One of the main ideas is to present the operator−∆p as a duality mapping between W01,p(Ω) and its dual W−1,p0(Ω), 1p + p10 = 1, corresponding to the normalization function ϕ(t) =tp−1

Nova S´erie

VARIATIONAL AND TOPOLOGICAL METHODS FOR DIRICHLET PROBLEMS

WITH p-LAPLACIAN

G. Dinca, P. Jebelean and J. Mawhin Presented by L. Sanchez

0 – Introduction

The aim of this paper is to obtain existence results for the Dirichlet problem (−∆pu=f(x, u) in Ω ,

u|_∂Ω = 0 . Here ∆p = _∂x^∂

i(|∇u|^p−2_∂x^∂u

i), 1< p <∞, is the so-called p-Laplacian and f : Ω×R→R is a Carath´eodory function which satisfies some special growth conditions. One of the main ideas is to present the operator−∆p as a duality mapping between W₀^1,p(Ω) and its dual W^−1,p0(Ω), ¹_p + _p¹0 = 1, corresponding to the normalization function ϕ(t) =t^p−1. This idea, coming from Lions’ book [23], proves to be a very fruitful one. The properties of the Nemytskii operator (Nfu)(x) = f(x, u(x)), generated by the Carath´eodory function f, the homo- topy invariance of the Leray–Schauder degree (under the form of a priori esti- mate, uniformly with respect to λ ∈ [0,1], of the solutions set of the equation u = λ(−∆p)⁻¹N_fu with (−∆p)⁻¹N_f: W₀^1,p(Ω) → W₀^1,p(Ω) compact), the well known Mountain Pass Theorem of Ambrosetti and Rabinowitz and the varia- tional characterization of the first eigenvalue of −∆p on W₀^1,p(Ω) are the other essential tools which are also used.

Received: November 24, 2000.

1 – Thep-Laplacian as duality mapping

The main idea of this paragraph is to present the operator −∆p, 1< p <∞, as duality mappingJ_ϕ: W₀^1,p(Ω)→W^−1,p0(Ω), ¹_p +_p¹0 = 1, corresponding to the normalization functionϕ(t) =t^p−1.

Originated in the well known book of Lions (see [23]), this presentation has the advantage of allowing to apply the general results known for the duality mapping to the particular case of thep-Laplacian. For example, the surjectivity of the duality mapping (itself an immediate consequence of a well known result of Browder (see e.g. [8])) achieves the existence of the W₀^1,p(Ω)-solution for the equation −∆pu = f, with f ∈ W^−1,p0(Ω). Note that if f ∈ W^−1,p0(Ω) is given, then an elementu∈W₀^1,p(Ω) is said to be solution of the Dirichlet problem

(−∆pu=f in Ω, u|∂Ω= 0 ,

if the equality−∆pu=f is satisfied in the sense ofW^−1,p0(Ω).

For the convenience of the reader we have considered to put away the def- initions and the results concerning the duality mapping, which will be used in the sequel. Because these results are already known, the proof is often omit- ted; however the proof is given when these results achieve specific properties for p-Laplacian.

1.1. Basic results concerning the duality mapping

Below, X always is a real Banach space, X^∗ stands for its dual and h·,·i is the duality betweenX^∗ and X. The norm on X and onX^∗ is denoted byk k.

Given a set valued operator A: X → P(X^∗),the range of A is defined to be the set

R(A) = ^[

x∈D(A)

Ax

whereD(A) ={x∈X|Ax6=∅} is the domain of A. The operatorA is said to bemonotone if

hx^∗₁−x^∗₂, x₁−x₂i ≥ 0 wheneverx₁, x₂ ∈D(A) and x^∗₁ ∈Ax₁,x^∗₂ ∈Ax₂.

A continuous function ϕ: R+→R+ is called a normalization function if it is strictly increasing,ϕ(0) = 0 andϕ(r)→ ∞ withr→ ∞.

By duality mapping corresponding to the normalization functionϕ, we mean the set valued operatorJϕ: X → P(X^∗) as following defined

J_ϕx = ⁿx^∗∈X^∗| hx^∗, xi=ϕ(kxk)kxk, kx^∗k=ϕ(kxk)^o forx∈X.

By the Hahn–Banach theorem, it is easy to check that D(Jϕ) =X.

Some of the main properties of the duality mapping are contained in the following

Theorem 1. Ifϕis a normalization function, then:

(i) for eachx∈X,J_ϕx is a bounded, closed and convex subset ofX^∗; (ii) Jϕ is monotone:

hx^∗₁−x^∗₂, x₁−x₂i ≥ ^³ϕ(kx1k)−ϕ(kx2k)^{´ ³}kx1k − kx2k^´ ≥ 0, for each x₁, x₂∈X and x^∗₁ ∈J_ϕx₁,x^∗₂ ∈J_ϕx₂;

(iii) for each x∈X, J_ϕx = ∂ψ(x), where ψ(x) =

kxkR

0

ϕ(t)dt and ∂ψ: X→ P(X^∗) is the subdifferential ofψin the sense of convex analysis, i.e.

∂ψ(x) = ⁿx^∗ ∈X^∗|ψ(y)−ψ(x)≥ hx^∗, y−xi for all y∈X^o.

For proof we refer to Beurling and Livingston [5], Browder [8], Lions [23], Cior˜anescu [9].

Remark 1. We recall that a functional f: X → R is said to be Gˆateaux differentiable atx∈X if there existsf⁰(x)∈X^∗ such that

t→0lim

f(x+t h)−f(x)

t = hf⁰(x), hi for allh∈X.

If the convex function f: X → R is Gˆateaux differentiable at x ∈ X, then it is a simple matter to verify that ∂f(x) consists of a single element, namely x^∗=f⁰(x).

This simple remark will be essentially used in the sequel.

The geometry of the spaceX(orX^∗) supplies further properties of the duality mapping. That is why we recall the following (see e.g. Diestel [11])

Definition 1. The spaceX is said to be:

(a) uniformly convex if for each ε ∈ (0,2], there exists δ(ε) > 0 such that if kxk=kyk= 1 and kx−yk ≥ε then kx+yk ≤2 (1−δ(ε));

(b) locally uniformly convex if from kxk=kxnk= 1 andkxn+xk →2 with n→ ∞, it results that x_n→x(strongly in X);

(c) strictly convex if for each x, y ∈ X with kxk = kyk = 1, x 6= y and λ∈(0,1), we have kλ x+ (1−λ)yk<1.

Theorem 2. The following implications hold:

X uniformly convex =⇒ X locally uniformly convex =⇒ X strictly convex . For proof we refer to Diestel [11].

Theorem 3 (Pettis–Milman). If X is uniformly convex then X is reflexive.

For proof see e.g. Br´ezis [6] or Diestel [11] — where the original proof of Pettis is given.

In the sequel, ϕwill be a normalization function.

Proposition 1.

(i) If X is strictly convex, then Jϕ is strictly monotone:

hx^∗₁−x^∗₂, x₁−x₂i > 0

for each x₁, x₂ ∈ X, x₁ 6= x₂ and x^∗₁ ∈ J_ϕx₁, x^∗₂ ∈ J_ϕx₂; in particular, J_ϕx₁∩J_ϕx₂=φ if x₁6=x₂.

(ii) If X^∗ is strictly convex, then card(Jϕx) = 1, for all x∈X.

Proof: (i) First, it is easy to check that (see e.g. James [18]) if X is strictly convex, then for each x^∗ ∈ X^∗\{0} there exists at most an element x∈ X with kxk= 1, such thathx^∗, xi=kx^∗k.

Now, supposing by contradiction that there exist x₁, x₂ ∈ X with x₁ 6= x₂ andx^∗₁∈J_ϕx₁,x^∗₂∈J_ϕx₂, satisfying

hx^∗₁−x^∗₂, x₁−x₂i = 0

we have

0 = hx^∗₁−x^∗₂, x₁−x₂i ≥ ^³ϕ(kx1k)−ϕ(kx2k)^{´ ³}kx1k − kx2k^´ ≥ 0 and so, we getkx1k=kx2k.

Remark thatx₁6=x₂,kx₁k=kx₂k implies x₁ 6= 0,x₂ 6= 0.

We obtain 0 =

¿

x^∗₁−x^∗₂, x₁

kx1k− x₂ kx2k

À

=

"

ϕ(kx₁k)−

¿

x^∗₁, x₂ kx2k

À#

+

"

ϕ(kx₂k)−

¿

x^∗₂, x₁ kx1k

À#

and both of the brackets being positive, it results kx^∗₁k = ϕ(kx1k) =

¿ x^∗₁, x₂

kx2k À

which together with

kx^∗₁k =

¿ x^∗₁, x₁

kx1k À

yields _¿

x^∗₁, x₂ kx2k

À

= kx^∗₁k =

¿ x^∗₁, x₁

kx1k À

.

By the above mentioned result of James, we have _kx^x1

1k = _kx^x2

2k i.e. x₁ = x₂ which is a contradiction.

(ii) It results from the fact that Jϕx is a convex part of ∂B(0, ϕ(kxk)) = {x^∗ ∈X^∗| kx^∗k=ϕ(kxk)}.

Proposition 2. If X is locally uniformly convex and Jϕ is single valued (Jϕ: X → X^∗), then Jϕ satisfies the (S+) condition: if xn * x (weakly in X) and lim sup

n→∞ hJϕxn, xn−xi ≤0 thenxn→x(strongly in X).

Proof: It is immediately that from xn* x and lim sup

n→∞ hJϕxn, xn−xi ≤0 it results that lim sup

n→∞ hJϕx_n−J_ϕx, x_n−xi ≤0.

By

0 ≤ ^³ϕ(kxnk)−ϕ(kxk)^{´ ³}kxnk − kxk^´ ≤ ^DJ_ϕx_n−J_ϕx, x_n−x^E

it follows _³

ϕ(kxnk)−ϕ(kxk)^{´ ³}kxnk − kxk|^´ → 0 and hence,kxnk → kxk.

Now, by a well known result (see e.g. Diestel [11]), in a locally uniformly convex space, fromx_n* x and kxnk → kxk it resultsx_n→x.

Proposition 3. IfXis reflexive andJ_ϕ: X→X^∗thenJ_ϕis demicontinuous:

ifx_n→x inX thenJ_ϕx_n* J_ϕxin X^∗.

Proof: By the boundedness of (xn) it follows that (Jϕx_n) is bounded inX^∗. SinceX^∗ is also reflexive, in order to prove that J_ϕx_n * J_ϕx it suffices to show that all subsequences of (Jϕx_n) which are weakly convergent have the same limit, namelyJ_ϕx.

Let x^∗ ∈X^∗ be the weak limit of a subsequence of (Jϕx_n), still denoted by (Jϕx_n).

By the weakly lower semicontinuity of the norm, we have:

kx^∗k ≤ lim inf

n→∞ kJϕx_nk = lim

n→∞ϕ(kxnk) = ϕ(kxk) . On the other hand, from xn→x and Jϕxn* x^∗, it follows that

hJϕx_n, x_ni → hx^∗, xi.

But,

hJϕxn, xni = ϕ(kxnk)kxnk →ϕ(kxk)kxk.

We gethx^∗, xi=ϕ(kxk)kxkand so,ϕ(kxk)≤ kx^∗k. Finallyhx^∗, xi=ϕ(kxk)kxk andϕ(kxk) =kx^∗k, which means x^∗ =J_ϕx.

Theorem 4. LetX be reflexive andJϕ: X →X^∗. Then R(Jϕ) =X^∗. Proof: The result follows from a well known theorem of Browder [8]: if X is reflexive andT: X → X^∗ is monotone, hemicontinuous and coercive, thenT is surjective.

In our case, J_ϕ is monotone by Theorem 1 (ii). The fact thatJ_ϕ is hemicon- tinuous means:

t→0lim D

J_ϕ(u+t v), w^E = hJϕu, wi

foru, v, w∈X, and it results from the demicontinuity ofJ_ϕ (Proposition 3).

Finally,

hJϕu, ui

kuk =ϕ(kuk)→ ∞ with kuk → ∞, henceJϕ is coercive.

Theorem 5. LetX be reflexive, locally uniformly convex andJ_ϕ: X→X^∗. Then J_ϕ is bijective with its inverseJ_ϕ⁻¹ bounded, continuous and monotone.

Moreover, it holds

J_ϕ⁻¹ = χ⁻¹J_ϕ^∗−1

where χ: X →X^∗∗, is the canonical isomorphism betweenX andX^∗∗andJ_ϕ^∗−1: X^∗ → X^∗∗ is the duality mapping on X^∗ corresponding to the normalization functionϕ⁻¹.

Proof: By Theorem 4,J_ϕ is surjective. The spaceX being locally uniformly convex, it is strictly convex (Theorem 2) and by Proposition 1 (i) we have that Jϕ is injective.

Let, now, χ be the canonical isomorphism between X and X^∗∗ (hχ(x), x^∗i= hx^∗, xi) and let J_ϕ^∗−1: X^∗ → X^∗∗ be the duality mapping corresponding to the normalization functionϕ⁻¹. It should be noticed that becauseXis reflexive and locally uniformly convex, so is X^∗∗; in particular X^∗∗ is strictly convex (Theo- rem 2) and, consequently,J_ϕ^∗−1: X^∗ →X^∗∗is single valued (Proposition 1 (ii)).

It is easy to see that:

J_ϕ⁻¹ = χ⁻¹J_ϕ^∗−1 . (1)

From (1) and because a duality mapping maps bounded subsets into bounded subsets, it is immediately thatJ_ϕ⁻¹ is bounded.

To see that J_ϕ⁻¹ is continuous, let x^∗_n→x^∗ inX^∗.

From (1) and by Proposition 3 we have that J_ϕ⁻¹x^∗_n * J_ϕ⁻¹x^∗. By the defini- tion of the duality mappingJϕ, it is easy to see that kJ_ϕ⁻¹x^∗_nk → kJ_ϕ⁻¹x^∗k. But the spaceX is assumed to be locally uniformly convex, and so,J_ϕ⁻¹x^∗_n→J_ϕ⁻¹x^∗. To prove the monotonicity of J_ϕ⁻¹, the spaceX is identified withX^∗∗ by the canonical isomorphismχ. Then, forx^∗₁, x^∗₂ ∈X^∗, we have:

Dχ(J_ϕ⁻¹x^∗₁)−χ(J_ϕ⁻¹x^∗₂), x^∗₁−x^∗₂^E = ^DJ_ϕ^∗−1x^∗₁−J_ϕ^∗−1x^∗₂, x^∗₁−x^∗₂^E

and we apply Theorem 1 (ii) withϕ⁻¹ instead ofϕand X^∗ instead ofX.

1.2. The functional framework

In the sequel, Ω will be a bounded domain in R^N, N ≥ 2 with Lipschitz continuous boundary andp∈(1,∞). We shall use the standard notations:

W^1,p(Ω) =

½

u∈L^p(Ω)| ∂u

∂x_i ∈L^p(Ω), i= 1, ..., N

¾

equipped with the norm:

kuk^p_W1,p(Ω) = kuk^p_0,p+ XN i=1

°°

°°

∂u

∂xi

°°

°°

p

0,p

wherek k0,p is the usual norm on L^p(Ω).

It is well known that (W^1,p(Ω),k kW^1,p(Ω)) is separable, reflexive and uni- formly convex (see e.g. Adams [1, Theorem 3.5]).

We need the space

W₀^1,p(Ω) = the closure of C₀^∞(Ω) in the space W^1,p(Ω)

= ⁿu∈W^1,p(Ω)| u|_∂Ω= 0^o

the value of u on ∂Ω being understood in the sense of the trace: there is a unique linear and continuous operator γ : W^1,p(Ω) → W^1−1p,p(∂Ω) such that γ is surjective and for u ∈ W^1,p(Ω)∩C(Ω) we have γ u = u|∂Ω. It holds W₀^1,p(Ω) = ker γ.

The dual space (W₀^1,p(Ω))^∗ will be denoted byW^−1,p0(Ω), where ¹_p +_p¹0= 1.

For each u∈ W^1,p(Ω), we put

∇u= µ ∂u

∂x₁, ..., ∂u

∂x_N

¶

, |∇u|= Ã_N

X

i=1

µ∂u

∂x_i

¶₂!¹₂

and let us remark that

|∇u| ∈ L^p(Ω), |∇u|^p−2 ∂u

∂xi

∈ L^p0(Ω) for i= 1, ..., N .

Therefore, by the theorem concerning the form of the elements ofW^−1,p0(Ω) (see Br´ezis [6] or Lions [23]) it follows that the operator−∆pmay be seen acting from W₀^1,p(Ω) intoW^−1,p0(Ω) by

h−∆pu, vi = Z

Ω

|∇u|^p−2∇u∇v for u, v∈W₀^1,p(Ω).

By virtue of the Poincar´e inequality

kuk_0,p ≤ Const(Ω, n)k|∇u|k_0,p for all u∈W₀^1,p(Ω), the functional

W₀^1,p(Ω)3u 7−→ kuk1,p: =k|∇u|k0,p

is a norm onW₀^1,p(Ω), equivalent withk k_W^1,p_(Ω).

Because the geometrical properties of the space are not automatically main- tained by passing to an equivalent norm, we give a direct proof of the following theorem

Theorem 6. The space(W₀^1,p(Ω),k k1,p) is uniformly convex.

Proof: First, let p ∈ [2,∞). Then (see e.g. Adams [1, pp. 36]) for each z, w∈R^N, it holds:

¯¯

¯¯ z+w

2

¯¯

¯¯

p

+

¯¯

¯¯ z−w

2

¯¯

¯¯

p

≤ 1 2

³|z|^p+|w|^p´ .

Let u, v ∈W₀^1,p satisfy kuk1,p =kvk1,p = 1 and ku−vk1,p ≥ ε∈ (0,2]. We have

°°

°° u+v

2

°°

°°

p 1,p

+

°°

°° u−v

2

°°

°°

p 1,p

= Z

Ω

ﯯ¯

∇u+∇v 2

¯¯

¯¯

p

+

¯¯

¯¯

∇u− ∇v 2

¯¯

¯¯

p!

≤ 1 2

Z

Ω

³|∇u|^p+|∇v|^p´ = 1 2

³kuk^p_1,p+kvk^p_1,p^´ = 1

which yields

°°

°° u+v

2

°°

°°

p 1,p

≤ 1− µε

2

¶p

(2) .

If p∈(1,2), then (see e.g. Adams [1, pp. 36]) for eachz, w∈R^N, it holds:

¯¯

¯¯ z+w

2

¯¯

¯¯

p⁰

+

¯¯

¯¯ z−w

2

¯¯

¯¯

p⁰

≤

·1 2

³|z|^p+|w|^p´¸

1 p−1 .

A straightforward computation shows that if v∈W₀^1,p(Ω) then|∇v|^p0∈L^p−1(Ω) andk|∇v|^p0k0,p−1 =kvk^p_1,p⁰ .

Let v₁, v₂ ∈ W₀^1,p(Ω). Then |∇v1|^p0,|∇v2|^p0 ∈ L^p−1(Ω), with 0 < p−1 < 1 and, according to Adams [1, pp. 25],

°°

°|∇v1|^p0+|∇v2|^p0°°_°

0,p−1 ≥ k|∇v1|^p0k0,p−1+k|∇v2|^p0k0,p−1 . Consequently,

°°

°°

v₁+v₂ 2

°°

°°

p⁰ 1,p

+

°°

°°

v₁−v₂ 2

°°

°°

p⁰ 1,p

=

°°

°°

¯¯

¯¯∇v₁+v₂ 2

¯¯

¯¯

p⁰°°°°

0,p−1

+

°°

°°

¯¯

¯¯∇v₁−v₂ 2

¯¯

¯¯

p⁰°°°°

0,p−1

≤

°°

°°

°

¯¯

¯¯∇v₁+v₂ 2

¯¯

¯¯

p⁰

+

¯¯

¯¯∇v₁−v₂ 2

¯¯

¯¯

p⁰°°°°°

0,p−1

=



 Z

Ω

ﯯ¯

∇v1+∇v2

2

¯¯

¯¯

p⁰

+

¯¯

¯¯

∇v1−∇v2

2

¯¯

¯¯

p⁰!p−1 



1 p−1

≤

"

1 2

Z

Ω

³|∇v₁|^p+|∇v₂|^p´

#_p−1¹

=

·1

2kv1k^p_1,p+ 1

2kv2k^p_1,p

¸_p−1¹ .

For u, v ∈ W₀^1,p(Ω) with kuk_1,p = kvk1,p = 1 and ku−vk_1,p ≥ ε∈ (0,2], we

get °°°°

u+v 2

°°

°°

p⁰ 1,p

≤ 1− µε

2

¶p⁰

(3) .

From (2) and (3), in either case there exists δ(ε) >0 such that ku+vk1,p ≤ 2 (1−δ(ε)).

Below, the space W₀^1,p(Ω) always will be considered to be endowed with the normk k1,p.

Theorem 7. The operator −∆p: W₀^1,p(Ω)→ W^−1,p0(Ω)is a potential one.

More precisely, its potential is the functional ψ: W₀^1,p(Ω)→R, given by ψ(u) = 1

pkuk^p_1,p and

ψ⁰ = −∆p = J_ϕ

where Jϕ: W₀^1,p(Ω) → W^−1,p0(Ω) is the duality mapping corresponding to the normalization functionϕ(t) =t^p−1.

Proof: Since ψ(u) =

kukR1,p

0

ϕ(t)dt, it is sufficient to prove that ψ is Gˆateaux differentiable and ψ⁰(u) = −∆pu for all u ∈ W₀^1,p(Ω) (see Theorem 1 (iii) and Remark 1).

If u ∈ W₀^1,p(Ω) is such that |∇u| = 0_L^p_(Ω) (this implies that kuk1,p = 0 i.e.

u = 0_W^1,p

0 (Ω)), then it is immediately that hψ⁰(u), hi = 0 for all h ∈ W₀^1,p(Ω).

Therefore, we may suppose that|∇u| 6= 0L^p(Ω).

It is obvious thatψcan be written as a product ψ=QP, where Q:L^p(Ω)→R is given byQ(v) = ¹_pkvk^p_0,p and P: W₀^1,p(Ω)→L^p(Ω) is given by P(v) =|∇v|.

The functional Qis Gˆateaux differentiable (see Vainberg [28]) and hQ⁰(v), hi = h|v|^p−1signv, hi

(4)

for allv, h∈L^p(Ω).

Simple computations show that the operator P is Gˆateaux differentiable atu and

P⁰(u)·v = ∇u∇v (5) |∇u|

for allv∈W₀^1,p(Ω).

Combining (4) and (5), we obtain that ψ is Gˆateaux differentiable atu and hψ⁰(u), vi = ^DQ⁰(P(u)), P⁰(u)·v^E

=

¿

|∇u|^p−1, ∇u∇v

|∇u|

À

= Z

Ω

|∇u|^p−2∇u∇v = h−∆pu, vi

for allv∈W₀^1,p(Ω).

Remark 2. Let k k∗ be the dual norm ofk k1,p. Then, we have k−∆puk∗ = kJϕuk∗ = ϕ(kuk1,p) = kuk^p−1_1,p .

Theorem 8. The operator −∆p defines a one-to-one correspondence be- tween W₀^1,p(Ω) and W^−1,p0(Ω), with inverse (−∆p)⁻¹ monotone, bounded and continuous.

Proof: It is obvious from Theorems 7, 6 and 5.

Remark 3. In fact, the above Theorem 8 asserts that for eachf ∈W^−1,p0(Ω), the equation−∆pu=f has a unique solution in W₀^1,p(Ω).

The properties of (−∆p)⁻¹ show how the solution u= (−∆p)⁻¹f depends on the dataf. These properties will be used in the sequel.

Since the elements of W₀^1,p(Ω) vanish on the boundary∂Ω in the sense of the trace, it is natural that the unique solution inW₀^1,p(Ω) of the equation−∆pu=f to be called solution of the Dirichlet problem

(−∆pu=f , u|∂Ω= 0 .

We shall conclude this section with two technical results which will be useful in the sequel.

We have seen (Theorem 7) that the functional ψ(u) = ¹_pkuk^p_1,p is Gˆateaux differentiable onW₀^1,p(Ω). Moreover, we have:

Theorem 9. The functional ψ is continuously Fr´echet differentiable on W₀^1,p(Ω).

For the proof we need the following lemma (see Glowinski and Marrocco [16]).

Lemma 1.

(i) If p∈[2,∞) then it holds:

¯¯

¯|z|^p−2z− |y|^p−2y^¯¯_¯ ≤ β|z−y|^³|z|+|y|^´p−2 for all y, z ∈R^N withβ independent of y andz;

(ii) If p∈(1,2], then it holds:

¯¯

¯|z|^p−2z− |y|^p−2y^¯¯_¯ ≤ β|z−y|^p−1 for all y, z ∈R^N withβ independent of y andz.

Proof of Theorem 9: Consider the product space X =^QN_i=1L^p0(Ω) endowed with the norm

[h]_0,p⁰ = µXN

i=1

khik^p_0,p⁰ 0

¶¹

p0

for h= (h₁, .., h_N)∈ X.

We define g= (g₁, ..., gN) : W₀^1,p(Ω)→ X by g(u) = |∇u|^p−2∇u , foru∈W₀^1,p(Ω).

Let us prove that g is continuous.

By the equivalence of the norms on R^N we can find a constant C₁ >0 such that

[h]^p_0,p⁰ 0 ≤ C₁ Z

Ω

|h|^p0 ,

for allh∈ X.

Let p∈(2,∞) and u, v∈W₀^1,p(Ω). By Lemma 1 (i) and by the H¨older inequality, we have:

hg(u)−g(v)^ip

0

0,p ≤ C₁ Z

Ω

¯¯

¯g(u)−g(v)^¯¯_¯^p

0

≤ C₂ Z

Ω

|∇u− ∇v|^p0³|∇u|+|∇v|^´p

0(p−2)

≤ C₂ku−vk^p_1,p^{0 °°}_°|∇u|+|∇v|^°°_°^p

0(p−2) 0,p

which yields

hg(u)−g(v)ⁱ

0,p ≤ Cku−vk^p_1,p^{0 ³}kuk_1,p+kvk_1,p^´p

0(p−2)

(6)

withC >0 constant independent of u and v.

If p∈(1,2] and u, v∈W₀^1,p(Ω), then from Lemma 1 (ii) it follows hg(u)−g(v)^ip

0

0,p⁰ ≤ C₂⁰ Z

Ω

|∇u− ∇v|^p0(p−1) = C₂⁰ ku−vk^p_1,p

or _h

g(u)−g(v)ⁱ

0,p⁰ ≤ C⁰ku−vk^p−1_1,p (7)

withC⁰>0 constant independent ofu and v.

From (6) and (7) the continuity of g is obvious.

On the other hand, it holds

°°

°ψ⁰(u)−ψ⁰(v)^°°_°

∗ ≤ K^hg(u)−g(v)ⁱ

0,p⁰

(8)

withK >0 constant independent of u, v∈W₀^1,p(Ω).

Indeed, by the H¨older inequality and by the equivalence of the norms on R^N, we successively have:

¯¯

¯ D

ψ⁰(u)−ψ⁰(v), w^E¯¯_¯ ≤ Z

Ω

¯¯

¯g(u)−g(v)^¯¯_¯|∇w|

≤ µ Z

Ω

¯¯

¯g(u)−g(v)^¯¯_¯^p

0¶¹

p0 µ Z

Ω

|∇w|^p

¶¹

p

≤ K µXN

i=1

°°

°gi(u)−gi(v)^°°_°^p

0

0,p⁰

¶¹

p0

kwk1,p

= K^hg(u)−g(v)ⁱ

0,p⁰kwk1,p

for u, v, w∈W₀^1,p(Ω), proving (8).

Now, by the continuity of g and (8), the conclusion of the theorem follows in a standard way: a functional is continuously Fr´echet differentiable if and only if it is continuously Gˆateaux differentiable.

Remark 4. Naturally, the Fr´echet differential of ψ at u∈ W₀^1,p(Ω) will be denoted byψ⁰(u) and it is clear that ψ⁰(u) =−∆pu.

Theorem 10. The operator −∆p satisfies the (S+) condition: if un * u (weakly inW₀^1,p(Ω)) and lim sup

n→∞ h−∆pun, un−ui ≤0, thenun→u(strongly in W₀^1,p(Ω)).

Proof: It is a simple consequence of Proposition 2, Theorems 6, 2 and 7.

2 – The problem −∆pu=f(x, u), u|_∂Ω = 0

In this paragraph we are interested about sufficient conditions on the right- hand memberfensuring the existence of someu∈W₀^1,p(Ω) such that the equality

−∆pu=f(x, u) holds in the sense ofW^−1,p0(Ω). Such anuwill be called solution of the Dirichlet problem

(−∆pu=f(x, u) in Ω , u|∂Ω = 0 .

(9)

Thus, first of all, appropriate conditions onf ensuring thatN_fu∈W^−1,p0(Ω) must be formulated,Nf being the well known Nemytskii operator defined by f, i.e. (Nfu)(x) =f(x, u(x)) forx∈Ω. Hence, we are guided to consider some basic results on the Nemytskii operator. Simple proofs of these facts can be found in e.g. de Figueiredo [14] or Kavian [20] (see also Vainberg [28]).

2.1. Detour on the Nemytskii operator

Let Ω be as in the beginning of Section 1.2 andf: Ω×R→Rbe aCarath´eodory function, i.e.:

(i) for eachs∈R, the function x7→f(x, s) is Lebesgue measurable in Ω;

(ii) for a.e.x∈Ω, the functions7→f(x, s) is continuous inR.

We make the convention that in the case of a Carath´eodory function, the assertion “x∈Ω” to be understood in the sense “a.e. x∈Ω”.

Let Mbe the set of all measurable function u: Ω→R.

Proposition 4. If f: Ω×R→R is Carath´eodory, then, for each u ∈ M, the functionN_fu: Ω→Rdefined by

(N_fu)(x) =f(x, u(x)) for x∈Ω is measurable inΩ.

In view of this proposition, a Carath´eodory function f: Ω×R→Rdefines an operatorN_f: M → M, which is called Nemytskii operator.

The proposition here below states sufficient conditions when a Nemytskii op- erator maps anL^p1 space into another L^p2 space.

Proposition 5. Suppose f: Ω×R→R is Carath´eodory and the following growth condition is satisfied:

|f(x, s)| ≤ C|s|^r+b(x) for x∈Ω, s∈R, whereC≥0 is constant, r >0and b∈L^q1(Ω),1≤q₁ <∞.

Then Nf(L^q1r(Ω)) ⊂L^q1(Ω). Moreover, Nf is continuous from L^q1r(Ω) into L^q1(Ω)and maps bounded sets into bounded sets.

Concerning the potentiality of a Nemytskii operator, we have:

Proposition 6. Suppose f: Ω×R→Ris Carath´eodory and it satisfies the growth condition:

|f(x, s)| ≤ C|s|^q−1+b(x) for x∈Ω, s∈R , whereC≥0 is constant, q >1,b∈L^q0(Ω), ¹_q+_q¹0 = 1.

Let F: Ω×R→Rbe defined byF(x, s) =^Rs

0

f(x, τ)dτ.

Then:

(i) the function F is Carath´eodory and there exist C₁ ≥ 0 constant and c∈L¹(Ω)such that

|F(x, s)| ≤ C₁|s|^q+c(x) for x∈Ω, s∈R; (ii) the functional Φ : L^q(Ω)→R defined by Φ(u) : =^R

Ω

NF u = ^R

Ω

F(x, u) is continuously Fr´echet differentiable andΦ⁰(u) =N_Fu for all u∈L^q(Ω).

It should be noticed that, under the conditions of the above Proposition 6, we haveN_f(L^q(Ω))⊂L^q0(Ω), N_F(L^q(Ω))⊂L¹(Ω), each of the Nemytskii oper- atorsN_f and NF being continuous and bounded (it is a simple consequence of Proposition 5). It should also be noticed that for each fixedu∈L^q(Ω), it holds Nfu= Φ⁰(u)∈L^q0(Ω).

Now, we return to problem (9).

First, let us denote by p^∗ the Sobolev conjugate exponent ofp, i.e.

p^∗ =







 N p

N−p if p < N ,

∞ if p≥N .

Below, the functionf: Ω×R→Rwill be always assumed Carath´eodory and satisfying the growth condition

|f(x, s)| ≤ C|s|^q−1+b(x) for x∈Ω, s∈R, (10)

whereC≥0 is constant, q∈(1, p^∗), b∈L^q0(Ω), ¹_q +_q¹0 = 1.

The restriction q ∈ (1, p^∗) ensures that the imbedding W₀^1,p(Ω) ,→ L^q(Ω) is compact. Hence, the diagram

W₀^1,p(Ω),→^Id L^q(Ω)^N→^f L^q0(Ω) ^I

∗

,→d W^−1,p0(Ω)

shows that N_f is a compact operator (continuous and maps bounded sets into relatively compact sets) fromW₀^1,p(Ω) intoW^−1,p0(Ω).

An element u∈W₀^1,p(Ω) is said to besolution of problem (9) if

−∆pu = N_fu (11)

in the sense ofW^−1,p0(Ω) i.e.

h−∆pu, vi=hNfu, vi for all v∈W₀^1,p(Ω)

or _Z

Ω

|∇u|^p−2∇u∇v = Z

Ω

f(x, u)v for all v∈W₀^1,p(Ω). (12)

At this stage, in the approach of problem (9), two strategies appear to be natural.

The first reduces problem (9) to a fixed point problem with compact operator.

Indeed, by Theorem 8, the operator (−∆p)⁻¹ : W^−1,p0(Ω) → W₀^1,p(Ω) is bounded and continuous.

Consequently, (11) can be equivalently written u = (−∆p)⁻¹N_fu (13)

with (−∆p)⁻¹N_f: W₀^1,p(Ω)→W₀^1,p(Ω) a compact operator.

The second is a variational one: the solutions of problem (9) appear as critical points of aC¹ functional on W₀^1,p(Ω).

To see this, we first have that−∆p =ψ⁰, where the functionalψ(u) = ¹_pkuk^p_1,p is continuously Fr´echet differentiable onW₀^1,p(Ω). On the other hand, under the basic condition (10) and taking into account that the imbedding W₀^1,p(Ω) → L^q(Ω) is continuous (in fact, compact), the functional Φ : W₀^1,p(Ω)→R defined by Φ(u) =^R

Ω

F(x, u) with F(x, s) =^Rs

0

f(x, τ)dτ, is continuously Fr´echet differen- tiable onW₀^1,p(Ω) and Φ⁰(u) =N_fu.

Consequently, the functional F: W₀^1,p(Ω)→Rdefined by F(u) = ψ(u)−Φ(u) = 1

pkuk^p_1,p− Z

Ω

F(x, u)

isC¹ inW₀^1,p(Ω) and

F⁰(u) = (−∆p)u−Nfu .

The search for solutions of problem (9) is, now, reduced to the search of critical points ofF, i.e. of thoseu∈ W₀^1,p(Ω) such thatF⁰(u) = 0.

2.2. Existence of fixed points for (−∆p)⁻¹N_f via a Leray–Schauder technique

In this section, the “a priori estimate method” will be used in order to es- tablish the existence of fixed points for the compact operator T = (−∆p)⁻¹Nf: W₀^1,p(Ω)→W₀^1,p(Ω) (see Dinca and Jebelean [13]).

For it suffices to prove that the set

S = ⁿu∈W₀^1,p(Ω)| u=α T u for someα∈[0,1]^o is bounded inW₀^1,p(Ω).

By (10), for arbitrary u∈W₀^1,p(Ω), it is obvious that kT uk^p_1,p = ^D(−∆p)T u, T u^E = hNfu, T ui =

Z

Ω

f(x, u)T u

≤ Z

Ω

³C|u|^q−1+b(x)^´|T u|.

Furthermore, for u∈ S i.e.u=α T u, with someα∈[0,1], we have kT uk^p_1,p ≤ C α^q−1kT uk^q_0,q +kbk_0,q⁰kT uk_0,q

≤ C α^q−1C₁^qkT uk^q_1,p+kbk0,q⁰C₁kT uk1,p

≤ C C₁^qkT uk^q_1,p+kbk_0,q⁰C₁kT uk1,p

the constantC₁ coming from the continuous imbedding W₀^1,p(Ω)→L^q(Ω).

Consequently, for each u∈ S, it holds

kT uk^p_1,p−K₁kT uk^q_1,p−K₂kT uk_1,p ≤ 0 (14)

withK₁, K₂ ≥0 constants.

Remark that if (14) would imply that there is a constant a ≥ 0 such that kT uk1,p ≤a, then the boundedness ofS would be proved, because we would have kuk1,p=αkT uk1,p≤a.

But this is obviously true if q ∈(1, p).

We have obtained

Theorem 11. If the Carath´eodory function f : Ω×R→R satisfies (10) with q ∈ (1, p) then the operator (−∆p)⁻¹N_f has fixed points in W₀^1,p(Ω) or equivalently, problem (9) has solutions. Moreover, the set of all solutions of problem (9) is bounded in the spaceW₀^1,p(Ω).

Remark 5. We shall see that if (10) holds with b ∈ L^∞(Ω) then the vari- ational approach allows to weaken the hypotheses of Theorem 11 and problem (9) still has solutions but the boundedness of the set of all solutions will not be ensured.

Remark 6. The conditionq∈(1, p) appear as a technical condition, needed in obtaining the boundedness ofS.

It is a natural question if the set S still remains bounded in case that q =p and it is a simple matter to see that if q = p and 1−K₁ > 0 then the above reasoning still works. This means that we are interested to work with “the best constants”C and C₁ such that 1−C·C₁^p be strictly positive.

There are situations when 1−C·C₁^p >0 fails. The example here below shows that thenS can be unbounded.

Let λbe an eigenvalue of −∆p in W₀^1,p(Ω) and u be a corresponding eigen- vector:

−∆pu = λ|u|^p−2u . It is clear that

λ = kuk^p_1,p kuk^p_0,p . (15)

Becausekvk0,p≤C₁kvk_1,pfor allv∈W₀^1,p(Ω), from (15), it results that 1−λC₁^p≤0.

Consider the Carath´eodory function f(x, s) = λ|s|^p−2s. Clearly, the growth condition (10) is satisfied with q = p and b = 0, c = λ. Consequently, (14) becomes

(1−λ C₁^p)kT uk^p_1,p ≤ 0

for allu∈ S and no conclusion on the boundedness ofS can be derived.

In fact, S is unbounded.

Indeed, we have −∆p(t u) =Nf(t u) i.e. t u= (−∆p)⁻¹Nf(t u) for all t ∈R, which means{t u|t∈R} ⊂ S and so,S is unbounded.

Remark 7. In the case f(x, s) = g(s) +h(x) with g: R→R continuous and h ∈ L^∞(Ω), the homotopy invariance of Leray–Schauder degree (but in a different functional framework) is used by Hachimi and Gossez [17] in order to prove the following result (see [17] Th 1.1):

If

(i) lim sup

s→±∞

g(s)

|s|^p−2s ≤ λ₁ and (ii) lim sup

s→±∞

pG(s)

|s|^p < λ₁

where G(s) = Rs 0

g(τ)dτ and λ₁ is the first eigenvalue of −∆p in W₀^1,p(Ω), then the problem

−∆pu = g(u) +h(x) in Ω, u= 0 on∂Ω has a solution inW₀^1,p(Ω)∩L^∞(Ω).

2.3. Existence results by a direct variational method

As we have already emphasized in Section 2.1, in the variational approach, under the growth condition (10) onf, the solutions of problem (9) are precisely the critical points of theC¹ functionalF: W₀^1,p(Ω)→R defined by

F(u) = ψ(u)−Φ(u) = 1

pkuk^p_1,p − Z

Ω

F(x, u)

whereF(x, u) = Rs 0

f(x, τ)dτ.

Remark that the compact imbedding W₀^1,p(Ω) ,→ L^q(Ω) implies that F is weakly lower semicontinuous inW₀^1,p(Ω) .

So, by a standard result, in order to derive sufficient conditions for (9) has solutions, a first suitable way is to ensure the coerciveness of F. Such results

were obtained by Anane and Gossez [4] even in more general conditions onf. It is not our aim to detail this direction. However, we depict a few such of results.

First we refer to a result of Anane and Gossez [4].

Let G: Ω×R→R be a Carath´eodory function, such that, for anyR >0, Ω 3 x→ sup

|s|≤R

|G(x, s)| ∈L¹(Ω). (16)

We write G(x, s) = ^λ1_p^|s|p +H(x, s), where λ₁ is the first eigenvalue of −∆p

on W₀^1,p(Ω) (see e.g. Anane [3], Lindqvist [22]) and let us define H^±(x) as the superior limit of ^H(x,s)_|s|p ass→ ±∞ respectively.

It holds (see Proposition 2.1 in Anane–Gossez [4]):

Theorem 12. Assume (16) and (i) H^±(x)≤0 a.e. uniformly inx;

(ii) H⁺(x) < 0 on Ω⁺ and H⁻(x) < 0 on Ω⁻ for subsets Ω^± of positive measure.

Then

G(u) = 1

pkuk^p_1,p− Z

Ω

G(x, u)

is well defined onW₀^1,p(Ω), takes values in]− ∞,+∞], is weakly lower semicon- tinuous and coercive.

We now return to problem (9).

By virtue of Proposition 6 (i), we have for any R >0 sup

|s|≤R

|F(x, s)| ≤ C₁R^q+c(x) ∈ L¹(Ω) showing that (16) is fulfilled withG(x, s) =F(x, s).

Clearly, in this case H(x, s) =F(x, s)−^λ1|s|_p ^p.

In order to extend a result of Mawhin–Ward–Willem [25] for the particular case p = 2 to the general case p ∈ (1,∞), suppose that there exists a function α(x)∈L^∞(Ω) withα(x)< λ₁, on a set of positive measure, such that

lim sup

s→±∞

p F(x, s)

|s|^p ≤ α(x) ≤ λ₁ uniformly in Ω . (17)

We obtain

H^±(x) = lim sup

s→±∞

H(x, s)

|s|^p = lim sup

s→±∞

µF(x, s)

|s|^p − λ₁ p

¶

=

= lim sup

s→±∞

F(x, s)

|s|^p −λ₁

p ≤ α(x)−λ₁ p

which yieldsH^±(x)≤0 uniformly in Ω, i.e. (i) in Theorem 12.

On the other hand, it is clear that

H^±(x) ≤ α(x)−λ₁ p < 0

on the set of positive measure Ω₁ ={x∈Ω|α(x)< λ₁} and (ii) in Theorem 12 is checked.

We have obtained

Theorem 13. Letf: Ω×R→Rbe a Carath´eodory function satisfying the growth condition (10). Suppose that there existsα(x) ∈L^∞(Ω)with α(x)< λ₁ on a set of positive measure such that (17) holds.

Then F is coercive; consequently problem (9) has solutions.

A direct proof of Theorem 13 can be given as it follows.

Define N: W₀^1,p(Ω)→R by

N(v) = kvk^p_1,p− Z

Ω

α(x)|v|^p

and let us prove that there existsε₀ >0 such that

N(v)≥ε₀ for all v∈W₀^1,p(Ω) with kvk_1,p= 1 . (18)

For, let us recall (see e.g. Anane [3]) that λ₁ = inf

(kvk^p_1,p

kvk^p_0,p | v∈W₀^1,p(Ω)\{0}

) (19)

the infimum being attained exactly whenv is multiple of some functionu₁>0.

By (17) and (19) it follows that N(v)≥0 for all v∈W₀^1,p(Ω).

Supposing, by contradiction, that there is a sequence (vn) in W₀^1,p(Ω) with kvnk1,p= 1 and N(vn)→ 0, we can find a subsequence of (vn), still denoted by (vn), and some v₀ ∈ W₀^1,p(Ω) with vn* v₀, weakly in W₀^1,p(Ω) and vn→ v₀, strongly inL^p(Ω).