A direct application of the main result in the meaningful subcase of the Heisenberg group is also presented

Two nonlinear days in Urbino 2017

Electronic Journal of Differential Equations, Conference 25 (2018), pp. 179–196.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CRITICAL DIRICHLET PROBLEMS ON H DOMAINS OF CARNOT GROUPS

GIOVANNI MOLICA BISCI, PATRIZIA PUCCI Dedicated to the memory of our beloved friend Anna

Abstract. The paper deals with the existence of at least one (weak) solution for a wide class of one-parameter subelliptic critical problems in unbounded domains Ω of a Carnot group G, which present several difficulties, due to the intrinsic lack of compactness. More precisely, when the real parameter is sufficiently small, thanks to the celebrated symmetric criticality principle of Palais, we are able to show the existence of at least one nontrivial solution.

The proof techniques are based on variational arguments and on a recent compactness result, due to Balogh and Krist´aly in [2]. In contrast with a persisting assumption in the current literature we do not require any longer the strongly asymptotically contractive condition on the domain Ω. A direct application of the main result in the meaningful subcase of the Heisenberg group is also presented.

1. Introduction

This paper constitutes the initial part of a project devoted to the study of nonlin- ear equations defined on possibly unbounded domains of Carnot groups. Differential problems involving a subelliptic operator on an unbounded domain Ω of stratified groups have been intensively studied in recent years by many authors, see, among others, the papers of Garofalo and Lanconelli [16], Maad [23, 24], Schindler and Tintarev [32], Tintarev [33] and references therein.

On the contrary, once a domain is not bounded the Folland-Stein spaceHW₀^1,2(Ω) maybe not be compactly embedded into a Lebesgue space. This lack of compactness produces several difficulties exploiting variational methods. To recover compactness on the unbounded case a persisting hypothesis in the above cited results was the strongly asymptotically contractive condition on Ω, introduced by Maad, see [23]

for details. Indeed, every bounded domain is strongly asymptotically contractive.

In the Euclidean setting unbounded domains were covered in the pioneering paper [12].

Now, we observe that a strongly asymptotically contractive domain Ω is geo- metrically thin at infinity. In presence of symmetries, by replacing the contractive

2010Mathematics Subject Classification. 35R03, 35A15.

Key words and phrases. Carnot groups; compactness results; subelliptic critical equations.

c

2018 Texas State University.

Published September 15, 2018.

179

assumption on Ω with a geometrical hypothesis, see condition (H) below, intro- duced recently by Balogh and Krist´aly in [2], we are able to treat here subelliptic critical equations, in which the domain is possibly large at infinity.

The purpose of the present paper is to establish the existence of (weak) solutions of the one-parameter problem

−∆_Gu+u=h(q)f(u) +λ|u|^2∗−2u in Ω,

u= 0 on∂Ω. (1.1)

More precisely, our strategy is to find a topological groupT, acting continuously on HW₀^1,2(Ω), such that theT-invariant closed subspaceHW_0,T^1,2(Ω) can be compactly embedded in suitable Lebesgue spaces. Successively, assuming the left invariance of the standard Haar measureµof the Carnot groupG, with respect to the action of the group ∗ : T ×HW₀^1,2(Ω) → HW₀^1,2(Ω), see Bourbaki [6, Chapter III §2 No 4] and Bourbaki [7, Chapter 7§1 No 1], the principle of symmetric criticality of Palais, see Lemma 3.5 below, can be applied to the associated energy Euler- Lagrange functionalIλ, allowing a variational approach of problem (1.1).

Moreover, as usual, when dealing with critical equations, one of the main difficul- ties appears since the Palais-Smale condition for the Euler-Lagrange functional Iλ

does not hold at any level, but just under a suitable threshold. Along this paper we overcome these difficulties, using some strategies considered in the literature also in context different than the one treated here, see, for instance, papers [5, 22, 28, 30].

Let us briefly introduce the structural setting of problem (1.1). LetG= (G,◦) be a Carnot group of step r and homogeneous dimension Q > 2, with neutral element denoted by e. LetT = (T,·) be a closed infinite topological group acting continuously and left-distributively onGby the map∗:T×G→G. Assume that T acts isometrically on the horizontal Folland-Stein space HW₀^1,2(G), where the action]:T×HW₀^1,2(G)→HW₀^1,2(G) is defined for every (τ, u)∈T×HW₀^1,2(G) by

(τ ]u)(q) =u(τ⁻¹∗q) for allq∈G.

In what followsdCC:G×G→R⁺0 denotes the Carnot-Carath´eodory distance on G, whileµis the natural Haar measure onGand “ lim inf ” is the Kuratowski lower limit of sets.

Let Ω be a nonempty open T-invariant subset of G, with boundary ∂Ω, and assume that

(H1) for every (qk)k⊂Gsuch that

k→∞lim d_CC(e, q_k) =∞ and µ lim inf

k→∞(q_k◦Ω)

>0,

whereqk◦Ω ={qk◦q:q∈Ω}, then there exist a subsequence (qk_j)jof (qk)k

and a sequence of subgroups (Tq_kj)j ofT, with cardinality card(Tq_kj) =∞, having the property that for allτ₁,τ₂∈T_qkj, withτ₁6=τ₂, it results

j→∞lim inf

q∈GdCC((τ1∗qk_j)◦q,(τ2∗qk_j)◦q) =∞.

A domain Ω ofG, for which condition (H1) holds, is simply calledHdomain.

In (1.1) thesubelliptic Laplacian operator∆_GonGis the second-order differential operator

∆_G=

m₁

X

k=1

X_k²,

where B ={X1, . . . , Xm₁} is a basis of the first graduated component G1 of the stratified Lie algebraG=⊕^r_k=1Gk associated toG; see Section 2.

Thecritical Sobolev exponent2^∗in the Carnot groupGis 2^∗= 2Q/(Q−2). The parameterλis a real number. The nonlinearityf :R→Ris a continuous function, with associated primitive

F(t) = Z t

0

f(ξ)dξ for everyt∈R, and satisfies

(H2) F >0 inR\ {0}, and there existC >0 ands∈(1,2) such that

|f(t)| ≤C|t|^s−1 for allt∈R; (H3) there exist a0>0, δ >0 ands1∈(1,2) such that

F(t)≥a0|t|^s1 for allt∈R, with|t| ≤δ.

SinceQ >2, by [18] we know that for allϕ∈C₀^∞(Ω)

kϕk2^∗≤CQ,2kD_Gϕk2, (1.2)

whereCQ,2 is a positive constant depending on the dimensionQand D_G= (X₁, . . . , X_m1)

denotes the horizontal gradient.

Concerning the functionhin (1.1), we assume thathsatisfies (H4) 0≤h∈L ²

∗

2∗ −s(Ω) and there exists a nonempty open set Ω0⊂Ω such that

q∈Ωinf0

h(q)>0.

Clearly, condition (H4) simply requires thathbe nontrivial and belong to a suitable Lebesgue space. Finally, suppose that

(H5) the functional Ψ :HW₀^1,2(Ω)→Rgiven by Ψ(u) =

Z

Ω

h(q)f(u)dµ(q) for allu∈HW₀^1,2(Ω)

isT-invariant, that is Ψ(τ ]u) = Ψ(u) for all (τ, u)∈T×HW₀^1,2(G).

In Section 2 we present the useful criterion Lemma 2.5 on the validity of assumption (H5). We are now able to state the main existence result for (1.1).

Theorem 1.1. Let Ωbe aHdomain of G. Assume thatf andhfulfil(H2)–(H5).

Then (1.1) admits at least one nontrivial solution uλ in the Folland-Stein space HW₀^1,2(Ω) for allλ≤0. Furthermore, ifλ >0 andhsatisfies

khk 2∗ 2∗ −s < 1

C

1 C_Q,2^(2−s)Q+2s

2^α Q

2−s1/2

, (1.3)

whereC ands are introduced in(H2),C_Q,2>0 in (1.2), and α= 2^∗(6−s)−8

(2^∗−2)(2−s),

then there exists λ^∗ > 0 such that problem (1.1) admits at least one nontrivial solution u_λ in HW₀^1,2(Ω)for all λ∈(0, λ^∗).

|z|

t

O

ψ2

ψ1

Ω_ψ

Figure 1. A simple prototype of Ω_ψ

Thanks to [2, Theorem 1.1] and Lemma 2.5 below, a direct application of Theo- rem 1.1 gives the existence of at least one solution for subelliptic equations defined on a special class of (unbounded) domains of the Heisenberg groupHⁿ=Cⁿ×R, n≥1. More precisely, let ψ1, ψ2 :R⁺0 → R, R⁺0 = [0,∞), be two functions that are bounded on bounded sets, withψ1(t)< ψ2(t) for everyt∈R⁺0. Define

Ωψ=

q∈Hⁿ:q= (z, t) withψ1(|z|)< t < ψ2(|z|) , where|z|=pPn

i=1|zi|²; see Figure 1.

Then the subelliptic problem (1.1) becomes

−∆_Hⁿu+u=h(q)f(u) +λ|u|^2∗−2u in Ωψ

u= 0 on∂Ω_ψ, (1.4)

where ∆_Hn the subelliptic Kohn-Laplace operator.

LetU(n) =U(n)× {1}, where U(n) =U(n,C) =

τ ∈GL(n;C) :hτ z, τ z⁰i_Cn=hz, z⁰i_Cn for allz, z⁰ ∈Cⁿ , that isU(n) is the usual unitary group. Hereh·,·i_Cⁿdenotes the standard Hermitian product onCⁿ, in other wordshz, z⁰i_Cn=Pn

k=1z_k·z_k⁰.

Hence,U(n) is the unitary group endowed with the natural multiplication law

·:U(n)×U(n)→U(n), which acts continuously and left-distributively on Hⁿ by the map∗:U(n)×Hⁿ→Hⁿ, defined by

bτ∗q= (τ z, t) for allbτ= (τ,1)∈U(n) and allq= (z, t)∈Hⁿ,

thanks to [2, Lemma 3.1]. TakingT =U(n), then Ωψ is U(n)-invariant and aH domain, as shown in the proof of Theorem 1.1 of [2]. Moreover,

HW_0,^1,2

U(n)(Ωψ) ={u∈HW₀^1,2(Ωψ) :u(z, t) =u(|z|, t) for allq= (z, t)∈Ωψ},

that isHW_0,^1,2

U(n)(Ωψ) = HW_cyl^1,2(Ωψ) is the space of cylindrically symmetric func- tions ofHW₀^1,2(Ωψ).

Finally,U(n) acts isometrically on the horizontal Folland-Stein spaceHW₀^1,2(Hⁿ), where the action ]: U(n)×HW₀^1,2(Hⁿ)→HW₀^1,2(Hⁿ) is defined for every (bτ , u) inU(n)×HW₀^1,2(Hⁿ) by

(bτ ]u)(q) =u(τ⁻¹z, t) for allq= (z, t)∈Hⁿ,

in view of [2, Lemma 3.2] A special case of Theorem 1.1 reads as follows.

Corollary 1.2. Let Ω_ψ be defined as above. Assume thatf andhfulfil(H2)–(H4), and h is cylindrically symmetric, that is h(q) = h(z, t) = h(|z|, t) for every q = (z, t)∈Ω_ψ. Then (1.1)admits at least one nontrivial solutionu_λ inHW_0,^1,2

U(n)(Ω_ψ) for allλ≤0.

Furthermore, if λ >0 andh satisfies also (1.3), then there exists λ^∗ >0 such that problem (1.1)admits at least one nontrivial solution uλ inHW_0,^1,2

U(n)(Ωψ)for allλ∈(0, λ^∗).

If the functionsψ₁andψ₂are bounded, the domain Ω_ψis strongly asymptotically contractive and the whole spaceHW₀^1,2(Ωψ) is compactly embedded inL^ν(Ωψ) for everyν∈(2,2^∗). We refer to [2, 24] for further details. In such a case Corollary 1.2 follows by using the embedding result proved by Garofalo and Lanconelli in [16].

See also Schindler and Tintarev [32].

On the Heisenberg setting, a Rubik-cube technique, see [2], applied to subgroups ofU(n) and suitable variational arguments allow us to obtain further multiplicity results that will be presented in the forthcoming paper [27].

The manuscript is organized as follows. In Section 2 we present the notations and recall some properties of the functional solution space of (1.1). In particular, in order to apply critical point methods to problem (1.1), we need to exploit some analytic properties of the closed subspaceHW_0,T^1,2(Ωψ), introduced above. Then, in the same section, we give the key Lemmas 2.1 and 2.3 which are particularly useful for the proof of Theorem 1.1. Finally, in Section 3 we describe the geometrical profile of the underlying functional in Lemmas 3.1 and 3.2 and we prove the existence result stated in Theorem 1.1.

For general references on the subject and on methods treated along the paper we refer to the monographs [4, 21] as well as [11, 25, 26, 36] and the references therein.

2. Notation and preliminaries

In this section we briefly recall some basic facts on Carnot groups and the func- tional Folland-Stein spaceHW₀^1,2(Ω). A Carnot groupG= (G,◦) is a connected, simply connected, nilpotent Lie group, whose Lie algebraGadmits a stratification, i.e.

G=⊕^r_k=1Gk,

where the integerris called thestep ofG, whileG_k is the linear subspace of finite dimensionmk ofGfor everyk∈ {1, . . . , r}, and

[G₁,G_k] =G_k+1 for allk, with 1≤k < r−1 and [G₁,G_r] ={O}.

In this context the symbol [G1,Gk] denotes the subalgebra of Ggenerated by the commutators [X, Y], whereX ∈G1andY ∈Gk, and where the last bracket denotes the Lie bracket of vector fields, that is [X, Y] =XY −Y X.

The left translation byq∈GonGis given by`q(p) =q◦pfor everyp∈G. Let Γ(TG) be the space of global sections of the tangent bundleTG onG. A vector fieldX ∈Γ(TG) is left invariant if for everyq∈Gone has

X(ϕ◦`q) = (Xϕ)◦`q, for anyϕ∈C^∞(G) andp∈G.

The Lie algebraGassociated toGis the Lie algebra of left invariant vector fields X onG. Moreover,Gis canonically isomorphic to the tangent spaceTeG.

Let

m=

r

X

k=1

mk

be thetopological dimension of the Carnot groupG.

The exponential map exp_G :G→Gis given by exp_G(X) =γ_X(1), whereγ_X is the unique integral curve associated to the left invariant vector field X such that γ_X(0) = e. In other words, the curve γ_X is the unique solution of the Cauchy problem

˙

γ_X(t) =X(γ_X(t)), γ_X(0) =e. (2.1) The curveγ_Xis defined for anyt∈R, that is left invariant vector fields are complete.

Indeed, γ_X(t+s) =γ_X(s)γ_X(t) by (2.1). Therefore, γ_X can be extended in the entireR.

SinceGis nilpotent, connected and simply connected Lie group, the exponential map exp_G is a smooth diffeomorphism fromGontoG.

Leth·,·i0be a fixed inner product on the first graduated componentG1ofG, with associated orthonormal basis B= {X1, X2, . . . , Xm₁}. From now on, we consider the extension of the inner producth·,·i0to the whole tangent bundleTGby group translation. The corresponding norm is denoted byk · k0. A left invariant vector fieldX ∈Gis said to behorizontal if

X(q)∈span{X1(q), . . . , Xm₁(q)}

for everyq∈G. Indeed,G₁is considered to be the horizontal direction, while the remaining layers G2,· · ·,Gr are viewed as the vertical directions. In particular, the last layer Gr is the center of the Lie algebra and the horizontal direction G1

generates in the sense of Lie algebras the wholeG. More precisely, G_k = [G₁,[G₁,[G₁, . . .[G₁,G₁]· · ·]]]

| {z }

ktimes

for allk= 2,· · ·, r.

Since the map exp_G is bijective, for every element q ∈Gthere exists a unique vector fieldX =Pm1

k=1xkXk+Pm

k=m1+1xkX_k⁰ ∈Gsuch that q= exp_G(X) = exp_GX^m1

k=1

xkXk+

m

X

k=m1+1

xkX_k⁰ ,

where {Xm₁+1, . . . , Xm} are non-horizontal vector fields that extend B to an or- thonormal basisB^∗ ofG.

Now, observe that G∼=R^m. Then, there exists a smooth map %such that the following diagram is commutative

R^m G=⊕^r_k=1G_k

G π⁻¹

% exp_G

whereπ⁻¹is the inverse of the canonical projectionπ:G→R^m such that R^m3(x1, . . . , xm₁, . . . , xm) X=Pm1

k=1x_kX_k+Pm

k=m1+1x_kX_k⁰ ∈G

q∈G π⁻¹

% exp_G

Thus, we often identify every element q ∈ G with its exponential coordinates (x1, . . . , xm1, xm1+1, . . . , xm)∈R^m respect to the basisB^∗ inG.

More precisely, it is possible to identify the Carnot group (G,◦) with (R^m, ?), where the expression of the group operation?is given by

x ? y=%⁻¹(%(x)◦%(y)) for allx, y∈R^m

and is explicitly determined by the Baker-Campbell-Hausdorff formula.

Whenever we are in presence of a stratification, it is possible to define a one- parameter group{∆η}η>0of dilatations of the algebra. More precisely, for a fixed real number η > 0 and all X ∈ G_k, we set ∆_η(X) = η^kX and extend the map

∆_η to the wholeGby linearity. Furthermore, the family{∆_η}_η>0induces a family {δ_η}_η>0 of the group automorphisms on Gby the exponential map such that the following diagram is commutative

G G=⊕^r_k=1Gk

G G=⊕^r_k=1G_k exp⁻¹_G

exp_G

δ_η ∆_η

that is

δ_η(q) = exp_G(∆_η(exp⁻¹

G (q))) for everyq∈G.

Thehomogeneous dimension Qof G, attached to the automorphisms{δη}η>0, is defined by

Q=

r

X

k=1

kdimGk =m₁+ 2m₂+· · ·+rm_r.

In particular, the above definition of Q and the fact that {δη}η>0 is a family of automorphisms on G imply that the Jacobian determinant of the dilation δ_η is constant inqand given byη^Q.

Moreover, letµdenote the push-forward of them-dimensional Lebesgue measure LmonGvia the exponential map. Then,dµdefines a biinvariant Haar measure on Gand

dµ(q◦δη) =η^Qdµ(q).

SinceGcan be identified with (R^m, ?) by using the exponential map, if E⊂Gis a measurable subset, its Haar measure is explicitly given byµ(E) =Lm(ρ⁻¹(E)).

Therefore, the same notation will be used for both measures.

Takeq1,q2∈Gand letHΓq₁,q₂(G) be the set of piecewise smooth curvesγ, such thatγ: [0,1]→G, ˙γ(t)∈G1 a.e. t∈[0,1], (γ(0), γ(1)) = (q1, q2) and

Z 1

0

kγ(t)k˙ 0dt <∞.

Since HΓ_q1,q2(G) 6= ∅ by the celebrated Chow-Rashevski˘ı theorem in [10], it is possible to define theCarnot-Carath´eodory distance onG, as follows

dCC(q1, q2) = inf

γ∈HΓ_q1,q2(G)

Z 1

0

kγ(t)k˙ 0dt.

The metricd_CC is left invariant onGand for everyη >0 it results dCC(δη(q1), δη(q2)) =η dCC(q1, q2),

for everyq₁, q₂∈G.

The Euclidean distance to the origin | · |onGinduces a homogeneous pseudo- norm | · |G onG and (via the exponential map) one on the group G. Indeed, for X ∈G, withX=Pr

k=1Xk, whereXk ∈Gk, define a pseudo-norm onGas follows

|X|G =X^r

k=1

|Xk|^2r!/k2r!

. The induced pseudo-norm onGhas the form

|q|_G=|exp⁻¹

G (q)|_G for allq∈G.

The function | · |_G is usually known as the non-isotropic gauge. It defines a pseudo-distance onGgiven by

d(p, q) =|p⁻¹◦q|_G for allp, q∈G, that is equivalent to theCarnot-Carath´eodory distance d_CC onG.

Thus, Carnot groups are endowed with the intrinsic Carnot-Carath´eodory ge- ometry. The adjective intrinsic is meant to emphasize a privileged role played by the horizontal layer and by group translations and dilations. It is worth stressing that the Carnot-Carath´eodory geometry is not Riemannian at any scale. In fact, Carnot groups can be seen as a particular case of more general structures, the so-calledsub-Riemannian spaces.

The most basic second-order partial differential operator in a Carnot groupGis thesub-Laplacian, or equivalently thehorizontal Laplacian in G, given by

∆_G=

m1

X

k=1

X_k².

We shall denote by D_G = (X1, . . . , Xm₁) the related horizontal gradient and set kD_Guk0= Pm1

k=1(Xku)^21/2 .

Obviously, Euclidean spaces are commutative Carnot groups, and, more pre- cisely, the only commutative Carnot groups. The simplest example of Carnot group of step two is provided by the Heisenberg groupHⁿof topological dimension m= 2n+ 1 and homogeneous dimensionQ= 2n+ 2, that is the Lie group whose underlying manifold isR²ⁿ⁺¹, endowed with the non-Abelian group law

q◦q⁰=

z+z⁰, t+t⁰+ 2

n

X

i=1

(y_ix⁰_i−x_iy_i⁰)

for allq,q⁰ ∈Hⁿ, with

q= (z, t) = (x1, . . . , xn, y1, . . . , yn, t), q⁰= (z⁰, t⁰) = (x⁰₁, . . . , x⁰_n, y₁⁰, . . . , y_n⁰, t⁰).

The vector fields forj= 1, . . . , n Xj= ∂

∂x_j + 2yj

∂

∂t, Yj= ∂

∂y_j −2xj

∂

∂t, ∂

∂t, (2.2)

constitute a basis B^∗ for the real Lie algebra H=G of left invariant vector fields onHⁿ. The basisB^∗ satisfies the Heisenberg canonical commutation relations for position and momentum [Xj, Yk] =−4δjk∂/∂t, all other commutators being zero.

Ifu∈C²(Hⁿ), then thehorizontal Laplacian inHⁿofu, called theKohn-Spencer Laplacian, is defined as follows

∆_Hnu=

n

X

j=1

(X_j²+Y_j²)u

=

n

X

j=1

∂²

∂x²_j + ∂²

∂y²_j + 4yj

∂²

∂x_j∂t−4xj

∂²

∂y_j∂t

u+ 4|z|²∂²u

∂t²,

and ∆_Hⁿishypoellipticaccording to the celebrated Theorem 1.1 due toH¨ormander in [19].

Turning back to (1.1), we need to introduce the suitable solution space. Let Ω be a nontrivial open subset of G. The Folland-Stein horizontal Sobolev space HW₀^1,2(Ω) is the completition ofC₀^∞(Ω), with respect to the Hilbertian norm

kuk=Z

Ω

kD_Guk²₀dµ(q) + Z

Ω

|u|²dµ(q)1/2

, hu, ϕi=

Z

Ω

hD_Gu, D_Gϕi0dµ(q) + Z

Ω

uϕdµ(q).

(2.3)

Of course, if Ω = G, then HW^1,2(G) = HW₀^1,2(G), where HW^1,2(G) denotes the horizontal Sobolev space of the functions u∈L²(G) such that D_Guexists in the sense of distributions and kD_Guk0 is in L²(G), endowed with the Hilbertian norm (2.3).

In particular, the embedding

HW₀^1,2(Ω),→L^ν(Ω) (2.4)

is continuous for any ν ∈ [2,2^∗]; see Folland and Stein [15]. Furthermore, by [17, 20, 35] we know that, ifOis a bounded open set ofG, the embedding

HW₀^1,2(O),→,→L^ν(O) (2.5)

is compact for allν, with 1≤ν <2^∗.

Let (G,◦) be a Carnot group, and (T,·) be a closed topological group, with neutral element . The groupT is said to act continuously onG, if there exists a map∗:T×G→Gsuch that the following conditions

(H6) j∗q=qfor every q∈G;

(H7) τ1∗(τ2∗q) = (τ1·τ2)∗q for everyτ1, τ2∈T andq∈G hold. In addition, the action∗ isleft distributed if

(H8) τ∗(p◦q) = (τ∗p)◦(τ∗q) for everyτ∈T andp, q∈G.

A set Ω⊂GisT-invariant, with respect to∗, ifT∗Ω = Ω.

We assume that T induces an action ]: T×HW₀^1,2(G)→HW₀^1,2(G), defined for every (τ, u)∈T×HW₀^1,2(G) by

(τ ]u)(q) =u(τ⁻¹∗q) for allq∈G. (2.6) The groupT actsisometrically onHW₀^1,2(Ω) if

kτ ]uk=kuk for all (τ, u)∈T×HW₀^1,2(G). (2.7) Let

HW_0,T^1,2(Ω) ={u∈HW₀^1,2(Ω) :τ ]u=ufor allτ∈T}

be the T-invariant subspace ofHW₀^1,2(Ω). Clearly, HW_0,T^1,2(Ω) is closed, since the action]ofT onHW₀^1,2(Ω) is continuous by (H6) and (H7).

The following compactness result is due to Balog and Krist´aly and given in [2, Theorem 3.1].

Lemma 2.1. Let G = (G,◦) be a Carnot group of step r and homogeneous di- mension Q > 2, with neutral element denoted by e. Let T = (T,·) be a closed infinite topological group acting continuously and left distributively on G by the map ∗:T×G→G. Assume furthermore thatT acts isometrically onHW₀^1,2(G), where the action ] : T ×HW₀^1,2(G) → HW₀^1,2(G) is defined in (2.6). Let Ω be a nonempty T-invariant open subset of G, satisfying condition (H1). Then the embedding

HW_0,T^1,2(Ω),→,→L^ν(Ω) is compact for everyν ∈(2,2^∗).

Remark 2.2. By (2.4) the embeddings

HW_0,T^1,2(Ω),→L^ν(Ω)

are continuous for everyν ∈[2,2^∗]. In particular, there exists a constantC_ν such that

kukν ≤C_νkuk for allu∈HW_0,T^1,p(Ω), (2.8) whereCν depends onν andQ.

We also notice that inequality (1.2) yields

kuk₂^∗≤C_Q,2kD_Hnuk₂ (2.9)

for allu∈HW_0,T^1,2(Ω).

Lemma 2.3. Let (uk)k be in HW_0,T^1,2(Ω), such thatuk * uweakly in HW_0,T^1,2(Ω), andu_k→ua.e. inΩ. Then

k→∞lim Z

Ω

|uk−u|^2∗dµ(q) = lim

k→∞

Z

Ω

|uk|^2∗dµ(q)− Z

Ω

|u|^2∗dµ(q),

k→∞lim Z

Ω

|u|^2∗−2u(uk−u)dµ(q) = 0,

k→∞lim Z

Ω

|uk|^2∗−2ukudµ(q) = Z

Ω

|u|^2∗dµ(q).

(2.10)

Proof. The first part of (2.10) is just the celebrated Brezis-Lieb lemma in [8].

For the second part of (2.10), it is enough to observe that u_k * u in L^2∗(Ω) by Lemma 2.1 and thatϕ7→R

Ω|u|^2∗−2uϕdµ(q) is a linear continuous functional on L^2∗(Ω). While the third limit is a consequence of [1, Proposition A.8].

A functionu∈HW₀^1,2(Ω) is said to be a (weak)solution of problem (1.1) if hu, ϕi=

Z

Ω

h(q)f(u)ϕdµ(q) +λ Z

Ω

|u|^2∗−2uϕdµ(q) (2.11) for anyϕ∈HW₀^1,2(Ω).

Problem (1.1) has a variational nature and the Euler-Lagrange functional I_λ associated to (1.1) is

I_λ(u) = 1 2kuk²−

Z

Ω

h(q)F(u)dµ(q)− λ 2^∗

Z

Ω

|u|^2∗dµ(q).

Clearly, the functional Iλ is well-defined in HW₀^1,2(Ω) and, thanks to (H2) and (H4), it is of classC¹(HW₀^1,2(Ω)). Moreover, for everyu∈HW₀^1,2(Ω)

hI_λ⁰(u), ϕi=hu, ϕi − Z

Ω

h(q)f(u)ϕdµ(q)−λ Z

Ω

|u|^2∗−2uϕdµ(q) (2.12) for allϕ∈HW₀^1,2(Ω). Hence, the critical points ofI_λinHW₀^1,2(Ω) are exactly the (weak) solutions of (1.1).

Letu∈HW_0,T^1,2(Ω) be a solution of problem (1.1) only in theHW_0,T^1,2(Ω) sense, that is

hu, ϕi= Z

Ω

h(q)f(u)ϕdµ(q) +λ Z

Ω

|u|^2∗−2uϕdµ(q) (2.13) for any ϕ∈HW_0,T^1,2(Ω). Then, u∈HW_0,T^1,2(Ω) is a solution of (1.1) in the whole spaceHW₀^1,2(Ω), that is in sense of definition (2.11), if theprinciple of symmetric criticality of Palais given in [29] holds. To prove this let us recall the well known principle of symmetric criticality of Palais stated in the general form proved in [13]

for reflexive strictly convex Banach spaces. For details and comments we refer to [9, Section 5].

More precisely, letX = (X,k · kX) be a reflexive strictly convex Banach space.

Suppose that G is a subgroup of isometries g : X → X, that is g is linear and kgukX =kukX for allu∈X. Consider theG-invariant closed subspace ofX,

ΣG={u∈X :gu=ufor allg∈ G}.

By [13, Proposition 3.1] we have

Lemma 2.4. Let X,G andΣbe as before and letI be aC¹ functional defined on X such thatI◦g=Ifor all g∈ G. Thenu∈ΣG is a critical point of Iif and only if uis a critical point of J =I|ΣG.

From now on we assume thatT satisfies the main structural conditions of The- orem 1.1 and that Ω is a nonempty open subset of G, which is T-invariant. We apply the principle of symmetric criticality to the Sobolev spaceHW_0,T^1,2(Ω) under the action]:T×HW₀^1,2(G)→HW₀^1,2(G) defined in (2.6). Clearly,

kτ ]uk=kuk for all (τ, u)∈T×HW₀^1,2(Ω), (2.14)

sinceT acts isometrically on HW₀^1,2(Ω) by assumption. Moreover, the functional Ψ :HW₀^1,2(Ω)→Ris T-invariant by assumption (H5). Thus,Iλ isT-invariant in HW₀^1,2(Ω).

Hence, the principle of symmetric criticality of Palais ensures thatu∈HW_0,T^1,2(Ω) is a solution of problem (1.1) if and only if uis a critical point of the functional Jλ:HW_0,T^1,2(Ω)→R, whereJλ=Iλ|_HW1,2

0,T(Ω).

We end the section by an essential lemma which shows when the key assumption (H5) is satisfied. To this aim, we need to introduce some facts well known in abstract group measure theory.

Lemma 2.5. Suppose that the action ∗ of the group T on the Carnot group G satisfies conditions(H6)–(H8). Assume furthermore that the natural Haar measure µ, defined on G, is left ∗ invariant, that is for all measurable subset E of Gand for allτ ∈T

µ(τ∗E) =µ(E), whereτ∗E={τ∗q:q∈E}.

IfhisT-invariant, that ish(τ∗q) =h(q)for allτ ∈T andq∈G, andf :R→R is a continuous function, then(H5) holds.

Proof. Fix τ ∈ T and u ∈ HW₀^1,2(Ω). Then, putting τ⁻¹∗q = p, we get by (H6)–(H8)

Ψ(τ ]u) = Z

Ω

h(q)f((τ ]u)(q))dµ(q) = Z

Ω

h(q)f(u(τ⁻¹∗q))dµ(q)

= Z

τ∗Ω

h(τ∗p)f(u(p))dµ(τ∗p)

= Z

Ω

h(p)f(u(p))dµ(p) = Ψ(u),

since Ω and h are T-invariant by assumption, and the left ∗ invariance of the measureµimplies

dµ(τ∗p) =dµ(p) for allp∈G,

which is exactly [7, formula (10)], being 1 the multiplier ofµ. See also [3, Chapter 4].

This shows that Ψ is T-invariant, that is Ψ satisfies (H5), and concludes the

proof.

3. Proof of Theorem 1.1

In this section we suppose that the assumptions of Theorem 1.1 are satisfied, without further mentioning. Thus, problem (1.1) has a variational structure and, as explained in Section 2, it is enough to study the critical points of the functional Jλ:HW_0,T^1,2(Ω)→R, defined by

Jλ(u) =1 2kuk²−

Z

Ω

h(q)F(u)dµ(q)− λ 2^∗

Z

Ω

|u|^2∗dµ(q) (3.1) for allu∈HW_0,T^1,2(Ω). We first show thatJ_λhas a useful geometrical profile, and recall that, whenλ >0, we require also (1.3) onh.

Lemma 3.1. For any parameter λ≤1 there exist positive numbers ρ0 andjsuch that Jλ(u)≥j for anyu∈ HW_0,T^1,2(Ω), withkuk =ρ0, and for any functionh of the type stated in Theorem 1.1. Moreover,

m_λ= inf

u∈Bρ0

Jλ(u)<0,

whereB_ρ0 ={u∈HW_0,T^1,2(Ω) :kuk< ρ₀}, and there exist a sequence (u_k)_k in B_ρ0 and a function uλ inBρ0 such that for all k,

ku_kk< ρ₀, m_λ≤ J_λ(u_k)≤m_λ+1 k, u_k * u_λ inHW_0,T^1,2(Ω), u_k→u_λ a.e. inΩ,

J_λ⁰(uk)→0 in[HW_0,T^1,2(Ω)]⁰.

(3.2)

Proof. Fixλ≤1. By (H2), Lemma 2.1 and (2.9) we obtain J_λ(u)≥1

2kuk²−C Z

Ω

h(q)|u|^sdµ(q)− λ 2^∗kuk²₂^∗∗

≥1

2kuk²−CC_Q,2^s khk 2∗

2∗ −skuk^s−λ⁺

2^∗C_Q,2^2∗ kuk^2∗,

(3.3)

for allu∈HW_0,T^1,2(Ω). Therefore, ifλ≤0, forρ0>0 sufficiently large we have Jλ(u)≥ρ^s₀h1

2ρ^2−s₀ −CC_Q,2^s khk 2∗ 2∗ −s

i

= >0 for allu∈HW_0,T^1,2(Ω), withkuk=ρ₀, since 1< s <2.

Inλ∈(0,1], then the Young inequality yields for anyε >0 CC_Q,2^s khk 2∗

2∗ −skuk^s≤εkuk²+ε^−2−ss

CC_Q,2^s khk 2∗ 2∗ −s

2−s²

, being 1< s <2. Thus, forε= 1/4 it follows that

Jλ(u)≥ 1

4kuk²−

2^sCC_Q,2^s khk 2∗ 2∗ −s

2/(2−s)

−C_Q,2^2∗ 2^∗ kuk^2∗, since 0< λ≤1. Let us consider the function

η(t) =1

4t²−C_Q,2^2∗

2^∗ t^2∗, t≥0.

Now the numberρ0= (2CQ,2)^2−21∗ >0 is such that η(ρ₀) = max

t≥0 η(t) = 1 2

1 2 − 1

2^∗

2C_Q,2^2∗ 2/(2−2^∗)

>0

because 2< 2^∗. Therefore, for any function h, satisfying (1.3), and for any u in HW_0,T^1,2(Ω), withkuk=ρ₀, we obtain

Jλ(u)≥η(ρ0)−

2^sCC_Q,2^s khk 2∗ 2∗ −s

2/(2−s)

=j>0, which concludes the proof of the first part.

Letq₀ ∈ Ω₀ and R >0 be so small that B ⊂Ω₀, where B =B(q₀,2R) is the open ball of center q₀ and radiusR and Ω₀ is given in (H4). Chooseϕ∈C₀^∞(B)

such that 0 ≤ ϕ ≤ 1, with kϕk ≤ ρ0, and R

Bϕ^s1dµ(q) > 0. Let δ > 0 be the number given in (H3). For allt∈(0, δ), then (H3) and (H4) yield

Jλ(tϕ)≤1

2ktϕk²− Z

Ω

h(q)F(tϕ)dµ(q)−λt^2∗ 2^∗

Z

Ω

ϕ^2∗dµ(q)

≤t² 2kϕk²−

Z

Ω

h(q)F(tϕ)dµ(q) +λ⁻t^2∗ 2^∗

Z

B

ϕ^2∗dµ(q)

≤t²

2ρ²₀−t^s1a₀ inf

q∈Ω0

h(q) Z

B

ϕ^s1dµ(q) +λ⁻t^2∗ 2^∗

Z

B

ϕ^2∗dµ(q).

Hence, Jλ(tϕ)< 0 for for t ∈ (0, δ) sufficiently small, since 1 < s1 <2 < 2^∗ by (H3). This shows thatmλ<0 and completes the proof.

Applying the Ekeland variational principle inB_ρ0and the first part of the lemma, there exists a sequence (u_k)_k inB_ρ0 such that

m_λ≤ Jλ(u_k)≤m_λ+ 1

k, Jλ(u)≥ Jλ(u_k)−1

kku−u_kk

for all u ∈ Bρ₀. A standard procedure gives that J_λ⁰(uk) → 0 in [HW_0,T^1,2(Ω)]⁰ as k → ∞ and, up to a subsequence, the bounded sequence (uk)k ⊂ Bρ₀ weakly converges to some uλ ∈ Bρ₀ and uk → uλ a.e. in Ω. This completes the proof

of (3.2) and of the lemma.

Clearly, (3.2) of Lemma 3.1 implies that the bounded sequence (uk)k is a Palais- Smale sequence ofJλ inHW_0,T^1,2(Ω) at levelmλ.

Lemma 3.2. There existsλ^∗∈(0,1]such that, up to a subsequence,(u_k)_kstrongly converges to some u_λ inHW_0,T^1,2(Ω)for all λ < λ^∗.

Proof. Fix λ ≤ 1. By (3.2) of Lemma 3.1, in addition to (2.9) and Lemma 2.3, passing up to a further subsequence, if necessary, (uk)k anduλ∈Bρ₀ satisfy (3.2) and

uk * uλ in HW_0,T^1,2(Ω), kukk →κλ, D_Guk* D_Gu in L²(Ω,R^m1),

u_k →u_λ in L^ν(Ω), u_k→u_λ a.e. in Ω, ku_k−u_λk²₂^∗∗→c_λ,

|uk|^2∗−2uk *|uλ|^2∗−2uλ in L^{2∗/(2∗−1)}(Ω),

(3.4)

whereκλ andcλ are nonnegative numbers, andν ∈(2,2^∗). We claim that Z

Ω

h(q)|uk−uλ|^sdµ(q)→0. (3.5) Since h ∈ L ²

∗

2∗ −s(Ω) and (uk)k is bounded in HW_0,T^1,2(Ω), by (1.2) for any ε > 0 there exists a measurable setE⊂Ω such that

Z

Ω\E

h(q)|uk−uλ|^sdµ(q)

≤Z

Ω\E

|h(q)|^{2∗/(2∗−s)}dµ(q)^(2∗−s)/2^∗

kuk−uλk²₂∗≤ ε 2. Furthermore, for any measurable subsetU ⊂E, by the H¨older inequality

Z

U

h(q)|uk−uλ|^sdµ(q)≤cZ

U

|h(q)|^{2∗/(2∗−s)}dµ(q)^(2∗−s)/2^∗

,

where c= sup_kkuk−uλk²₂∗. Hence,{h(q)|uk−uλ|^s}k is equi-integrable and uni- formly bounded inL¹(E), thanks to (H4). Thus by (3.4) and the Vitali convergence theorem, for allε >0 there existsk0>0 such that

Z

E

h(q)|uk−uλ|^sdµ(q)≤ ε 2 for allk≥k₀. Therefore,

Z

Ω

h(q)|uk−uλ|^sdµ(q)≤ Z

Ω\E

h(q)|uk−uλ|^sdµ(q) + Z

E

h(q)|uk−uλ|^sdµ(q)≤ε for allk≥k₀. This proves the claim and (3.5).

Now (H2) and the H¨older inequality give

Z

Ω

h(q)f(uk)(uk−uλ)dµ(q) ≤C

Z

Ω

h(q)|uk|^s−1|uk−uλ|dµ(q)

≤C˜Z

Ω

h(q)|uk−uλ|^sdµ(q)^1/s , for a suitable constant ˜C >0. Thus, by (3.5) it follows that

k→∞lim Z

Ω

h(q)f(uk)(uk−uλ)dµ(q) = 0. (3.6) Similarly, by using again (H4) and (H2) we have ask→ ∞

Z

Ω

h(q)f(uk)ϕdµ(q)→ Z

Ω

h(q)f(uλ)ϕdµ(q), (3.7) for anyϕ∈HW_0,T^1,2(Ω).

By (3.2), (3.4)–(3.7) we see thatuλ is a solution of (1.1), that isuλ is a critical point ofJλ inHW_0,T^1,2(Ω). In particular, ask→ ∞

o(1) =hJ_λ⁰(uk)− J_λ⁰(uλ), uk−uλi= (κ²_λ− kuλk²)− kukk²₂^∗∗+kuλk²₂^∗∗+o(1).

Consequently, by (3.4) and the Br´ezis-Lieb lemma [8] we get the main formula

k→∞lim kuk−uλk²=λ lim

k→∞kuk−uλk²₂^∗∗=λ cλ. (3.8) Let us first consider the caseλ≤0. Then, (3.8) gives at once thatku_k−u_λk=o(1) ask→ ∞, that is (uk)_k strongly converges tou_λ inHW_0,T^1,2(Ω), as stated.

Let us now consider the caseλ∈(0,1]. By using (2.9), withu=u_k−u_λ, we get λ c_λ≥C_Q,2^2∗ c^2/2_λ ^∗ (3.9) for allλ∈(0,1]. Let us define

λ^∗=

(inf{λ∈(0,1] :c_λ>0}, if there existsλ∈(0,1] such thatc_λ>0, 1, ifcλ= 0 for allλ∈(0,1].

We claim thatλ^∗>0 if there existsλ >0 such thatcλ>0. Otherwise, there exists a sequence (λk)k, with cλ_k >0, such thatλk →0 ask → ∞. Thus, (3.9) implies that

λkc^1−2/2_λ ^∗

k ≥C_Q,2^2∗ >0.

This is an obvious contradiction since {cλ}_λ∈(0,1] is uniformly bounded above by (2.9). Indeed, (u_k)_k ⊂ B_ρ0, u_λ ∈ B_ρ0 and ρ₀, given in Lemma 3.1, is inde- pendent ofλ.

Hence,cλ= 0 for anyλ∈(0, λ^∗). Therefore, for allλ∈(0, λ^∗), lim

k→∞kuk−u_λk2^∗= 0.

Now (3.8) implies

k→∞lim kuk−uλk= 0.

In conclusion,uk→uλ ask→ ∞in HW_0,T^1,2(Ω) for allλ < λ^∗, as stated.

Proof of Theorem 1.1. LetJλbe the restriction of the energy functional Iλ to the subspace HW_0,T^1,2(Ω). For any λ≤1 Lemma 3.1 and the Ekeland variational prin- ciple give the existence of a Palais-Smale sequence (uk)k in HW_0,T^1,2(Ω) of Jλ at level m_λ < 0. Moreover, by Lemma 3.2 there exists λ^∗ > 0 such that, up to a subsequence, (uk)k strongly converges to some uλ in HW_0,T^1,2(Ω) for all λ < λ^∗. Furthermore,mλ =Jλ(uλ)<0 andJ_λ⁰(uλ) = 0 for allλ < λ^∗. Consequently, the functionuλ∈HW_0,T^1,2(Ω) is a nontrivial critical point of the functionalJλ. Now, as observed in Section 2, since the action]:T×HW₀^1,2(Ω)→HW₀^1,2(Ω) given in (2.6) is supposed to be isometric, the functionalI_λ is T-invariant by assumption (H5).

Hence, the principle of symmetric criticality of Palais, recalled in (2.4), implies that u_λ∈HW_0,T^1,2(Ω) is a nontrivial critical point also forI_λ in HW₀^1,2(Ω), that is a nontrivial solution for (1.1) in the sense of definition (2.11). This completes the

proof.

Acknowledgments. The authors were partly supported by the Italian MIUR projectVariational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT 009), and are members of theGruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of theIsti- tuto Nazionale di Alta Matematica (INdAM).

The manuscript was realized within the auspices of the INdAM-GNAMPA Project 2018 denominated Problemi non lineari alle derivate parziali Prot U-UFMBAZ- 2018-000384), and of the Fondo Ricerca di Base di Ateneo - Esercizio 2015of the University of Perugia, namedPDEs e Analisi Nonlineare.

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