On Nonuniformly Subelliptic Equations of Q−sub-Laplacian Type with Critical Growth in the Heisenberg Group

On Nonuniformly Subelliptic Equations of Q−sub-Laplacian Type with Critical Growth in the Heisenberg Group

Advanced Nonlinear Studies 12 (2012), 659–681 On Nonuniformly Subelliptic Equations of Q−sub-Laplacian Type with Critical Growth in the Heisenberg Group ∗ Nguyen Lam, Guozhen Lu† Department of Mathematics Wayne State University, Detroit, Michigan 48202 e-mail: nguyenlam@wayne.edu, gzlu@math.wayne.edu Hanli Tang School of Mathematical Sciences Beijing Normal University, Beijing, China 100875 e-mail: rainthl@163.com Received in revised form 05 April 2012 Communicated by Susanna Terracini Abstract n 2n Let H = R × R be the n−dimensional Heisenberg group, ∇Hn be its subelliptic gradient ( )1/4 operator, and ρ (ξ) = |z|4 + t2 for ξ = (z, t) ∈ Hn be the distance function in Hn Denote n H = H , Q = 2n + and Q′ = Q/(Q − 1) Let Ω be a bounded domain with smooth boundary in H Motivated by the Moser-Trudinger inequalities on the Heisenberg group, we study the existence of solution to a nonuniformly subelliptic equation of the form   + εh(ξ) in Ω − divH (a (ξ, ∇H u (ξ))) = f (ξ,u(ξ))   ρ(ξ)β   1,Q ,  u ∈ W (Ω)\ {0}     u ≥ in Ω ( ′) where f : Ω × R → R behaves like exp α |u|Q when |u| → ∞ In the case of Q−subLaplacian ( )  (ξ,u)  − divH |∇H u|Q−2 ∇H u = fρ(ξ)  β + εh(ξ) in Ω    1,Q ,  u ∈ W (Ω)\ {0}     u ≥ in Ω we will apply minimax methods to obtain multiplicity of weak solutions 2010 Mathematics Subject Classification 42B37, 35J92, 35J62 Key words Moser-Trudinger inequality, Heisenberg group, subelliptic equations, Q-subLaplacian, Mountain-Pass theorem, Palais-Smale sequences, existence and multiplicity of solutions ∗ Research is partly supported by a US NSF grant DMS–0901761 Author: Guozhen Lu at gzlu@math.wayne.edu † Corresponding 659 660 N Lam, G Lu, H Tang Introduction Let Ω ⊂ Rn be an open and bounded set and W01,p (Ω) ( n ≥ 2) be the completion of C0∞ (Ω) under the norm  1/p ∫ ∫    p p ∥u∥W 1,p (Ω) :=  |∇u| dx + |u| dx   Ω Ω Then the Sobolev embeddings say that np W01,p (Ω) ⊂ L n−p (Ω) if ≤ p < n n W01,p (Ω) ⊂ C 1− p (Ω) if n < p The case p = n can be seen as the limit case of these embeddings and it is known that W01,n (Ω) ⊂ Lq (Ω) for ≤ q < ∞ However, by some easy examples, we can conclude that W01,n (Ω) * L∞ (Ω) It is showed by Judovich [18], Pohozaev [38] and Trudinger [43] independently that W01,n (Ω) ⊂ ( ) Lφn (Ω) where Lφn (Ω) is the Orlicz space associated with the Young function φn (t) = exp |t|n/(n−1) − n−1 , where ωn−1 Extending this result, J Moser [37] finds the largest positive real number βn = nωn−1 is the area of the surface of the unit n−ball, such that if Ω is a domain with finite n−measure in Euclidean n−space Rn , n ≥ 2, then there is a constant c0 depending only on n such that ∫ ( n ) exp β |u| n−1 dx ≤ c0 |Ω| Ω ∫ 1,n for any β ≤ βn , any u ∈ W0 (Ω) with Ω |∇u|n dx ≤ Moreover, this constant βn is sharp in the meaning that if β > βn , then the above inequality can no longer hold with some c0 independent of u Such an inequality is nowadays known as Moser-Trudinger type inequality However, when Ω has infinite volume, the result of J Moser is meaningless In this case, the sharp Moser-Trudinger type inequality was obtained by B Ruf [39] in dimension two and Y.X Li-Ruf [29] in general dimension The Moser-Trudinger type inequalities has been extended to many different settings: high order derivatives by D Adams which is now called Adams type inequalities [1, 20, 23, 35, 40, 42]; compact Riemannian manifolds without boundary by Fontana [17] (see [28]); singular Moser-Trudinger inequalities which are the combinations of the Hardy inequalities and Moser-Trudinger inequalities are established in [3, 5, 23] It is also worthy to note that the Moser-Trudinger type inequalities play an essential role in geometric analysis and in the study of the exponential growth partial differential n equations where, roughly speaking, the nonlinearity behaves like eα|u| n−1 as |u| → ∞ Here we mention Atkinson-Peletier [6], Carleson-Chang [8], Flucher [16], Lin [30], Adimurthi et al [2, 3, 4, 5], ´ [13], de Figueiredo- O-Ruf ´ Struwe [41], de Figueiredo-Miyagaki-Ruf [12], J.M O [11], Y.X Li [26, 27], Lu-Yang [33, 34], Lam-Lu [19, 21, 22] and the references therein Now, let us discuss the Moser-Trudinger type inequalities on the Heisenberg group For some notations and preliminaries about the Heisenberg group, see the next section In the setting of the Heisenberg group, although the rearrangement argument is not available, Cohn and the second author of this paper [9] can still set up a sharp Moser-Trudinger inequality for bounded domains on the Heisenberg group: Theorem A Let H = Hn be a n−dimensional Heisenberg group, Q = 2n + 2, Q′ = )Q′ −1 ( Q −1 −1 )Γ( ) Γ(n) Then there exists a constant C0 deQ/(Q − 1), and αQ = Q 2πn Γ( 21 )Γ( Q−1 2 pending only on Q such that for all Ω ⊂ Hn , |Ω| < ∞, ∫ ′ exp(αQ |u(ξ)|Q )dξ ≤ C0 < ∞ (1.1) sup |Ω| Ω u∈W 1,Q (Ω), ∥∇H u∥ Q ≤1 L Subelliptic equations with critical growth in the Heisenberg group Hn 661 If αQ is replaced by any larger number, the integral in (1.1) is still finite for any u ∈ W 1,Q (H), but the supremum is infinite It is clear that when |Ω| = ∞, Theorem A is not meaningful In the case, we have the following version of the Moser-Trudinger type inequality (see [10]): Theorem B Let α∗ be such that α∗ = αQ /c∗ Then for any pair β, α satisfying ≤ β < Q, < α ≤ α∗ , and αα∗ + Qβ ≤ 1, there holds ∫ } ) ( { Q/(Q−1) (α, u) − S 1, the integral if further ααQ + Qβ > α α∗ |u|kQ/(Q−1) k! in (1.2) is still finite for any u ∈ W 1,Q (H), but the supremum is infinite Here, c∗ is defined as follows: Let u : H → R be a nonnegative function in W 1,Q (H), and u∗ be the decreasing rearrangement of u, namely u∗ (ξ) := sup {s ≥ : ξ ∈ {u > s}∗ } where {u > s}∗ = Br = {ξ : ρ (ξ) ≤ r} such that |{u > s}| = |Br | It is known from a result of Manfredi and V Vera De Serio [36] that there exists a constant c ≥ depending only on Q such that ∫ ∫ ∗Q |∇H u|Q dξ |∇H u | dξ ≤ c H H for all u ∈ W 1,Q (H) Thus we can define { } ∫ ∫ Q ∗ 1/(Q−1) ∗Q 1,Q c = inf c : |∇H u | dξ ≤ c |∇H u| dξ, u ∈ W (H) ≥ H (1.3) H We notice that in Theorem B, we cannot exhibit the best constant α∗ (1 − Qβ ) due to the loss of the non-optimal rearrangement argument in the Heisenberg group Nevertheless, the first two authors recently used a completely different but much simpler approach, namely a rearrangementfree argument, to set up the sharp Moser-Trudinger type inequality on Heisenberg groups in [24] Moreover, we have developed in [25] a rearrangement-free method to establish the sharp Adams and n singular Adams inequalities on high order Sobolev spaces W m, m (Rn ) with arbitrary orders (including fractional orders) This extends those results in [40] and [20, 23] in full generality The main result on sharp Moser-Trudinger inequality on the Heisenberg group proved in [24] is as follows Theorem C Let τ be any positive real number Then for any pair β, α satisfying ≤ β < Q and < α ≤ αQ (1 − Qβ ) , there holds ∫ } ) ( { (1.4) exp α |u|Q/(Q−1) − S Q−2 (α, u) < ∞ sup β ∥u∥1,τ ≤1 H ρ (ξ) When α > αQ (1 − infinite Here β Q ), the integral in (1.4) is still finite for any u ∈ W 1,Q (H), but the supremum is ∥u∥1,τ = [∫ H Q |∇H u| + τ ∫ |u| H Q ]1/Q 662 N Lam, G Lu, H Tang In this paper, we will prove the critical singular Moser-Trudinger inequality on bounded domains (see Lemma 4.1) and study a class of partial differential equations of exponential growth by using the Moser-Trudinger type inequalities on the Heisenberg group More precisely, we consider the existence of nontrivial weak solutions for the nonuniformly subelliptic equations of Q−sub-Laplacian type of the form:   − divH (a (ξ, ∇H u (ξ))) = f (ξ,u(ξ)) + εh(ξ) in Ω   ρ(ξ)β   1,Q (NU)  u ∈ W (Ω)\ {0}     u ≥ 0in Ω where Ω is a bounded domain with smooth boundary in H, ) ( |a (ξ, τ)| ≤ c0 h0 (ξ) + h1 (ξ) |τ|Q−1 ′ ∞ (Ω), ≤ β < Q, , f : Ω × R → R for any τ in RQ−2 and a.e ξ in Ω, h0 ∈ LQ (Ω) and h0 ∈ Lloc ) ( 1,Q )∗ ( ′ Q behaves like exp α |u| when |u| → ∞, h ∈ W0 (Ω) , h , and ε is a positive parameter The main features of this class of problems are that it involves critical growth and the nonlinear operator Q−sub-Laplacian type In spite of a possible failure of the Palais-Smale (PS) compactness condition, in this article we apply minimax method, in particular, the mountain-pass theorem to obtain the weak solution of (NU) in a suitable subspace of W01,Q (Ω) Moreover, in the case of Q−sub-Laplacian, i.e., a (ξ, ∇H u) = |∇H u|Q−2 ∇H u, we will apply minimax methods, more precisely, the mountain-pass theorem combined with minimization and the Ekeland variational principle, to obtain the existence of at least two weak solutions to the nonhomogeneous problem ( )  (ξ,u)  − divH |∇H u|Q−2 ∇H u = fρ(ξ)  β + εh(ξ) in Ω    1,Q (NH)  u ∈ W (Ω)\ {0}     u ≥ 0in Ω Our paper is organized as follows: In Section 2, we give some notations and preliminaries about the Heisenberg group We also provide the assumptions on the nonlinearity f in this section We will discuss the variational framework and state our main results in Section In Section 4, we proof critical singular Moser-Trudinger inequality on bounded domains and also, some basic lemmas that are useful in our paper We will investigate the existence of nontrivial solution to Eq (NU) (Theorem 3.1) in Section The last section (Section 6) is devoted to the study of multiplicity of solutions to equation (NH) (Theorems 3.2 and 3.3) Preliminaries and assumptions First, we provide some notations and preliminary results Let Hn = R2n × R be the n−dimensional Heisenberg group Recall that the Heisenberg group Hn is the space R2n+1 with the noncommutative law of product ( ) ( (⟨ ⟩ ⟨ ⟩)) (x, y, t) · x′ , y′ , t′ = x + x′ , y + y′ , t + t′ + y, x′ − x, y′ , where x, y, x′ , y′ ∈ Rn , t, t′ ∈ R, and ⟨·, ·⟩ denotes the standart inner product in Rn The Lie algebra of Hn is generated by the left-invariant vector files T= ∂ ∂ ∂ ∂ ∂ , Xi = + 2yi , Yi = − 2xi , i = 1, , n ∂t ∂xi ∂t ∂yi ∂t Subelliptic equations with critical growth in the Heisenberg group Hn 663 We will fix some notations: ( )1/4 z = (x, y) ∈ R2n , ξ = (z, t) ∈ Hn , ρ (ξ) = |z|4 + t2 , where ρ (ξ) denotes the Heisenberg distance between ξ and the origin Denote H = Hn , Q = 2n + 2, ∫ 1/(Q−1) Q αQ = QσQ , σQ = ρ(z,t)=1 |z| dµ We now use |∇H u| to express the norm of the subelliptic gradient of the function u : H → R:   n )1/2 ∑ (   (Xi u) + (Yi u)  |∇H u| =  i=1 Now, we will provide conditions on the nonlinearity of Eq (NU) and (NH) Motivated by the Moser-Trudinger inequalities (Theorems A and B), we consider here the maximal growth on the nonlinear term f (ξ, u) which allows us to treat Eq.(NU) and (NH) variationally in a suitable subspace of W01,Q (Ω) We assume that f : Ω × R → R is continuous, f (ξ, 0) = and f behaves like ) ( exp α |u|Q/(Q−1) as |u| → ∞ More precisely, we assume the following growth conditions on the nonlinearity f (ξ, u): ( f 1)There exists α0 > such that lim | f((ξ,s)|Q′ ) = uniformly on ξ ∈ Ω, ∀α > α0 and lim | f((ξ,s)|Q′ ) |s|→∞ exp α|s| |s|→∞ exp α|s| = ∞ uniformly on ξ ∈ Ω, ∀α < α0 ( f 2) There exist constant R0 , M0 > such that for all ξ ∈ Ω and s ≥ R0 , F(ξ, s) = ∫s f (ξ, τ)dτ ≤ M0 f (ξ, s) Since we are interested in nonnegative weak solutions, it is convenient to define f (ξ, u) = for all (ξ, u) ∈ Ω × (−∞, 0] We note that conditions ( f 1), ( f 2) imply that: (a) F(ξ, s) ≥ 0, ∀ (ξ, s) ∈ Ω × R (b) There is a positive constant C such that ( ) u ∀s ≥ R0 , ∀ξ ∈ Ω : F(ξ, s) ≥ C exp M0 ( f 3) There exists p > Q and s1 > such that for all ξ ∈ Ω and s > s1 , < pF(ξ, s) ≤ s f (ξ, s) Let A be a measurable function on Ω × RQ−2 such that A(ξ, 0) = and a(ξ, τ) = ∂A(ξ,τ) is a ∂τ Caratheodory function on Ω × RQ−2 Assume that there are positive real numbers c0 , c1 , k0 , k1 and ∞ (Ω) , h0 ∈ LQ/(Q−1) (Ω) , h1 (ξ) ≥ two nonnegative measurable functions h0 , h1 on Ω such that h1 ∈ Lloc for a.e ξ in Ω and the following condition holds: ) ( (A1) |a(ξ, τ)| ≤ c0 h0 (ξ) + h1 (ξ) |τ|Q−1 , ∀τ ∈ RQ−2 , a.e.ξ ∈ Ω (A2) c1 |τ − τ1 |Q ≤ ⟨a(ξ, τ) − a(ξ, τ1 ), τ − τ1 ⟩ , ∀τ, τ1 ∈ RQ−2 , a.e.ξ ∈ Ω (A3) ≤ a(ξ, τ) · τ ≤ QA (ξ, τ) , ∀τ ∈ RQ−2 , a.e.ξ ∈ Ω (A4) A (ξ, τ) ≥ k0 h1 (ξ) |τ|Q , ∀τ ∈ RQ−2 , a.e.ξ ∈ Ω Then A verifies the growth condition: ) ( A (ξ, τ) ≤ c0 h0 (ξ) |τ| + h1 (ξ) |τ|Q , ∀τ ∈ RQ−2 , a.e.ξ ∈ Ω Q ∞ (Ω) For examples of A, we can consider A (ξ, τ) = h(ξ) |τ|Q where h ∈ Lloc (2.5) 664 N Lam, G Lu, H Tang Variational framework and main results We introduce some notations: { } ∫ E = u ∈ W01,Q (Ω) : Ω h1 (ξ) |∇H u|Q dξ < ∞ ( ( ′ )) ∫1 M = lim k exp k t Q − t dt k→∞ d is the radius of the largest open ball centered at contained in Ω We notice that M is well-defined and is a real number greater than or equal to (see [13]) Under our conditions, we can see that E is a reflexive Banach space when endowed with the norm )1/Q (∫ Q h1 (ξ) |∇H u| dξ ∥u∥ = Ω Furthermore, since h1 (ξ) ≥ for a.e ξ in Ω :     Q     ∥u∥   λ1 (Q) = inf  : u ∈ E \ > for any ≤ β < Q {0}  ∫ Q   |u|     β dξ (3.6) Ω ρ(ξ) Now, from ( f 1), we obtain for all (ξ, u) ∈ Ω × R, ) ( | f (ξ, u)| , |F (ξ, u)| ≤ b3 exp α1 |u|Q/(Q−1) for some constants α1 , b3 > Thus, by the Moser-Trudinger type inequalities, we have F (ξ, u) ∈ L1 (Ω) for all u ∈ W01,Q (Ω) Define the functional E, T, Jε : E → R by ∫ E (u) = ∫ Ω A(ξ, ∇H u)dξ dξ T (u) = Ω F(ξ,u) ρ(ξ)β ∫ Jε (u) = E (u) − T (u) − ε Ω hudξ then the functional Jε is well-defined Moreover, Jε is a C functional on E with ∫ ∫ ∫ f (ξ, u)v a (ξ, ∇H u) ∇H vdξ − hvdξ, ∀u, v ∈ E DJε (u) v = dξ − ε β Ω ρ (ξ) Ω Ω We next state our main results Theorem 3.1 Suppose that (f1)-(f3) are satisfied Furthermore, assume that (f4) lim sup s→0+ F(ξ, s) < λ1 (Ω) uniformly in ξ ∈ Ω k0 |s|Q Then there exists ε1 > such that for each < ε < ε1 , (NU) has a weak solution of mountain-pass type Theorem 3.2 In addition to the hypotheses in Theorem 3.1, assume that Q−1 ) ( αQ (Q − β)Q (f5) lim s f (ξ, s) exp −α0 |s|Q/(Q−1) ≥ β0 > Q−1 Q−β s→∞ Q r MαQ−1 ωQ−1 uniformly on Ω Then, there exists ε2 > 0, such that for each < ε < ε2 , problem (NH) has at least two weak solutions and one of them has a negative energy In the case where the function h does not change sign, we have Theorem 3.3 Under the assumptions in Theorems 3.1 and 3.2, if h(ξ) ≥ (h(ξ) ≤ 0) a.e., then the solutions of problem (NH) are nonnegative (nonpositive) Subelliptic equations with critical growth in the Heisenberg group Hn 665 Some lemmas First, we will prove the following critical singular Moser-Trudinger inequality: Lemma 4.1 (The critical singular Moser-Trudinger inequality) Let H = Hn be a n−dimensional Heisenberg group,∫ Ω ⊂ Hn , |Ω| < ∞, Q = 2n + 2, Q′ = Q/(Q − 1), ≤ β < Q, and αQ = Qσ1/(Q−1) , σQ = ρ(z,t)=1 |z|Q dµ Then there exists a constant C0 depending only on Q and β such Q that ∫ exp(α (1 − β ) |u(ξ)|Q′ )dξ Q Q ≤ C0 < ∞ sup β 1− ρ (ξ)β Ω u∈W01,Q (Ω), ∥∇H u∥LQ ≤1 |Ω| Q ( ) If αQ − Qβ is replaced by any larger number, then the supremum is infinite Recall in [9] that for < α < Q, we will say that a non-negative function g on H is a kernel of ξ is a point on the unit sphere We are order α if g has the form g(ξ) = ρ (ξ)α−Q g (ξ∗ ) where ξ∗ = ρ(ξ) also assuming that for every δ > and < M < ∞ there are constants C(δ, M) such that ∫ ∫M ( ∗ ∗ −1 ) ∗ ∗ ds g ξ (sζ ) − g (ξ ) dµ (ξ ) ≤ C(δ, M) s Σδ ∗ for all ζ ∈ Σ = {ξ ∈ H : ρ (ξ) = 1}, where Σδ is the subset of the unit sphere given by } { Σδ = ξ∗ ∈ Σ : δ ≤ g (ξ∗ ) ≤ δ−1 Note that we will choose g (ξ) = result: |z|Q−1 |ξ|2Q−2 in the proof of Lemma 4.1 First, we will prove the following Lemma 4.2 Suppose g is an allowed kernel of order α, Q − pα = 0, Ω ⊂ H, |Ω| < ∞, f ∈ L p and ≤ β < Q Let Q A(g, p) = Aα (g) = ∫ ′ |g (ξ∗ )| p dµ Σ Here p′ = p/(p − 1) Then there exists a constant C0 depending only on Q and β such that [ ( ) ( f ∗g(ξ) ) p′ ] β ∫ exp A(g, p) − Q ∥ f ∥ p dµ ≤ C0 β ρ (ξ)β |Ω|1− Q Ω Proof Set u (ξ) = f ∗ g(ξ) ϕ(s) = |Ω|1/p f ∗ (|Ω| e−s )e−s/p Then ∫ f (ξ) dξ = ∫|Ω| = ∫∞ p Ω f ∗ (t) p dt ϕ(s) p ds By the Hardy-Littlewood inequality, the O’Neil’s lemma, noting that with h(ξ) = ( CQ ) Qβ , where C Q is the volume of the unit ball, we have t , ρ(ξ)β then h∗ (t) = 666 N Lam, G Lu, H Tang ∫ exp (A (g, p) (1 − β ) |u(ξ)| p′ ) Q ρ (ξ)β Ω β Q ≤ CQ ∫|Ω| ( β eA(g,p) 1− Q ) ′ u∗ (t) p β dξ dt tQ β β β Q 1− Qβ = C QQ |Ω|1− Q ∫∞ ) ) ] ( [( ( ) p′ β β A (g, p) u∗ |Ω| e−s − − s ds exp − Q Q ≤ C Q |Ω| ×   p′   |Ω|e  ∫|Ω| ∫ −s    ) ( 1    β  − p′ − p′ ∗ ∗ −s  f (z)z dz  f (z)dz + exp  − Q  p (|Ω| e ) ∞     ∫  |Ω|e−s ) ] [( ds exp − Qβ s   p′   ∫∞ ∫s (    ) w  ′    β  − p′ s/p ϕ(w)e dw + ϕ(w)  exp  − Q  pe ∞    ∫  β s Q 1− Qβ ) ] [( ds = C Q |Ω| exp − Qβ s β Q β = C Q |Ω|1− Q ∫∞ ] [ exp −F(1− β ) (s) ds Q where   p′ ) ∫∞   β β   a(s, t)ϕ(s)ds , F(1− β ) (s) = − t− 1− Q  Q Q  −∞   for < s < t    (t−s)/p′ pe for t < s < ∞ a(s, t) =     for − ∞ < s ≤ ( ) ( Using Lemma 3.1 in [23], we get our desired result Proof (of Lemma 4.1) Now, using Lemma 4.2, noting that by Theorem 1.2 in [9], we have that Q−1 ( ) |u (ξ)| ≤ σQ −1 |∇H u| ∗ g (ξ) , where g (ξ) = |ξ||z|2Q−2 , we can derive the result of Lemma 4.1 Note that the sharpness of the constant ) ( αQ − Qβ comes from the test functions in [9] Subelliptic equations with critical growth in the Heisenberg group Hn 667 Using the critical singular Moser-Trudinger inequality, we can prove the following two lemmas (see [14] and [10]): Lemma 4.3 For κ > 0, q > and ∥u∥ ≤ M = M (β, κ) with M sufficiently small, we have ∫ exp (κ |u|Q/(Q−1) ) |u|q dξ ≤ C (Q, κ) ∥u∥q ρ (ξ)β Ω Proof By Holder inequality and Lemma 4.1, we have ∫ exp (κ |u|Q/(Q−1) ) |u|q dξ ρ (ξ)β Ω (  ( ) ′ ) 1/r ∫ exp κr ∥u∥Q′ |u| Q  (∫ )1/r′ ∥u∥   r′ q  dξ ≤  dξ |u|  ρ (ξ)rβ  Ω  Ω ≤ C(Q) (∫ r′ q |u| dξ Ω )1/r′ if r > sufficiently close to Here r′ = r/(r − 1) By the Sobolev embedding, we get the result ( ) ′ Lemma 4.4 If u ∈ E and ∥u∥ ≤ N with N sufficiently small (κN Q < αQ − Qβ ), then ∫ exp (κ |u|Q/(Q−1) ) Ω ρ (ξ)β |v| dξ ≤ C (Q, M, κ) ∥v∥ s for some s > Proof The proof is similar to Lemma 4.3 We also have the following lemma (for Euclidean case, see [31]): Lemma 4.5 Let {wk } ⊂ W01,Q (Ω), ∥∇H wk ∥LQ (Ω) ≤ If wk → w , weakly and almost everywhere, exp{α|wk |Q/(Q−1) } is bounded in L1 (Ω) for ∇H wk → ∇H w almost everywhere, then ρ(ξ)β ) ( ( )−1/(Q−1) β αQ − ∥∇H w∥Q 0 small enough: < α (1 + δ) ∥∇H wk − ∇H w∥Q/(Q−1) LQ (Ω) ) ( β αQ < 1− Q Now, noting that for some C(δ) > : δ |wk |Q/(Q−1) ≤ (1 + ) |wk − w|Q/(Q−1) + C(δ) |w|Q/(Q−1) (4.7) 668 N Lam, G Lu, H Tang by Holder inequality, we have ∫ exp {α |w |Q/(Q−1) } k dξ ρ (ξ)β ∫ exp {α(1 + δ ) |w − w|Q/(Q−1) + αC(δ) |w|Q/(Q−1) } k Ω ≤ ρ (ξ)β Ω dξ } 1/q′ { { } 1/q ∫ ∫  exp αq(1 + 2δ ) |wk − w|Q/(Q−1)   exp αq′C(δ) |w|Q/(Q−1)  ≤  dξ  dξ β β ρ (ξ) ρ (ξ) Ω Ω { } 1/q ∫  exp α(1 + δ) |wk − w|Q/(Q−1)   ≤ C  dξ  ρ (ξ)β Ω where we choose q = is bounded in L1 (Ω) 1+δ 1+ 2δ and q′ = q q−1 Now, by (4.7) and Lemma 4.1, we have that exp{α|wk |Q/(Q−1) } ρ(ξ)β The existence of solution for the problem (NU) The existence of nontrivial solution to Eq (NU) will be proved by a mountain-pass theorem without a compactness condition such like the one of the (PS) type This version of the mountain-pass theorem is a consequence of the Ekeland’s variational principle First, we will check that the functional Jε satisfies the geometric conditions of the mountain-pass theorem Lemma 5.1 Suppose that ( f 1) and ( f 4) hold Then there exists ε1 > such that for < ε < ε1 , there exists ρε > such that Jε (u) > if ∥u∥ = ρε Furthermore, ρε can be chosen such that ρε → as ε → Proof From ( f 4), there exist τ, δ > such that |u| ≤ δ implies F(ξ, u) ≤ k0 (λ1 (Q) − τ) |u|Q (5.8) for all ξ ∈ Ω Moreover, using ( f 1) for each q > Q, we can find a constant C = C(q, δ) such that ( ) F(ξ, u) ≤ C |u|q exp κ |u|Q/(Q−1) (5.9) for |u| ≥ δ and ξ ∈ Ω From (5.8) and (5.9) we have ) ( F(ξ, u) ≤ k0 (λ1 (Q) − τ) |u|Q + C |u|q exp κ |u|Q/(Q−1) for all (ξ, u) ∈ Ω × R Now, by (A4), Lemma 4.3, (3.6) and the continuous embedding E ֒→ LQ (Ω), we obtain ∫ |u|Q Q (Q) (λ dξ − C ∥u∥q − ε ∥h∥∗ ∥u∥ − τ) Jε (u) ≥ k0 ∥u∥ − k0 β Ω ρ (ξ) ) ( (λ1 (Q) − τ) ≥ k0 − ∥u∥Q − C ∥u∥q − ε ∥h∥∗ ∥u∥ λ1 (Q) Thus [ ( ] ) (λ1 (Q) − τ) Q−1 q−1 − C ∥u∥ − ε ∥h∥∗ (5.10) Jε (u) ≥ ∥u∥ k0 − ∥u∥ λ1 (Q) ( ) Since τ > and q > Q, we may choose ρ > such that k0 − (λ1λ(Q)−τ) ρQ−1 − Cρq−1 > Thus, if (Q) ε is sufficiently small then we can find some ρε > such that Jε (u) > if ∥u∥ = ρε and even ρε → as ε → Subelliptic equations with critical growth in the Heisenberg group Hn 669 Lemma 5.2 There exists e ∈ E with ∥e∥ > ρε such that Jε (e) < inf Jε (u) ∥u∥=ρε Proof Let u ∈ E \ {0} , u ≥ with compact support Ω′ = supp(u) By ( f 2) and ( f 3), we have that for p > Q, there exists a positive constant C > such that ∀s ≥ 0, ∀ξ ∈ Ω : F (ξ, s) ≥ cs p − d Then by (2.5), we get ∫ ∫ Q Q p Jε (γu) ≤ Cγ h0 (ξ) |∇H u| dξ + Cγ ∥u∥ − Cγ Ω Ω (5.11) ∫ |u| p hudξ dξ + C + εγ ρ (ξ)β Ω Since p > Q, we have Jε (γu) → −∞ as γ → ∞ Setting e = γu with γ sufficiently large, we get the conclusion In studying this class of sub-elliptic problems involving critical growth, the loss of the (PS) compactness condition raises many difficulties In the following lemmas, we will analyze the compactness of (PS) sequences of Jε Lemma 5.3 Let (uk ) ⊂ E be an arbitrary (PS) sequence of Jε , i.e., Jε (uk ) → c, DJε (uk ) → in E ′ as k → ∞ Then there exists a subsequence of (uk ) (still denoted by (uk )) and u ∈ E such that  f (ξ,uk ) (ξ,u)  → fρ(ξ)  β  ρ(ξ)β      ∇H uk (ξ) → ∇H u(ξ)     a (ξ, ∇H uk ) ⇀ a (ξ, ∇H u)     u ⇀u k (Ω) strongly in Lloc almost everywhere in Ω ( Q/(Q−1) )Q−2 (Ω) weakly in Lloc weakly in E Furthermore u is a weak solution of (NU) In order to prove this lemma, we need the following two lemmas that can be found in [10], [13], [14] and [32]: Lemma 5.4 Let Br (ξ∗ ) be a Heisenberg ball centered at (ξ∗ ) ∈ Ω with radius r Then there exists a positive ε0 depending only on Q such that ∫ ) ( sup exp ε0 |u|Q/(Q−1) dξ ≤ C0 ∫ ∫ ∗ |Br (ξ )| Br (ξ∗ ) Q ∗ |∇H u| dξ≤1, ∗ udξ=0 Br (ξ ) Br (ξ ) for some constant C0 depending only on Q Lemma 5.5 Let (uk ) in L1 (Ω) such that uk → u in L1 (Ω) and let∫ f be a continuous function Then (ξ,u) f (ξ,uk (ξ)) f (ξ,uk ) k (ξ))uk (ξ)| → fρ(ξ) ∈ L1 (Ω) ∀k and Ω | f (ξ,uρ(ξ) dξ ≤ C1 β in L (Ω), provided that β ρ(ξ)β ρ(ξ)β Now we are ready to prove Lemma 5.3 Proof The proof is similar to Lemma 3.4 in [10] For the completeness, we sketch the proof here By the assumption, we have ∫ ∫ ∫ F(ξ, uk ) k→∞ A(ξ, ∇H uk )dξ − huk dξ → c (5.12) dξ − ε β (ξ) ρ Ω Ω Ω 670 and N Lam, G Lu, H Tang ∫ ∫ ∫ f (ξ, uk )v hvdξ ≤ τk ∥v∥ dξ − ε a (ξ, ∇H uk ) ∇H vdξ − β Ω Ω ρ (ξ) Ω (5.13) for all v ∈ E, where τk → as k → ∞ Choosing v = uk in (5.13) and by (A3), we get ∫ ∫ ∫ f (ξ, uk )uk A (ξ, ∇H uk ) ≤ τk ∥uk ∥ hu dξ − Q dξ + ε k ρ (ξ)β Ω Ω Ω This together with (5.12), ( f 3) and (A4) leads to ( ) p − ∥uk ∥Q ≤ C (1 + ∥uk ∥) Q and hence ∥uk ∥ is bounded and thus ∫ ∫ F(ξ, uk ) f (ξ, uk )uk dξ ≤ C, dξ ≤ C β β ρ (ξ) Ω ρ (ξ) Ω (5.14) Note that the embedding E ֒→ Lq (Ω) is compact for all q ≥ 1, by extracting a subsequence, we can assume that uk → u weakly in E and for almost all ξ ∈ Ω Thanks to Lemma 5.5, we have f (ξ, uk ) f (ξ, u) → in L1 (Ω) β ρ (ξ) ρ (ξ)β (5.15) Now, similar to that in [10], up to a subsequence, we define an energy concentration set for any fixed δ > 0, { } ∫ ) ( Q ′ Q Σδ = ξ ∈ Ω : lim lim |uk | + |∇H uk | dξ ≥ δ r→0k→∞ Br (ξ) Since (uk ) is bounded in E, Σδ must be a finite set For any ξ∗ ∈ Ω r Σδ , there exist r : < r < dist (ξ∗ , Σδ ) such that ∫ ) ( lim |uk |Q + |∇H uk |Q dξ < δ k→∞ so for large k : ∫ Br (ξ∗ ) Br (ξ∗ ) ( ) |uk |Q + |∇H uk |Q dξ < δ By results in [10], we can prove that ∫ | f (ξ, uk )| |uk − u| dξ ∗ ρ (ξ)β Br (ξ ) 1/q′ f (ξ, uk ) ≤ ∥uk − u∥Lq′ s′ ≤ C ∥uk − u∥Lq′ s′ → ρ (ξ)β/q q ρ (ξ)β Ls L and for any compact set K ⊂⊂ Ω \ Σδ , ∫ | f (ξ, uk ) uk − f (ξ, u) u| lim dξ = k→∞ K ρ (ξ)β So now, we will prove that for any compact set K ⊂⊂ Ω \ Σδ , ∫ lim |∇H uk − ∇H u|Q dξ = k→∞ K (5.16) (5.17) (5.18) (5.19) Subelliptic equations with critical growth in the Heisenberg group Hn 671 It is enough to prove for any ξ∗ ∈ Ω \ Σδ , and r given by (5.16), there holds ∫ lim |∇H uk − ∇H u|Q dξ = k→∞ (5.20) Br/2 (ξ∗ ) For this purpose, we take ϕ ∈ C0∞ (Br (ξ∗ )) with ≤ ϕ ≤ and ϕ = on Br/2 (ξ∗ ) Obviously ϕuk is a bounded sequence Choose v = ϕuk and v = ϕu in (5.13), we have: ∫ ϕ (a (ξ, ∇H uk ) − a (ξ, ∇H u)) (∇H uk − ∇H u) dξ Br (ξ∗ ) ∫ a (ξ, ∇H uk ) ∇H ϕ (u − uk ) dξ ≤ B (ξ∗ ) ∫ r ∫ f (ξ, uk ) dξ + ϕa (ξ, ∇H u) (∇H u − ∇H uk ) dξ + ϕ (uk − u) ∗ ∗ ρ (ξ)β Br (ξ ) Br (ξ ) ∫ ϕh (uk − u) dξ + τk ∥ϕuk ∥ + τk ∥ϕu∥ + ε Br (ξ∗ ) Note that by Holder inequality and the compact embedding of E ֒→ LQ (Ω), we get ∫ a (ξ, ∇H uk ) ∇H ϕ (u − uk ) dξ = lim k→∞ Since ∇H uk ⇀ ∇H u and uk ⇀ u, there holds ∫ ∫ lim ϕa (ξ, ∇H u) (∇H u − ∇H uk ) dξ = and lim k→∞ (5.21) Br (ξ∗ ) k→∞ Br (ξ∗ ) ϕh (uk − u) dξ = (5.22) Br (ξ∗ ) The Holder inequality and (5.17) implies that ∫ ϕ (uk − u) f (ξ, uk ) dξ = lim k→∞ So we can conclude that lim k→∞ ∫ Br (ξ∗ ) ϕ (a (ξ, ∇H uk ) − a (ξ, ∇H u)) (∇H uk − ∇H u) dξ = Br (ξ∗ ) and hence we get (5.20) by (A2) So we have (5.19) by a covering argument Since Σδ is finite, it follows that ∇H uk converges to ∇H u almost everywhere This immediately implies, up to a subse)Q−2 ( Q/(Q−1) (Ω) Let k tend to infinity in (5.13) and quence, a (ξ, ∇H uk ) ⇀ a (ξ, ∇H u) weakly in Lloc combine with (5.15), we obtain ⟨DJε (u), h⟩ = ∀h ∈ C0∞ (Ω) This completes the proof of the Lemma 5.1 The proof of Theorem 3.1 Proposition 5.1 Under the assumptions (f1)-(f4), there exists ε1 > such that for each < ε < ε1 , the problem (NU) has a solution u M via mountain-pass theorem Proof For ε sufficiently small, by Lemma 5.1 and 5.2, Jε satisfies the hypotheses of the mountainpass theorem except possibly for the (PS) condition Thus, using the mountain-pass theorem without the (PS) condition, we can find a sequence (uk ) in E such that Jε (uk ) → c M > and ∥DJε (uk )∥ → where c M is the mountain-pass level of Jε Now, by Lemma 5.3, the sequence (uk ) converges weakly to a weak solution u M of (NU) in E Moreover, u M , since h , 672 N Lam, G Lu, H Tang The multiplicity results to the problem (NH) In this section, we study the problem (NH) Note that Eq (NH) is a special case of the problem (NU) where |τ|Q , Q )1/Q (∫ |∇H u|Q dξ ∥u∥ = A (ξ, τ) = Ω As a consequence, there exists a nontrivial solution of standard ”mountain-pass” type as in Theorem 3.1 Now, we will prove the existence of the second solution Lemma 6.1 There exists η > and v ∈ E with ∥v∥ = such that Jε (tv) < for all < t < η In particular, inf Jε (u) < ∥u∥≤η Proof Let v ∈ E be a solution of the problem ( ) { − divH |∇H v|Q−2 ∇H v = h in Ω v = on ∂Ω ∫ Then, for h , 0, we have Ω hv = ∥v∥Q > Moreover, d Jε (tv) = t Q−1 ∥v∥Q − dt ∫ Ω f (ξ, tv) v dξ − ε ρ (ξ)β ∫ hvdξ Ω for t > Since f (ξ, 0) = 0, by continuity, it follows that there exists η > such that for all < t < η and thus Jε (tv) < for all < t < η since Jε (0) = Next, we define the Moser Functions (see [10, 19]):        ek (ξ, r) = 1/Q  m   σQ    ( log k log )(Q−1)/Q r ρ(ξ) 1/Q (log k) if ρ(ξ) ≤ if r k d dt Jε (tv) and β0 > be such that ( lim s f (ξ, s) exp −α0 |s| s→∞ Q′ ) Q−1 αQ (Q − β)Q ≥ β0 > Q−1 Q−β Q r MαQ−1 ωQ−1 Subelliptic equations with critical growth in the Heisenberg group Hn 673 ek (ξ, r) Then we have uniformly for almost every ξ ∈ Ω and B (0, r) ⊂ Ω Let mk (ξ) = m mk ∈ W01,Q (Br ) and ∥mk ∥ = Suppose, by contradiction, that for all k we get { tQ − max t≥0 Q ∫ } ( )Q−1 F (ξ, tmk ) Q − β αQ dξ ≥ Q Q α0 ρ (ξ)β Ω For each k, we can find tk > such that tkQ − Q Thus ∫ Ω } { Q ∫ t F (ξ, tk mk ) F (ξ, tmk ) dξ = max dξ − t≥0 Q ρ (ξ)β ρ (ξ)β Ω tkQ − Q ∫ Ω ( )Q−1 F (ξ, tk mk ) Q − β αQ dξ ≥ Q Q α0 ρ (ξ)β From F(ξ, u) ≥ 0, we obtain tkQ ≥ Since at t = tk we have ( Q − β αQ Q α0 )Q−1 (6.24) ) ( ∫ F (ξ, tmk ) d tQ dξ = − dt Q ρ (ξ)β Ω it follows that tkQ = ∫ Ω tk mk f (ξ, tk mk ) dξ = ρ (ξ)β ∫ ρ(ξ)≤r tk mk f (ξ, tk mk ) dξ ρ (ξ)β (6.25) Using hypothesis ( f 5), given τ > there exists Rτ > such that for all u ≥ Rτ and ρ (ξ) ≤ r, we have ) ( (6.26) u f (ξ, u) ≥ (β0 − τ) exp α0 |u|Q/(Q−1) From (6.25) and (6.26), for large k, we obtain ) ( ∫ exp α0 |tk mk |Q/(Q−1) Q dξ tk ≥ (β0 − τ) ρ (ξ)β ρ(ξ)≤ kr ( ) ωQ−1 ( r )Q−β = (β0 − τ) exp α0 tkQ/(Q−1) σ−1/(Q−1) log k Q Q−β k ( ) ωQ−1 Q−β α0 Q/(Q−1) r exp tk Q log k − Q log k + β log k = (β0 − τ) Q−β αQ Thus, setting Lk = α0 Q log k Q/(Q−1) tk − Q log tk − (Q − β) log k αQ we have ≥ (β0 − τ) ωQ−1 Q−β r exp Lk Q−β Consequently, the sequence (tk ) is bounded Moreover, by (6.24) and tkQ      α0 tkQ/(Q−1)  ωQ−1 Q−β r exp Q − (Q − β) log k ≥ (β0 − τ) Q−β αQ 674 N Lam, G Lu, H Tang it follows that ( k→∞ tkQ → Q − β αQ Q α0 )Q−1 (6.27) Set Ak = {ξ ∈ Br : tk mk ≥ Rτ } and Bk = Br \ Ak From (6.25) and (6.26) we have tkQ ≥ (β0 − τ) ∫ − (β0 − τ) ∫ ρ(ξ)≤r ) ( exp α0 |tk mk |Q/(Q−1) β ( ρ (ξ) exp α0 |tk mk | Bk Q/(Q−1) ρ (ξ)β ) dξ + ∫ Bk tk mk f (ξ, tk mk ) dξ ρ (ξ)β (6.28) dξ Notice that mk (ξ) → and the characteristic functions χBk → for almost everywhere ξ in Br Therefore the Lebesgue dominated convergence theorem implies ∫ tk mk f (ξ, tk mk ) dξ → ρ (ξ)β Bk and ∫ ) ( exp α0 |tk mk |Q/(Q−1) ρ (ξ) Bk β Moreover, using ( k→∞ tkQ → ≥ dξ → Q − β αQ Q α0 ωQ−1 Q−β r Q−β )Q−1 , we have ∫ ρ(ξ)≤r ≥ = ∫ ∫ ) ( exp α0 |tk mk |Q/(Q−1) ρ (ξ)β exp ρ(ξ)≤r ( Q−β Q exp ρ(ξ)≤r/k dξ αQ |mk |Q/(Q−1) ρ (ξ)β ( Q−β Q ) dξ ) αQ |mk |Q/(Q−1) ρ (ξ)β dξ + ∫ exp r/k≤ρ(ξ)≤r ( Q−β Q αQ |mk |Q/(Q−1) ρ (ξ)β ) dξ and ∫ ρ(ξ)≤r/k exp ( Q−β Q αQ |mk |Q/(Q−1) ρ (ξ)β ) [ ] Q−β −1/(Q−1) exp αQ σQ log k dξ dξ = Q ρ(ξ)≤r/k ωQ−1 ( r )Q−β (Q−β) k = Q−β k ωQ−1 Q−β = r Q−β ∫ Now, using the change of variable log ζ= ( ) r s log k Subelliptic equations with critical growth in the Heisenberg group Hn by straightforward computation, we have ) ( ∫ Q/(Q−1) α exp Q−β | |m k Q Q ρ (ξ)β r/k≤ρ(ξ)≤r = ωQ−1 r Q−β log k ∫1 which converges to obtain ωQ−1 Q−β M Q−β r ( 675 dξ [ ( )] exp (Q − β) log k ζ Q/(Q−1) − ζ dζ as k → ∞ Finally, taking k → ∞ in (6.28) and using (6.27), we Q − β αQ Q α0 )Q−1 ≥ (β0 − τ) ωQ−1 r Q−β M Q−β which implies that β0 ≤ Q−1 αQ (Q − β)Q , QQ−1 r Q−β MαQ−1 ωQ−1 but it is a contradiction and the proof is complete Corollary 6.1 Under the hypotheses (f1)-(f5), if ε is sufficiently small then } )Q−1 ( { Q ∫ ∫ F (ξ, tmk ) Q − β αQ t εhmk dξ < − dξ − t max Jε (tmk ) = max t≥0 t≥0 Q Q Q α0 ρ (ξ)β Ω Ω ∫ Proof Since Ω εhmk dξ ≤ ε ∥h∥∗ , taking ε sufficiently small, the result follows Note that we can conclude by inequality (5.10) and Lemma 6.1 that −∞ < cε = inf Jε (u) < (6.29) ∥u∥≤ρε Next, we will prove that this infimum is achieved and generate a solution In order to obtain convergence results, we need to improve the estimate of Lemma 6.2 Corollary 6.2 Under the hypotheses (f1)-(f5), there exist ε2 ∈ (0, ε1 ] and u ∈ W01,Q (Ω) with compact support such that for all < ε < ε2 , ( )Q−1 Q − β αQ for all t ≥ Jε (tu) < cε + Q Q α0 ε→0 Proof It is possible to raise the infimum cε by reducing ε By Lemma 5.1, ρε → Consequently, ε→0 cε → Thus there exists ε2 > such that if < ε < ε2 then, by Corollary 6.1, we have )Q−1 ( Q − β αQ max Jε (tmk ) < cε + t≥0 Q Q α0 Taking u = mk ∈ W01,Q (Ω), the result follows Lemma 6.3 If (uk ) is a (PS) sequence for Jε at any level with lim inf ∥uk ∥ < N k→∞ (6.30) with N sufficiently small, then (uk ) possesses a subsequence which converges strongly to a solution u0 of (NH) 676 N Lam, G Lu, H Tang Proof Extracting a subsequence of (uk ) if necessary, we can suppose that lim inf ∥uk ∥ = lim ∥uk ∥ < N k→∞ k→∞ By Lemma 5.3, we have that uk ⇀ u0 where u0 is a weak solution of (NH) Writing uk = u0 + wk , it follows that wk ⇀ in E Thus wk → in Lq (Ω) for all ≤ q < ∞ Using the Brezis-Lieb Lemma in [7], we have (6.31) ∥uk ∥Q = ∥u0 ∥Q + ∥wk ∥Q + ok (1) We claim that ∫ Ω f (ξ, uk )u0 k→∞ dξ → ρ (ξ)β ∫ Ω f (ξ, u0 )u0 dξ ρ (ξ)β (6.32) In fact, using u0 ∈ E, given τ > 0, there exists φ ∈ C0∞ (Ω) such that ∥φ − u0 ∥ < τ We have that ∫ ∫ ∫ f (ξ, uk ) (u0 − φ) f (ξ, uk )u0 f (ξ, u0 )u0 dξ − dξ ≤ dξ Ω ρ (ξ)β ρ (ξ)β ρ (ξ)β Ω Ω ∫ ∫ f (ξ, u0 ) (u0 − φ) | f (ξ, uk ) − f (ξ, u0 )| + ∥φ∥∞ dξ + dξ β ρ (ξ) ρ (ξ)β suppφ Ω Since |DJε (uk ) (u0 − φ)| ≤ τk ∥u0 − φ∥ with τk → 0, we have ∫ f (ξ, uk ) (u0 − φ) Q−1 dξ ∥u0 − φ∥ ≤ τk ∥u0 − φ∥ + ∥∇H uk ∥ β (ξ) ρ Ω + ∥εh∥∗ ∥u0 − φ∥ ≤ C ∥u0 − φ∥ < Cτ, where C is independent of k and τ Similarly, using that DJε (u0 ) (u0 − φ) = 0, we have ∫ f (ξ, u0 ) (u0 − φ) dξ < Cτ ρ (ξ)β Ω Since f (ξ,uk ) ρ(ξ)β → f (ξ,u0 ) ρ(ξ)β strongly in L1 (Ω) and by the previous inequalities, we conclude that ∫ ∫ f (ξ, u0 )u0 f (ξ, uk )u0 dξ − dξ < 2Cτ lim k→∞ Ω ρ (ξ)β ρ (ξ)β Ω and this shows the convergence (6.32) because τ is arbitrary From (6.31) and (6.32), we can write ∫ f (ξ, uk ) wk Q dξ + ok (1) DJε (uk ) uk = DJε (u0 ) u0 + ∥wk ∥ − ρ (ξ)β Ω that is ∥wk ∥Q = ∫ Ω f (ξ, uk ) wk dξ + ok (1) ρ (ξ)β From ( f 1), Lemma 4.1, Lemma 4.4 and the compact embedding E ֒→ Lt (Ω) for t ≥ 1, we have ∫ f (ξ, uk ) wk dξ → ρ (ξ)β Ω from which ∥wk ∥ → and the result follows Subelliptic equations with critical growth in the Heisenberg group Hn 677 6.1 Proof of Theorem 3.2 The proof of the existence of the second solution of (NH) follows by a minimization argument and Ekeland’s variational principle Proposition 6.1 There exists ε2 > such that for each ε with < ε < ε2 , Eq (NH) has a minimum type solution u0 with Jε (u0 ) = cε < 0, where cε is defined in (6.29) Proof Let ρε , N be as in Lemma 5.1 and Lemma 6.3 Note that we can choose ε2 > sufficiently small such that ρε < N Since Bρε is a complete metric space with the metric given by the norm of E, convex and the functional Jε is of class C and bounded below on Bρε , by the Ekeland’s variational principle there exists a sequence (uk ) in Bρε such that Jε (uk ) → cε = inf Jε (u) and ∥DJε (uk )∥ → ∥u∥≤ρε Observing that ∥uk ∥ ≤ ρε < N by Lemma 6.3, it follows that there exists a subsequence of (uk ) which converges to a solution u0 of (NH) Therefore, Jε (u0 ) = cε < Remark 6.1 By Corollary 6.2, we can conclude that )Q−1 ( Q − β αQ < c M < cε + Q Q α0 Proposition 6.2 If ε2 > is enough small, then the solutions of (NH) obtained in Propositions 5.1 and 6.1 are distinct Proof By Proposition 5.1 and 6.1, there exist sequences (uk ), (vk ) in E such that uk → u0 , Jε (uk ) → cε < 0, DJε (uk ) uk → and vk ⇀ u M , Jε (vk ) → c M > 0, DJε (vk ) vk → 0, ∇H vk (ξ) → ∇H u M (ξ) a.e Ω Now, suppose by contradiction that u0 = u M Then we have that vk u0 As in the proof of Lemma 5.3 we obtain f (ξ, vk ) f (ξ, u0 ) (6.33) → in L1 (Ω) β ρ (ξ) ρ (ξ)β From this, we have by ( f 2), ( f 3) and the Generalized Lebesgue’s Dominated Convergence Theorem: F(ξ, vk ) F(ξ, u0 ) → in L1 (Ω) β β (ξ) (ξ) ρ ρ Now, we have that Jε (uk ) = ∥uk ∥Q − Q ∫ ∥vk ∥Q − Q ∫ and Jε (vk ) = Ω Ω F(ξ, u0 ) −ε ρ (ξ)β F(ξ, u0 ) −ε ρ (ξ)β ∫ ∫ hu0 + o(1) = cε + o(1) Ω hu0 + o(1) = c M + o(1) Ω 678 N Lam, G Lu, H Tang which implies ∥uk ∥Q − ∥vk ∥Q → Q (cε − c M ) < (6.34) Also, since (uk ) , (vk ) are both bounded Palais-Smale sequences, we have ∫ ∫ f (ξ, uk ) Q uk − ε huk → DJε (uk ) uk = ∥uk ∥ − β Ω Ω ρ (ξ) ∫ ∫ f (ξ, vk ) DJε (vk ) vk = ∥vk ∥Q − hvk → v −ε β k Ω ρ (ξ) Ω and then ( Q Q ) ∫ [ ( ) ] f (ξ, uk ) f (ξ, uk ) − f (ξ, vk ) (uk − vk ) + vk ρ (ξ)β ρ (ξ)β ∥uk ∥ − ∥vk ∥ − Ω ∫ [h (uk − u0 ) − h (vk − u0 )] −ε Ω tends to as k → ∞ But it is clear that ∫ [h (uk − u0 ) − h (vk − u0 )] → (6.35) Ω ( )∗ since h ∈ W01,Q (Ω) and uk → u0 , vk ⇀ u0 in W01,Q (Ω) Also, notice that ∫ f (ξ, uk ) (uk − vk ) → β Ω ρ (ξ) (6.36) Indeed, let wk = vk − u0 Thus wk ⇀ and lim ∥wk ∥ > since wk Again, using Holder k→∞ inequality, Lemma 4.1 and Theorem A, note that ∥uk ∥ is small, we have ∫ ∫ ( )q 1/q ∫ 1/q′ f (ξ, uk ) f (ξ, uk ) ′ q (uk − vk ) ≤ dξ (uk − vk ) β Ω ρ (ξ)β Ω Ω ρ (ξ) ≤ C ∥uk − vk ∥q′ → So, it remains to show that ∫ ( Ω ) f (ξ, uk ) − f (ξ, vk ) vk dξ → ρ (ξ)β (6.37) The left hand side of (6.37) can be written as ) ) ∫ ( ∫ ( f (ξ, uk ) − f (ξ, vk ) f (ξ, uk ) − f (ξ, vk ) u dξ + wk dξ ρ (ξ)β ρ (ξ)β Ω Ω Arguing as in Lemma 6.3, we can conclude that ) ∫ ( f (ξ, uk ) − f (ξ, vk ) u0 dξ → ρ (ξ)β Ω Now, again by Holder inequality, Theorem A and the Sobolev embedding, we get ∫ f (ξ, uk ) wk dξ ≤ C ∥wk ∥q′ → β Ω ρ (ξ) So now, we just need to prove ∫ Ω f (ξ, uk ) wk dξ → ρ (ξ)β (6.38) (6.39) (6.40)