Ωrn K(z)|v|2 ∗ d2∗β hZ, νi dσ =Q − 2 − 2β 2 ˆ Ωrn d−2β|∇αv|2dz − 1 2 ˆ ∂Ωrn d−2β|∇αv|2hZ, νi dσ + ˆ ∂Ωrn d−2βh∇αv, ναi Zv dσ. (5.13) On the other hand, multiplying equation (5.1) by v and integrating by parts on Ωrn, we have ˆ Ωrn K(z)|v|2 ∗ d2∗β dz = ˆ Ωrn d−2β|∇αv|2dz − ˆ ∂Ωrn d−2βvh∇αv, ναi dσ. (5.14)
So, by (5.13) and (5.14), and since 2Q∗ − β −Q−2−2β2 = 0, we get 1 2∗ ˆ Ωrn ZK |v|2 ∗ d2∗β dz − 1 2∗ ˆ ∂Ωrn K(z)|v|2 ∗ d2∗βhZ, νi dz =Q − 2 − 2β 2 ˆ ∂Ωrn d−2βvh∇αv, ναi dσ − 1 2 ˆ ∂Ωrn d−2β|∇αv|2hZ, νi dσ + ˆ ∂Ωrn d−2βh∇αv, ναiZv dσ. (5.15)
Now, let us pass to the limit as rn → 0 in (5.15). From the integrability of ZK|v|
2∗ d2∗β, it follows that 1 2∗ ˆ Ωrn ZK |v|2 ∗ d2∗β dz −→ 1 2∗ ˆ Ω ZK |v|2 ∗ d2∗β dz, as rn→ 0. (5.16)
Moreover, we verify that the boundary integrals on ∂Brn in (5.15) vanish as rn → 0. Indeed, observe that, since ν = − ∇d
|∇d| on ∂Brn, then
hZ, νi = − Zd |∇d| = −
d
|∇d| on ∂Brn. From this, and using (5.4), we have
ˆ ∂Brn 1 2∗K(z) |v|2∗ d2∗β − 1 2d −2β|∇ αu|2 |hZ, νi| dσ = rn ˆ ∂Brn 1 2∗K(z) |v|2∗ d2∗β − 1 2d −2β|∇ αu|2 1 |∇d|dσ −→ 0, as rn→ 0. (5.17)
For the first integral in the right-hand side of (5.15), observe that
|h∇αv, ναi| = |h∇αv,|∇d|∇αdi| ≤ ψ|∇|∇d|αv| on ∂Brn. (5.18) Then, by (5.18), taking into account that ψ ∈ L∞ and using H¨older’s inequality, we get
ˆ ∂Brn d−2βvh∇αv, ναi dσ ≤ C ˆ ∂Brn d−2βv|∇αv|2 |∇d| dσ ≤ C ˆ ∂Brn 1 |∇d|dσ !2∗−2 2·2∗ ˆ ∂Brn |v|2∗ d2∗β 1 |∇d|dσ !1 2∗ ˆ ∂Brn |∇αv|2 d2β 1 |∇d|dσ !1 2 = C rn ˆ ∂Brn |v|2∗ d2∗β 1 |∇d|dσ !1 2∗ rn ˆ ∂Brn |∇αv|2 d2β 1 |∇d|dσ !1 2 = o(1), as rn→ 0, (5.19)
where we have used the fact that, by Federer’s co-area formula [7], for r > 0, ˆ
∂Br 1
|∇d|dσ = cQr
Q−1 (5.20)
and we have concluded by means of (5.4). Finally, concerning the integral ´∂B
rnd
−2βh∇
αv, ναiZv dσ, note that the vector field
Z can be expressed as
Z = d
ψ2Aα∇d (5.21)
as pointed out in [14, formula (2.13)] (see also [16, Proposition 3.1]). So, by (5.21) we get
|Zv| = |hZ, ∇vi| = |h d
ψ2∇αd, ∇αvi| ≤
d
ψ|∇αv| in Ω \ {0}. (5.22)
Hence, by (5.18) and (5.22), and using again (5.4) we obtain that ˆ ∂Brn d−2βh∇αv, ναiZv dσ ≤ rn ˆ ∂Brn d−2β|∇αv| 2 |∇d| dσ −→ 0, as rn→ 0. (5.23)
So, letting rn → 0 in (5.15) and using (5.16), (5.17), (5.19) and (5.23), the conclusion
follows under the assumption v ∈ C2(Ω \ {0}). To complete the proof for u ∈ Γ2(Ω \ {0}), we can argue by an approximation argument as in [15, page 77]. We omit the details. As a consequence of the above Pohozaev-type identity, we prove the following non-existence result on bounded starshaped domains of RN.
Theorem 5.2. Let Ω ⊂ RN be a connected bounded domain with C1-boundary, δ
λ
-starshaped about the origin. Assume that K ∈ C1(Ω) and that ZK ≤ 0 in Ω. Then,
the problem −divα d−2β∇αv = K(z)v2 ∗−1 d2∗β in Ω, v = 0 on ∂Ω (5.24)
Proof. Denoted by Π := {x = 0} the degeneration set of the Grushin vector fields, we
have that v ∈ C1(Ω \ Π), since X
jv ∈ C1(Ω \ {0}) by assumption and the coefficients of
the vector fields Xj are C1 and different from 0 in RN \ Π. Thus, the condition v = 0 on
∂Ω implies
∇v = ∂v
∂νν on ∂Ω \ Π.
This gives that ˆ ∂Ω d−2βh∇αv, ναiZv dσ = ˆ ∂Ω d−2β∂v ∂νh∇αv, ναih Z, νi dσ = ˆ ∂Ω d−2β|∇αv|2hZ, νi dσ.
Hence, in this case, identity (5.2) reduces to 1 2∗ ˆ Ω ZK v 2∗ d2∗β dz = 1 2 ˆ ∂Ω d−2β|∇αv|2hZ, νi dσ.
So, from the assumption ZK ≤ 0, we get ˆ ∂Ω d−2β ∂v ∂ν 2 |να|2hZ, νi dσ ≤ 0.
Then, since hZ, νi ≥ 0 on ∂Ω by the starshapedness of Ω, we have
∂v ∂ν
2
hZ, νi = 0 at any point of ∂Ω.
On the other hand, since Ω is bounded, there exists a bounded and connected open set
U ⊂ RN\ Π such that hZ, νi > 0 on ∂Ω ∩ U . Thus ∇v = ∂v
∂νν ≡ 0 on ∂Ω ∩ U. (5.25)
Setting v ≡ 0 in (RN\ Ω) ∩ U , we then obtain a nonnegative weak solution to
−divα
d−2β∇αv
= V v in U,
where V = K(z)vd2∗−22∗β ∈ L∞(U ). So, by Proposition 2.5 in [20], v ≡ 0 in U . Finally, a
connectedness argument gives v ≡ 0 in Ω.
Proof of Theorem 1.4 The thesis follows from Theorem 5.2, taking into account the
definition of the auxiliary function v in (3.3).
Finally, we prove a necessary condition for the existence of entire solutions to problem (5.1), involving the radial derivative ZK of the coefficient K. The result extends to the Grushin context the Euclidean result in [10].
Theorem 5.3. Let v ∈ Dα1,2(RN, d−2βdz) be a positive solution to −divα d−2β∇αv = K(z)v2 ∗−1 d2∗β in RN, (5.26)
where K ∈ L∞(RN) ∩ C1(RN) and ZK ∈ L∞(RN). Then
ˆ RN
ZK v2
∗
d2∗β dz = 0. (5.27)
Proof. Let v be a positive solution to (5.26). We shall apply identity (5.2) to v on an
appropriate sequence of balls BRn, with Rn→ ∞. To this aim, note that
ˆ ∞ 0 ds ˆ ∂Bs K(z) v2 ∗ d2∗β + |∇αv|2 d2β 1 |∇d|dσ = ˆ RN K(z)v2 ∗ d2∗β + |∇αv|2 d2β dz < ∞. (5.28) Therefore, there exists a sequence Rn→ +∞ such that
Rn ˆ ∂BRn K(z) v2 ∗ d2∗β + |∇αv|2 d2β 1 |∇d|dσ −→ 0, as n → ∞. (5.29)
Now, reasoning as in the previous proof, identity (5.2) on the balls BRn takes the form 1 2∗ ˆ BRn ZK v2 ∗ d2∗β dz − Rn 2∗ ˆ ∂BRn K(z)v2 ∗ d2∗β 1 |∇d|dσ =Q − 2 − 2β 2 ˆ ∂BRn d−2βvh∇αv,∇|∇d|αdi dσ −Rn 2 ˆ ∂BRn d−2β|∇αv|2 1 |∇d|dσ + Rn ˆ ∂BRn d−2β 1 ψ2(h∇αv, ∇αdi) 2 1 |∇d|dσ. (5.30) Hence, estimating the boundary integrals in (5.30) as in the previous proof and passing
to the limit as Rn→ ∞, the thesis follows by means of (5.29).
Proof of Theorem 1.5 The result follows from Theorem 5.3, taking into account the
definition of v in (3.3).
Acknowledgements The author is a member of Gruppo Nazionale per l’Analisi
Mate-matica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and she is partially supported by the GNAMPA research project
2018 “Metodi di analisi armonica e teoria spettrale per le equazioni dispersive”.
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