Compact embedding of a degenerate Sobolev space and existence of entire solutions to a semilinear equation for a Grushin-type operator

Compact Embedding of a Degenerate Sobolev Space and Existence of Entire Solutions to a Semilinear

Equation for a Grushin-type Operator.

FRANCESCA LASCIALFARI - DAVID PARDO(*)

ABSTRACT - We establish a compactness embedding result for suitable Sobolev subspaces naturally arising in the study of a Grushin-type operator inR^N. As an application, we study the solvability of a semilinear problem involving the above operator and a subcritical nonlinear term.

1. Introduction.

It is well known that, under a variational point of view, every positive solution of the problem

./

´

2Du4f(x,u)

u_N¯V40 f non linear (1)

on a bounded domainV’R^N, is a critical point of an Euler-type function- al associated to (1).

Under a set of assumptions on the functionf, such a critical point re- ally exists so that equation (1) is solvable. We point out that a similar re- sult relies on compact embeddings for Sobolev spaces, namely the Rel- lich-Kondrachov Theorem. This approach does not work in general ifV is unbounded, say V4R^N, because of the lack of compactness of the above embeddings. Nevertheless, the geometry of the Laplace operator

(*) Indirizzo degli AA.: Dipartimento di Matematica, Università degli Studi di Bologna - P.zza Porta S. Donato 5, 40127 Bologna, Italy. E-mail: lasciaHdm.uni- bo.it, pardoHdm.unibo.it

and the invariance ofR^Nunder rotations suggest to study the problem in a restricted functional setting.

For instance, Béréstycki and Lions in [1] studied problem (1) onR^N, with a functionfwhich depends onuandNxN. The symmetry offwith re- spect to NxN plays a crucial role since it allows to recover compactness even if the domain is unbounded. The use of subspaces of spherical func- tions was first introduced by Strauss in [9] where equation (1) is studied in the whole space.

The aim of the present paper is to establish a similar result in a de- generate elliptic case.

Precisely, for lD0 we study the following problem:

. /´

2D_Gu1lu4u^q21 uD0

on R^N on R^N (2)

where DG stands for the operator:

D_G4D_x1NxN^2aD_y, aD0 , xRⁿ, yR^m, n1m4N, nF2 , the power q is superlinear and subcritical for DG, that is 2EqE2_G*4 4_Q^2Q

22, andQ4n1(a11 )m. In the sequel we shall refer toD_Gas the Grushin operator.

The Dirichlet problem for equation (2) has been investigated by Tri in [11] in the case of starshaped bounded subsets ofR²and a zero bound- ary data. Among the authors who faced similar matters for degenerate operators we mention S. Biagini. In her work [2] she dealt with a semi- linear problem involving the Heisenberg operator and used a technique that can be fitted in our context.

Now let us make some remarks and introduce useful notations.

The Grushin operatorD_Gcan be written as the divergence of amodi- fied gradient, namely

DG4div (˜_G), ˜_G4(˜_x,NxN^a˜_y), (x,y)Rⁿ3R^m. Moreover, if we denote with A the N3N matrix

A(x)4

g

h

an easy computation shows that

˜_G4A(x)

u

y

v

(3)

If V’R^N, we define the Sobolev space

S^{1 , 2(V)}4 ]uL²(V) :˜_GuL²(V)(. We remark that S^{1 , 2}(V) endowed with the inner product

au,vbS^{1 , 2}(V)4

V

(uv1˜GuQ˜Gv)dx dy is a Hilbert space. Besides, if V is bounded, the embedding

S^{1 , 2(V)%}_K^Lq(V) (4)

is compact if 2EqE_Q^2Q

22, whereas the embedding S^{1 , 2(RN)%}_K^Lq(RN) (5)

for every q

k

l

We introduce the following closed subspaces of S^{1 , 2(RN):}

S_{4 ]}^uS^{1 , 2(RN) :u(x,y)}₄^W(_N^x_N^,y)₍^, Sr4 ]uS^{1 , 2(RN) :u(x,y)}4W(NxN,NyN)(, and the cone

V^»_{4 ]}^uSr(R^N) :W is non-increasing inNyN(

for which the following trivial inclusions holds:

V^%Sr%S^%S^{1 , 2(RN) .}

We stress that the requested cylindrical symmetry in the definition ofSr

is suggested by the structure of the Grushin operator.

DEFINITION 1.1. A function uS^{1 , 2(RN)}will be called a weak sol- ution of (2) if it satisfies the following identity:

R^N

˜_GuQ˜_Gf1l

R^N

uf4

R^N

u^q21f for every choice of fC₀^Q(R^N).

We are ready to state the main results of this paper:

THEOREM 1.1 (COMPACTEMBEDDING).If 2EqE2_G* ,then the restric- tion to the cone V of embedding S^{1 , 2}(R^N)^%KL^q(R^N) is compact.

THEOREM 1.2 (EXISTENCE AND REGULARITY). Let q]2 , 2_G* [. Then there exists a weak solution uC_loc^u (R^N)OC_loc^{2 ,u}(Rⁿ0]0( 3R^m) of prob- lem (2).

Moreover, u is radially symmetric with respect to each group of variables, that is u(x,y)4u(NxN,NyN).

The plan of the paper is the following.

In section 2 we prove Theorem 1.1 mainly exploiting a decay Lemma for functions belonging to V^.

In section 3 we deal with problem (2). The existence of a positive sol- ution will be achieved by applying Theorem 1.1 and rearrangements techniques. We prove also the regularity properties for the solution.

2. Compact embedding.

The purpose of this section is the proof of Theorem 1.1. We expect that a good decay of functions at infinity may help to recover compact- ness of embeddings (5), although the domain is unbounded. The struc- ture of V assures the following result:

LEMMA 2.1. Let nF2 , mF1 , and suppose uV. The following pointwise estimate holds:

u(x,y)G k_N NxN

n21 2 NyN

m 2

VuV_L1 /22(R^N)V˜_xuV_L1 /22(R^N)

where k_N is a dimensional constant.

PROOF. If uV, there exists a two-variable function W such that u(x,y)4W(NxN,NyN). Sinceuis non-increasing with respect toNyN, for a fixed sD0 we have

W(s,t)FW(s,t) (t[ 0 , t] .

We now multiply both terms by t^m21 and integrate over [ 0 , t].

Then:

0 t

W(s,t)t^m21dtF

0 t

W(s,t)t^m21dt4 t^m

m W(s,t) . (6)

Furthermore, an easy computation shows that d

ds

y

u

0 t

W(s,t)t^m21dt

v

z

F2sⁿ²¹

0 t

W(s,t)t^m21dtQ

0 t

¯W

¯s(s,t)t^m21dt. Let sD0; then, by integrating with respect to s[s,1Q] we ob- tain:

sⁿ²¹

u

0 t

W(s,t)t^m21dt

v

G2

s 1Q

N

0 t

W(s,t)t^m21dt

N

N

0 t

¯W

¯s(s,t)t^m21dt

N

G2

0 1Q

N

0 t

W(s,t)t^m21s

n21

2 dt

N

N

0 t

¯W

¯s(s,t)t^m21s

n21

2 dt

N

Apply Hölder inequality in the ds-integral:

sⁿ²¹

u

0 t

W(s,t)t^m21dt

v

^y

0 1Q

u

0 t

W(s,t)t^m21dt

v

^z

3

y

0 1Q

u

0 t¯W

¯s (s,t)t^m21dt

v

z

and the same in the dt-integral:

sⁿ²¹

u

0 t

W(s,t)t^m21dt

v

G2

y

0 1Q

u

0 t

W²(s,t)t^m21dt

vu

0 t

t^m21dt

v

z

3

y

0 1Q

u

0

t

g

h

v u

0 t

t^m21dt

v

z

Then we get:

sⁿ²¹

u

0 t

W(s,t)t^m21dt

v

By extracting the square roots of both members, and taking in account of (6), we find:

W(s,t)G k_m NsN

n21 2 NtN

m 2

VWV_L1 /22(R^N)

QV˜_xWV_L1 /22(R^N). r

REMARK 2.1. Suppose that the function W is non-increasing with respect to its first variable, that is u non-increasing in NxN instead of NyN. If nF1 ,mF2 , the same argument of the proof of the previous Lemma leads to the following estimate:

W(s,t)G k_n NsN

n1a 2 NtN

m21 2

VWV_L1 /22(R^N)

QVNxN^a˜_yWV_L1 /22(R^N).

We point out that in this case we need the additional hypotheses aEⁿ

2.

In order to prove compactness of embedding V^%_K^{Lq(RN) with q} ]2 , 2*_G[ we must show that every bounded sequence inVis precompact.

We divide R^N into suitable subsets Q_i, i41 ,R, 4 , defined as fol- lows:

Q₁»4 ](x,y)Rⁿ3R^m: NxN GR; NyN GR( Q₂»4 ](x,y)Rⁿ3R^m: NxN GR; NyN FR( Q3»4 ](x,y)Rⁿ3R^m: NxN FR; NyN GR( Q4»4 ](x,y)Rⁿ3R^m: NxN FR; NyN FR( where R is a constant to be chosen later.

The convergence of a bounded sequence in Q1 (enventually up to a subsequence) readily comes from embedding (4); in Q4 it is a conse- quence of Lemma 2.1. The main difficulties are represented by the par- ticular unboundedness of Q₂ and Q₃, where one variable is arbitrarily large.

For overcoming this obstacle, we shall take advantage of a technique, used by P. L. Lions in [7], that will be explained during the proof.

PROOF OF THEOREM 1.1. Let (uk)kF1 be a bounded sequence in V. Then there existsu0L^qand a subsequence, that we still denote by (u_k), such that:

u_kKu₀

u_k(x,y)Ku₀(x,y)

in L_loc^q OL² a.e. in R^N. Remark that

R^N

Nuk2u0N^q4

!

j41 4

Ij where Ij»4

Qj

Nu_k2u₀N^q. Due to embedding (4), we get I1K0 as kK 1Q.

About I₄ we write:

I44

Q4

Nu_k2u₀N^qGC_q

u

Q4

Nu_kN^q22Nu_kN²1

Q4

Nu₀N^q22Nu₀N²

v

By Lemma 2.1 one gets:

I₄GC_q C

R^{(q22 )(n212 1m2)}

u

Q₄

Nu_kN²1

Q₄

Nu₀N²

v

whereC is a constant depending only on the norms of the functions u_k. Then I₄K0 as RK Q, uniformly in kN.

It remains to prove the convergence of I2and I3 to 0 askK Q. For this purpose we define

fk(x)»4

NyN FR

Nu_k(x,y)N^qdy.

The sequence (f_k)_kF1is bounded inL¹(B), whereB»4 ]NxNGR(%Rⁿ; indeed the sequence (u_k)_kF1 is bounded in L^q. In addition, Lemma 2.1 implies that

NyNFR

Nu_k(x,y)N^qdyK

NyN FR

Nu₀(x,y)N^qdy4: f(x)

as kK Q, by compact embedding in L_loc^q . Then f_k(x)Kf(x) almost everywhere. As B is a compact set, the sequencefk converges to f in L¹(B). Moreover:

V˜_xf_kV_L1(B)4

NxN ER

N˜_xf_k(x)Ndx4

Q₂

N˜_x(u_k(x,y)N)^qdxdyG

Gq

Q₂

Nu_k(x,y)N^q21N˜_xu_k(x,y)Ndx dy.

Apply Hölder inequality and get:

V˜_xfkV_L1(B)GCV˜_xu_kV_L2(Q₂)Vu_kV_L2(q21 )(Q₂)

q21 .

If 2(q21 )G_Q^2Q

22, then

Vu_kV_L2(q21 )(Q2) q21

GCV˜_Gu_kV_L2(Q2) q21

, that implies V˜_xfkV_L1(B)GCN˜_Gu_kV_L2(Q2)

q21

. We can conclude that the se- quence (f_k)_kF1 is bounded in W^{1 , 1}(B), so that Vu_kV_Lq(Q₂)KVu₀V_Lq(Q₂).

Finally, we prove that I₃K0. Define, as above:

ck(y)»4

NxN FR

Nu_k(x,y)N^qdx.

By a similar proceeding, it is enough to observe that, for almost everyy,

˜_yck(y)4q

NxNDR

u_k^q21(x,y)

NxN^a NxN^a˜_yu_k(x,y)dxG

Gq

u

NxNDR

u_k^{2(q21 )}(x,y) NxN^2a

dx

v

u

NxNDR

NxN^2aN˜_yu_k(x,y)N²dx

v

G q R^a

u

NxNDR

u_k^{2(q21 )}(x,y)dx

v

u

NxNDR

N˜_Gu_k(x,y)N²

v

Then the sequence (ck) is bounded in W^{1 , 1}(NyN GR) and VukV_Lq(Q₃)K KVu₀V_Lq(Q3). Thus Theorem 1.1 is proved.

3. A semilinear problem for the Grushin operator.

This section is devoted to the proof of Theorem 1.2.

As announced in the Introduction, our approach to problem (2) is variational. Therefore we define, for every uS^{1 , 2(RN}), the func- tional:

J(u)4 1 2

R^N

(

)

Our goal is to find a critical point ofJas the minimum on a suitable mani- fold. We first prove the following

LEMMA 3.1. Let vS^{and let v*} denote the spherical decreasing re- arrangement of v with respect to the variable y. It holds

J(v* )GJ(v) . (8)

PROOF. We first recall some useful properties of the rearrange- ments we are dealing with. If vL^p, uL^p8, and ¹

p1_p¹

841 , then:

(i) VvV_Lp4Vv*V_Lp

(ii)

R^m

u(x,y)v(x,y)dyG

R^m

u* (x,y)v* (x,y)dy a.e. xRⁿ

(iii)

R^m

N˜_yv*N^pdyG

R^m

N˜_yvN^pdy for all pD1 .

Properties (i) and (ii) are a trivial consequence of the fact that, for a fixedx,u* (x,Q),v* (x,Q) are nothing but the Schwartz rearrangements ofu(x,Q),v(x,Q) (see for instance [6]). The third property follows direct- ly from the Polya-Szegö inequality. We also remark that property (iii) implies

R^N

NxN^2aN˜_yv*N²dx dyG

R^N

NxN^2aN˜_yvN²dx dy.

In order to prove (8) it remains to show the following inequality:

V˜_xv*V_L2GV˜_xvV_L2.

Letj41 ,R,nand lethc0. From the above properties (i) and (ii) we get:

R^m

Nv* (x1he_j,y)2v* (x,y)N²

h² dy4

4

R^m

1

h²

(

)

G

R^m

1

h²

(

)

4

R^m

Nv(x1he_j,y)2v(x,y)N²

h² dy

4

R^m

N

0 1

˜_xjv(x1the_j,y)dt

N

G

R^m

0 1

N˜_xjv(x1the_j,y)Ndt²dy. Then, by integrating with respect to x:

R^N

Nv* (x1he_j,y)2v* (x,y)N²

h² dx dyGV¯_xjvV²_L2(R^N). This proves that the family

w_h»4

g

h

is bounded inL²by theL²-norm of¯_xjv. Then, for a sequenceh_j70 ,w_hj weakly converges to a certain win L² such thatVwV_L2GV¯_xjvV_L2. It fol- lows that w is the weak derivative of v* with respect to x_j and V¯_xjv*V_L2GV¯_xjvV_L2. Since this inequality holds for any j41 ,R,n, we get

V˜_xv*V_L2GV˜_xvV_L2. r

PROOF OF THEOREM 1.2.. Let M_{4 ]}^uS^{: VuV}L^q41( and let J be the functional defined in (7). Since JD0 on M, there exists inf

M

J(u)4 4J₀F0. Our goal is to prove thatJ₀actually is a minimum forJ, so that J₀D0. Let (u_k)%M be a minimizing sequence. By definition, J(u)4 4J(NuN), then we can assume ukF0 for every k.

Moreover:

1. Vu_kV

S^{1 , 2}(R^N)4J(u_k)4J₀1o( 1 ), therefore (u_k)_k is bounded and there exists a function u₀S ^{such that u}ku₀ up to a subsequence.

2. Lemma 3.1 allows us to assume u_k radially symmetric and de- creasing with respect to NyN without loss of generality.

3. The sequence (u_k)_k is precompact inL^q by Theorem 1.1. Hence ukKu0 in L^q, and Vu0V_Lq41.

On the other hand, by the semicontinuity of J with respect to the

weak convergence, we have

J(u0)Glim inf

kK Q

J(uk)4J0.

Then J(u0)4J0. In addition, by Lagrange-Lusternik multiplier Theo- rem, u0 verifies the following integral identity:

0 1Q

R^m

g

¯r (r,y)2 n21 r

¯u0

¯r (r,y)W(r,y)

h

2

0 1Q

R^m

(

)

4m

0 1Q

R^m

[u0(r,y) ]^q21W(r,y)rⁿ²¹dy dr

for everyW4W(r,y)C₀^Q(]0 ,1Q[3R^m). Herem is a Lagrange multi- plier. SinceJ(u₀)D0 it must bemD0. In a standard way, we can rescale u₀in order to getm41. Thenu₀is a weak solution in (Rⁿ0]0()3R^m to the elliptic equation

2¯²W(r,y)

¯r² 2n21 r

¯

¯rW(r,y)2r^2aDyW(r,y)1lW4W^q21. A classical bootstrap argument shows that isu0 is a pointwise solution of

2DGu1lu4u^q21 in (R^m0]0()3Rⁿ.

Now we are going to prove thatu₀is a weak solution on the whole space, i.e. u₀ is a weak solution of problem (2).

It suffices to establish that for all fC₀^Q(R^N) one can find a se- quence ej70 verifying

R^N0Cej

˜_Gu₀Q˜_Gf2

R^N0Cej

(u₀^q21f2lu₀f)K0 as jK Q (9)

where

C_ej»4 ](x,y)Rⁿ3R^m:NxN Ge_j; NyN GR(

andRmust be chosen in a way that suppf%Rⁿ3B_R^m. Here we denote by B_R^m the set ]yR^m:NyNGR(.

Now fixeD0. Sinceu0is a classical solution inR^N0C_eandff0 when NyN 4R, by divergence Theorem we obtain

R^N0Ce

(˜_Gu₀Q˜_Gf2u₀^q21f2lu₀f)4

Ge

fA˜u₀Qn ds(x) (10)

where we denoted with G_e the set

](x,y)Rⁿ3R^m:NxN 4e; NyN GR( and with n4

g

h

Since u0 is symmetric in NxN and NyN, we have u₀(x,y)4n(NxN,NyN)

for a suitable 2-variable functionn. Hence˜u04

g

NyN

h

turns out that A˜u₀Qn4nx(NxN,NyN) on Ge. So

G_e

fA˜u₀Qn ds(x)4

G_e

fn_x(NxN,NyN)ds(x)dy.

Since f is bounded

N

e

fA˜nQn ds(x)dy

N

e

Nn_xNds(x)dy. (11)

The Hölder inequality yields:

G_e

Nn_x(NxN,NyN)Nds(x)dyGk

u

G_e

Nn_xN²ds(x)dy

v

(12)

On the other hand, since nx belongs to L²

0 1

u

Ge

NnxN²ds(x)dy

v

NyNER

u

NxNE1

NnxN²ds(x)

v

G

R²ⁿ¹¹

NnxN²dx dyE 1Q.

Then there exists a sequence ej70 such that ej

G_ej

NnxN²ds(x)dyK0 as jK Q.

Using this result in (12), (11) and (10), we get (9) completing the proof.

Finally, the local Hölder regularity of u0 can be proved applying the Moser iteration technique as presented in [5] and extended to the Grushin-type operators in [3] and [4].

Acknowledgements. The authors thank E. Lanconelli for suggesting the problem and for the hints raised up during the lively meetings.

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