Composition operators on Sobolev spaces*

A. Ukhlov

Ben-Gurion University of the Negev, Israel

(*joint works with V. Gol’dshtein and S. K. Vodop’yanov)

The Dirichlet problem and composition operators

We study composition operators on Sobolev spaces of weakly

differentiable functions. Namely, we study operators

ϕ∗ : L1p(Ω

′)→ L1q(Ω), 1 ≤ q ≤ p ≤ ∞,

which are defined by the composition rule:

ϕ∗(f) = f ϕ, f ∈ L1p(Ω

′).

Here ϕ is a mapping ϕ : Ω → Ω′ of the Euclidean domains

Ω, Ω′ ⊂ Rn, n ≥ 2.

The Sobolev space L1p(Ω), 1 ≤ p ≤ ∞, is a seminormed space

of locally summable weakly differentiable functions f : Ω → R

equipped with the following seminorm:

‖f | L1p(Ω)‖ =

(∫

Ω

|∇f |p(x) dx

)1/p

, 1 ≤ p <∞,

‖f | L1∞(Ω)‖ = ess sup

x∈Ω|∇f |

where ∇f is the weak gradient of the function f .

1

The composition operators theory is closely connected with the

elliptic equations theory.

For example, consider the classical Dirichlet problem for the

Laplace operator in the plane domain Ω ⊂ R2.

∆u = 0,

u|∂Ω = f.

u ∈ C2(Ω), ∆u = ∂2u∂x2 + ∂2u

∂y2 .

It is well known that for the unit disc D ⊂ R2 we have the unique

solution of this problem

u(r0, θ0) =1

2π

2π∫

0

1− r20

1 + r20 − 2r0 cos(θ − θ0)

f(θ) dθ

(the Poisson formula), obtained by means composition of the av-

erage value function with the conformal mapping ϕ : D→ D.

2

If Ω be a connected simply connected plane domain with non-

empty boundary then by the Riemannian Mappings Theorem there

exists a conformal homeomorphism ϕ : Ω → D. Recall that a

mapping

ϕ = u(x, y) + iv(x, y) : Ω→ D

is called a conformal homeomorphism if ϕ is differentiable (ana-

lytic) in Ω and ϕ′(z) 6= 0, z ∈ Ω.

It is well known that a composition of an analytic function and

harmonic function is an harmonic function again. So, if a function

u is a solution of the Dirichlet problem for the unit disc, then a

function u ϕ is a solution of the Dirichlet problem in the domain

Ω ⊂ R2 (with a corresponding boundary value).

Let Ω = (x, y) : y > 0 (upper half of the plane R2) and

u(x, 0) = f(x).

The conformal mapping w = ϕ(z) = z−z0

z−z0: Ω → D. Then the

solution is given by

u(x, y) =1

π

+∞∫

−∞

y

(t− x)2 + y2f(x)dx.

3

In analytical terms the condition of conformity we can rewrite

as: ϕ : Ω→ D, is a smooth mapping such that

|Dϕ(x)|2 = |J(x, ϕ)|, x ∈ Ω.

Here Dϕ is the Jacobi matrix of ϕ and J(x, ϕ) = det(Dϕ(x)).

Note, that conformal mappings preserve the Dirichlet integral:∫

Ω

|∇u ϕ)|2 dz =

∫

D

|∇u|2 dw

and we can consider solution of the Dirichlet problem as an applica-

tion of the Dirichlet’s principle to the energy integral (R. Courant.

”Dirichlet principle, Conformal mappings and Minimal Surfaces”).

In another words we can say that: Conformal mappings generate

a bounded composition operator (isomorphism)

ϕ∗ : L12(D)→ L1

2(Ω), ϕ∗(u) = u ϕ.

Here L12(Ω) is a semi-normed Sobolev space of weakly differentiable

functions with finite Dirichlet integral.

4

The boundary value problem for the p-Laplace operator

∆pu = div(|∇u|p−2∇u)

leads to composition operators in more general Sobolev spaces,

namely, L1p(Ω). Such type composition operators were studied by

many authors (V. Gol’dshtein, S. Hencl, P. Koskela, I. Markina,

V. G. Maz’ya, Yu. G. Reshetnyak, A. S. Romanov, A. Ukhlov,

S. K. Vodop’yanov ...).

The theory of composition operators on Sobolev spaces arises

to the Yu. G. Reshetnyak problem (1968) about description of all

isomorphisms ϕ∗ of homogeneous Sobolev spaces L1n, generated by

quasiconformal mappings ϕ of Euclidean spaces Rn, n ≥ 2, by the

composition rule ϕ(f) = f ϕ.

S. K. Vodop’yanov and V. Gol’dshtein (1975) proved that a

homeomorphism ϕ : D → D′ induces a composition operator ϕ∗

from L1n(D

′) to L1n(D) if and only if ϕ is a quasiconformal home-

omorphism. Because a homeomorphism inverse to quasiconformal

is also quasiconformal one then ϕ∗ is an isomorphism of L1n(D

′)

and L1n(D).

5

Recall that a homeomorphism ϕ : Ω→ Ω′ is called quasiconfor-

mal if ϕ belongs to the Sobolev class W 1,1loc (Ω) and there exists a

constant 1 ≤ K <∞ such that

|Dϕ(x)|n ≤ K|J(x, ϕ)| for almost all x ∈ Ω.

For p 6= n a homeomorphisms ϕ : D → D′ induces an iso-

morphisms of Sobolev spaces W 1p (D′) and W 1

p (D) if and only

if ϕ it is a bi-Lipschitz one. This result was proved for p > n

S. K. Vodop’yanov and V. Gol’dshtein (1975), for n− 1 < p < n

by V. Gol’dshtein and A. S. Romanov (1984) and for 1 ≤ p < n

by I. Markina (1990).

Homeomorphisms that induce bounded composition operators

from Sobolev space L1p(D

′) to L1p(D) were studied S. K. Vodop’ya-

nov (1988). A geometric description of such homeomorphisms was

obtained for p > n− 1 by V. Gol’dshtein, L. Gurov and A. S. Ro-

manov (1995).

The multipliers theory was applied to the composition operators

theory by V. G. Maz’ya and T. O. Shaposhnikova (1986).

6

New effects arise when we study composition operators with de-

creasing summability. In 1993 by A. Ukhlov was proved:

A homeomorphism ϕ : Ω → Ω′ generates by the composition

rule ϕ∗f = f ϕ the bounded operator

ϕ∗ : L1p(Ω

′)→ L1q(Ω), 1 ≤ q ≤ p <∞,

if and only if ϕ ∈ L11,loc(Ω), has finite distortion, and

Kp,q(f) =

(∫

Ω

(

|Dϕ|p

|J(x, ϕ)|

)q

p−q

dx

)p−qpq

<∞.

A mapping ϕ : Ω → Rn belongs to L1

1,loc(Ω) if its coordinate

functions ϕj belong to L11,loc(Ω), j = 1, . . . , n. In this case formal

Jacobi matrix Dϕ(x) =(

∂ϕi∂xj

(x))

, i, j = 1, . . . , n, and its determi-

nant (Jacobian) J(x, ϕ) = det Dϕ(x) are well defined at almost all

points x ∈ Ω. The norm |Dϕ(x)| of the matrix Dϕ(x) is the norm

of the corresponding linear operator Dϕ(x) : Rn → R

n defined by

the matrix Dϕ(x). We will use the same notation for this matrix

and the corresponding linear operator.

A mappings ϕ : Ω→ Rn has a finite distortion if Dϕ(x) = 0 for

almost all points x that belongs to set Z = x ∈ D : J(x, ϕ) = 0.

7

Necessity of studying of Sobolev mappings with integrable distor-

tion arises in problems of the non-linear elasticity theory. J. M. Ball

(1976, 1981) introduced classes of mappings, defined on bounded

domains Ω ∈ Rn:

A+p,q(Ω) =

ϕ ∈ W 1p (Ω) : adj Dϕ ∈ Lq(Ω), J(x, ϕ) > 0 a. e. in Ω,

p, q > n, where adj Dϕ is the formal adjoint matrix to the Jacobi

matrix Dϕ:

adj Dϕ(x) ·Dϕ(x) = I ·J(x, ϕ).

Mappings that generate composition operators on Sobolev spaces

are mappings of finite distortion. The theory of mappings of fi-

nite distortion is under intensive development at the last decades.

In series of works the geometrical and topological properties of

these mappings were studied (S. Hencl, J. Heinonen, I. Holopainen,

T. Iwaniec, P. Koskela, J. Maly, J. Manfredi, G. Martin, O. Mar-

tio, P. Pankka, V. Ryazanov, U. Srebro, V. Sverak, E. Villamor,

E. Yakubov).

8

Mappings that generate bounded composition operators on So-

bolev spaces

ϕ∗ : L1p(Ω

′)→ L1q(Ω), 1 ≤ q ≤ p ≤ ∞,

are natural generalization of (quasi)conformal mappings (mappings

of bounded distortion and we call them mappings of bounded

(p, q)-distortion.

In the description of composition operators on Sobolev spaces

the significant role belongs the additive set function which allows

to localize the composition operators.

Let a mapping ϕ : Ω → Ω′, where Ω, Ω′ are domains of Eu-

clidean space Rn, generates a bounded composition operator

ϕ∗ : L1p(Ω

′)→ L1q(Ω), 1 ≤ q < p ≤ ∞.

Then

Φ(A′) = supf∈L1

p(A′)∩C0(A′)

(∥

∥ϕ∗f | L1q(Ω)

∥

∥

∥

∥f | L1p(A

′)∥

∥

)r

,

where number r is defined according to 1/r = 1/q − 1/p, is a

bounded monotone countably additive function defined on open

bounded subsets A′ ⊂ Ω′.

9

The description of the composition operators is based on the

following local estimate:

|Dϕ(x)|q ≤(

Φ′(ϕ(x)))

p−qp |J(x, ϕ|

where Φ′ is the volume derivative of the set function Φ.

Let a monotone finitely additive set function Φ be defined on

open subsets of a domain Ω ⊂ Rn. Then for almost all points

x ∈ Ω the volume derivative

Φ′(x) = limδ→0,Bδ∋x

Φ(Bδ)

|Bδ|

is finite and for any open set U ⊂ Ω, the inequality∫

U

Φ′(x) dx ≤ Φ(U)

is valid.

10

It is well known that the mapping inverse to a quasiconformal

homeomorphism is quasiconformal also. We have the similar result

for homeomorphisms with bounded (p, q)-distortion.

Let ϕ : Ω → Ω′ be a homeomorphism with bounded (p, q)-

distortion, q > n − 1. Then the inverse mapping ϕ−1 : Ω′ → Ω

is a mapping with bounded (q′, p′)-distortion, q′ = q/(q − n + 1)

and p′ = p/(p− n + 1).

The main difficult here is to prove that the mapping which is

inverse to a homeomorphism with bounded (p, q)-distortion is the

Sobolev mapping.

11

The very interesting case in the composition operators theory

arises when p =∞. In this case we have

A homeomorphism ϕ : Ω → Ω′ between two domains Ω, Ω′ ⊂R

n belongs to the Sobolev space L1p(Ω), 1 ≤ p ≤ ∞ if and only if

the composition operator

ϕ∗ : L1∞(Ω′)→ L1

p(Ω)

is bounded.

The proof of this fact is based on

Let a function f : Ω → R belong to the Sobolev space L1∞(Ω).

Then there exists a sequence of smooth functions fk ∈ L1∞(Ω)

such that the sequence ∇fk weakly converges to ∇f in L∞(Ω).

12

The next theorem asserts regularity of the inverse mapping in

this case:

Let a homeomorphism ϕ : D → D′ between two domains D and

D′ ⊂ Rn, n ≥ 2, generate by the composition rule ϕ∗f = f ϕ a

bounded operator

ϕ∗ : L1∞(D′)→ L1

n−1(D),

possess the Luzin N -property and have finite distortion. Then

the inverse mapping ϕ−1 : D′ → D generates the composition

operator

(ϕ−1)∗ : L1∞(D)→ L1

1(D′)

and belongs to the Sobolev space L11(D

′).

The theorem is closely connected with the problem of regularity

of mappings inverse to Sobolev homeomorphisms which is inten-

sively studied at the last time (M. Csornyei, S. Hencl, P. Koskela,

Y. Maly).

The proof of the theorem is based on the lower estimate of the

composition operator

‖f | L11(D

′)‖ ≤ ‖ϕ | L1n−1(D)‖n−1 · ‖ϕ∗f | L1

∞(D)‖

13

As an application of composition operators theory we study the

Sobolev type embedding theorems in weighted Sobolev spaces.

This part of my talk is based on recent article, joint with V.

Gol’dshtein and V. V. Motreanu.

Let Ω be a domain in the Euclidean space Rn. For p ∈ (1, +∞),

q ∈ [1, +∞) we define the two-weighted Sobolev space

W 1,q,p(Ω, v, w) = u ∈ Lq(Ω, v) : ∇u ∈ Lp(Ω, w)

equipped with the norm

‖u‖W 1,q,p(Ω,v,w) = ‖u‖Lq(Ω,v) + ‖ |∇u| ‖Lp(Ω,w),

where ∇u is the weak gradient of the function u, i. e.

∇u = (∂u

∂x1, ...,

∂u

∂xn).

14

We have the following motivation for these spaces: if a function u

belongs to the usual Sobolev space W 1,p(Ω), 1 ≤ p < n, and Ω ⊂R

n is a regular domain, then by the classical Sobolev embedding

theorem we have that u ∈ Lq(Ω), q = np/(n−p). It means that in

the case of regular domains the space W 1,p(Ω), 1 ≤ p < n, really

is the space W 1,q,p(Ω). So, the main question here is the existence

of embedding theorems of W 1,p(Ω, w) into Lq(v, Ω).

We will use also the seminormed space

L1p(Ω, w) = u ∈ L1

loc(Ω) : ∇u ∈ Lp(Ω, w)

equipped with the seminorm

‖u‖L1p(Ω,w) = ‖ |∇u| ‖Lp(Ω,w).

15

We suppose that weight functions v : Rn → R and w : R

n → R

satisfying the Muckenhoupt Aq-condition and Ap-condition respec-

tively. Recall that the weight v satisfying the Muckenhoupt Aq-

condition if

supB ball⊂Rn

(

1

|B|

∫

B

w dx

)(

1

|B|

∫

B

w1

1−q dx

)q−1

< +∞.

For such weights the Lebesgue space Lq(Ω, v) and Sobolev space

W 1,q,p(Ω, v, w) are Banach spaces.

Let C∞c (Ω) be the space of C∞ functions with compact support

in Ω. Note that it is a subspace of W 1,q,p(Ω, v, w). Then, we define

W 1,q,p0 (Ω, v, w) as the closure of C∞c (Ω) in W 1,q,p(Ω, v, w).

In the particular case where w = w0 and q = p, we retrieve the

usual weighted Sobolev spaces W 1,p(Ω, w) := W 1,p,p(Ω, w, w) and

W 1,p0 (Ω, w) := W 1,p,p

0 (Ω, w, w), studied for instance by T. Kilpelai-

nen, A. Kufner and B. O. Turesson.

16

Our approach to the weighted embedding theorems is based on

the systematic application of the composition operators theory on

weighted Sobolev spaces to the embedding theory. This approach

was suggested by V. Gol’dshtein and V. Gurov for classical Sobolev

spaces and can be briefly described with the help of the following

diagram for weighted spaces:

W 1,p(Ω, w)ϕ∗

−→ W 1,p1(D)

↓ ↓

Lq(Ω, v)(ϕ−1)∗

←− Lq1(D)

Here the operator ϕ∗f = fϕ is a bounded composition operator

on weighted Sobolev spaces

ϕ∗ : W 1,p(Ω)→W 1,p1(D)

induced by a homeomorphism ϕ : D → Ω that maps smooth

domain (an embedding domain) D ⊂ Rn onto non smooth do-

main Ω ⊂ Rn. Suppose that its inverse homeomorphism ϕ−1 :

Ω→ D induces a bounded composition operator of corresponding

Lebesgue spaces. If the Sobolev space W 1p1

(D) permits a bounded

(compact) embedding operator into Lq1(D) then, using the corre-

sponding compositions, we can construct the embedding operator

of the weighted Sobolev space W 1,p(Ω, w) into Lq(Ω, v).

17

Composition operators on weighted spaces

Let D and Ω be domains in Euclidean space Rn, n ≥ 2. Re-

member that a homeomorphism ϕ : D → Ω belongs to Sobolev

class W 1,1loc (D) if its coordinate functions belong to W 1,1

loc (D). De-

note by Dϕ the weak differential of ϕ. The norm |Dϕ(x)| is

the standard norm of the linear operator defined by Dϕ(x) and

J(x, ϕ) = det(Dϕ).

Let D and Ω be domains in Euclidean space Rn, n ≥ 2 and

homeomorphism ϕ : D → Ω. The composition operator

ϕ∗ : L1p(Ω, w)→ L1

q(D, v), 1 ≤ q ≤ p < +∞,

is bounded if and only if ϕ ∈ W 1,1loc (D) has finite distortion and

(∫

D

(

|Dϕ(x)|v(x)pq (x)

|J(x, ϕ)|w(ϕ(x))

)

qp−q

dx

)

p−qpq

= K < +∞.

In the case v = 1 we call such mappings ϕ : D → Ω as w-

weighted (p, q)-quasiconformal homeomorphisms.

18

Embedding theorems

On the base of composition operators on weighted Sobolev spaces

we prove the embedding theorem for two-weighted Sobolev spaces.

Let a domain D ⊂ Rn be an embedding domain and there exists

a w-weighted (p, p1)-quasiconformal homeomorphism ϕ : D → Ω

of domain D onto bounded domain Ω.

If for some p ≤ q ≤ q1 <∞ the following inequality is correct

∫

D

(

|J(x, ϕ)|v(ϕ(x))

)

q1q1−q

dx < +∞,

then an embedding operator

i : W 1,p(Ω, w) → Lq(Ω, v)

is bounded, if q ≤ q1 ≤ np1/(n− p1), and is compact if q ≤ q1 <

np1/(n− p1).

19

Examples of domains with embedding properties

We study a family of domains containing as particular cases

domains with anisotropic Holder singularities.

Assume n ≥ 2. Let b > 0. Let gi : [0, +∞) → [0, +∞) (for

i = 1, . . . , n− 1) be C1 functions with the following properties:

(a) gi(0) = 0 and gi(t) > 0 for t > 0;

(b) there are constants t0 ∈ (0, 1), γ, δ > 1 and M1, M2 > 0 such

that

M1tγ−1 ≤

n−1∏

i=1

gi(t) ≤M2tδ−1 for all t ∈ (0, t0).

A typical example for gi is gi(t) = tγi (for i = 1, . . . , n − 1)

where γi ≥ 1. In this case, the above properties are satisfied with

γ = δ =∑N−1

i=1 γi + 1.

20

Denoting g = (g1, . . . , gn−1), we consider the domain

Hg,b = x ∈ Rn : 0 < xn < b, 0 < xi < gi(xn), i = 1, . . . , n−1.

For g1(t) = . . . = gn−1(t) = t we will use the notation Hb

instead of Hg,b. Every domain Hg,b (in particular, Hb) satisfies an

embedding property since it is bounded.

If the weight is polynomial, i.e. w(x) := |x|α, then w ∈ Ap(Rn)

if −n < α < n(p− 1).

Let p ∈ (1, +∞), q ∈ [1, +∞). Suppose that −n < α < n(p−1), max−n,−δ < β < n(q − 1) (resp. max−n,−δ < β ≤ 0

for q = 1) and p < minα + γ, n. Then the embedding

(L1,p0 (Hg,b, |x|

α), ‖ · ‖L1,p(Hg,b,|x|α)) ⊂ Lq(Hg,b, |x|

β)

is compact whenever q < min

npn−p,

(β+δ)pα+γ−p

. In particular, for

such q, the embedding

(W 1,q,p0 (Hg,b, |x|

β, |x|α), ‖ · ‖L1,p(Hg,b,|x|α)) ⊂ Lq(Hg,b, |x|

β)

is compact.

Note that, in the classical case (i.e., α = β = 0 and γ = δ = n)

we retrieve the condition q < p∗ = npn−p

.

21

The proof of the theorem is based on the mappings

ϕa : Hb → Hg,b

given by

ϕa(x) = (x1

xnga

1(xn), . . . ,xn−1

xnga

n−1(xn), xan).

Dirichlet problems

We study the existence of weak solutions for the following two

problems:

for p, q ∈ (1, +∞),

(P1)

−∆p,w(u) = v(x)|u|q−2u in Ω,

u ∈ W 1,q,p0 (Ω, v, w),

and for p ∈ (1, +∞), q ∈ [1, +∞),

(P2)

−∆p,w(u) = f in Ω,

u ∈W 1,q,p0 (Ω, 1, w),

where f ∈ Lq′(Ω) (with q′ = qq−1 if q > 1, and q′ = +∞ if q = 1).

Such type problems arise in the non-linear elasticity theory.

22

The operator ∆p,w entering problems (P1), (P2) is a w-weighted

p-Laplacian:

〈−∆p,w(u), f〉 =

∫

Ω

w(x)|∇u|p−2∇u · ∇f dx,

∀u, f ∈W 1,q,p0 (Ω, v, w).

We say that u ∈ W 1,q,p0 (Ω, v, w) is a weak solution of problem

(P1) if∫

Ω

w(x)|∇u|p−2∇u · ∇f dx =

∫

Ω

v(x)|u|q−2uf dx,

∀f ∈W 1,q,p0 (Ω, v, w),

and we say that u ∈W 1,q,p0 (Ω, w) is a weak solution of problem

(P2) if∫

Ω

w(x)|∇u|p−2∇u · ∇f dx =

∫

Ω

fv dx, ∀f ∈ W 1,q,p0 (Ω, w).

23

In order to show the existence of weak solutions for problems

(P1), (P2), we will assume that the space W 1,q,p0 (Ω, v, w)) is com-

pactly embedded in the Lebesgue space Lq(Ω, v) (with v = 1 in

the case of problem (P2)). Under this assumption, we show that

problem (P1) admits at least two nontrivial weak solutions of op-

posite constant sign, and that problem (P2) admits a unique weak

solution. The proofs follow the reasoning in the classical paper by

M. Otani, based on the Rellich-Kondrachov embedding theorem

for the usual Sobolev space W 1,p(Ω).

We solve the Dirichlet boundary value problems (P1) and (P2)

under the assumption that the domain Ω satisfies a suitable em-

bedding property. More precisely, we show:

Let p > 1, q ≥ 1 such that p 6= q. Assume w ∈ Ap(Rn) and

v ∈ Aq(Rn). Suppose that we have a compact embedding

(W 1,q,p0 (Ω, v, w), ‖ · ‖L1,p(Ω,w)) ⊂ Lq(Ω, v).

Then, the problem

(P1)

−∆p,w(u) = v(x)|u|q−2u in Ω,

u ∈ W 1,q,p0 (Ω, v, w),

has at least two nontrivial weak solutions u1 ≥ 0, u2 ≤ 0.

24

Let p > 1, q ≥ 1, f ∈ Lq′(Ω) and let w ∈ Ap(Rn). Suppose

that we have a compact embedding

(W 1,q,p0 (Ω, w), ‖ · ‖L1,p(Ω,w)) ⊂ Lq(Ω).

Then, the problem

(P2)

−∆p,w(u) = f in Ω,

u ∈W 1,q,p0 (Ω, w),

has a unique weak solution.

The proof of these theorem is based on the theorem (M. Otani).

Consider X a real Banach space and let X∗ be its topological

dual space. By 〈·, ·〉 we denote the duality brackets for the pair

(X, X∗). Let Φ(X) be the set of lower semicontinuous, convex

functions φ : X → R∪+∞ which are not identically +∞. The

effective domain of φ ∈ Φ(X) is

D(φ) = v ∈ X : φ(v) < +∞,

and the subdifferential ∂φ of φ is

∂φ(v) = v∗ ∈ X∗ : φ(w)− φ(v) ≥ 〈v∗, w − v〉, ∀w ∈ D(φ)

with the domain D(∂φ) = v ∈ X : ∂φ(v) 6= ∅. Note that, if ϕ

is of class C1 at v ∈ X , then ∂φ(v) = ϕ′(v).

25

Theorem. (M. Otani) Let X be a real Banach space and

φ1, φ2 ∈ Φ(X) be two nonnegative functions satisfying:

(i) there exist α1, α2 ≥ 1, α1 6= α2 such that

φi(λv) = λαiφi(v), ∀λ > 0, ∀v ∈ D(φi), i = 1, 2;

(ii) there is a constant C > 0 such that

φ2(v)1/α2 ≤ C φ1(v)1/α1, ∀v ∈ D(φ1).

Suppose there is u ∈ D(∂φ2) such that

(iii) u gives the best constant for (ii), i.e.,

R(u) = maxR(v) : v ∈ D(φ1), φ1(v) 6= 0,

where R(v) := φ2(v)1/α2

φ1(v)1/α1;

(iv) α1φ1(u) = α2φ2(u).

Then u ∈ D(∂φ1) and ∂φ2(u) ⊂ ∂φ1(u). In particular, if ∂φ1 is

single-valued, then u is a nontrivial solution of ∂φ1(u) = ∂φ2(u).

For the solution of (P1) we will apply Theorem (M. Otani) with

X = W 1,q,p0 (Ω, v, w) endowed with the norm ‖ · ‖L1,p(Ω,w), and the

functions φ1, φ2 : X → R given by

φ1(v) =1

p‖∇v‖pLp(Ω,w) and φ2(v) =

1

q‖v‖qLq(Ω,w0)

.

Note that

〈ϕ′1(u), f〉 =

∫

Ω

w(x)|∇u|p−2∇u·∇f dx, ∀u, f ∈W 1,q,p0 (Ω, v, w).

and

〈ϕ′2(u), f〉 =

∫

Ω

v(x)|u|q−2uf dx, ∀u, f ∈W 1,q,p0 (Ω, v, w).

26

For the solution of (P2) we will apply Theorem (M. Otani) with

X = W 1,q,p0 (Ω, w) endowed with the norm ‖ · ‖L1,p(Ω,w) and the

functions φ1, φ2 : X → R given by

φ1(v) =1

p‖∇v‖Lp(Ω,w) and φ2(v) =

∣

∣

∣

∣

∫

Ω

fv dx

∣

∣

∣

∣

.

Functionals associated to problems (P1) and (P2)

Let J1 : W 1,q,p0 (Ω, v, w)→ R and J2 : W 1,q,p

0 (Ω, w)→ R be the

functionals associated to problems (P1) and (P2) respectively, that

is

J1(u) =1

p

∫

Ω

w(x)|∇u|p dx−1

q

∫

Ω

w0(x)|u|q dx,

∀u ∈W 1,q,p0 (Ω, v, w),

J2(u) =1

p

∫

Ω

w(x)|∇u|p dx−

∫

Ω

fu dx, ∀u ∈ W 1,q,p0 (Ω, w).

27