MULTIPLE SOLUTIONS TO OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS ON ORLICZ-SOBOLEV SPACES VIA THE MOUNTAIN PASS THEOREM

on his 70th birthday

MULTIPLE SOLUTIONS TO OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS

ON ORLICZ-SOBOLEV SPACES VIA THE MOUNTAIN PASS THEOREM

GEORGE DINC€ and PAVEL MATEI

LetX be a real reexive smooth Banach space having the Kade£-Klee property, compactly imbedded in a real Banach spaceV and letG:V →Ra dierentiable functional. By using theZ²-version of the Mountain Pass Theorem (Rabinowitz [17]), the multiplicity of solutions to operator equationJϕu=G⁰(u), whereJϕis the duality mapping onX, corresponding to the gauge function,ϕis studied. Sim- ilar results are obtained in [12] and [13] by using the fountain theorem (Bartsch [3]) and the dual fountain theorem (Bartsch-Willem [4]), respectively. Equations of the above form withJϕ a duality mapping on Orlicz-Sobolev spaces, are con- sidered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces may be derived.

AMS 2000 Subject Classication: 35B38, 47J30.

Key words: critical point, mountain pass theorem, duality mapping, Orlicz- Sobolev space.

1. INTRODUCTION

This paper is concerned with multiplicity results for equations of the type

(1.1) J_ϕu=G⁰(u),

where

(i) Xis a real reexive and smooth Banach space having the Kade£-Klee property, compactly imbedded in the real Banach space V;

(ii) J_ϕ : X → X^∗ is a duality mapping corresponding to the gauge function ϕ(see Denition 1, below);

(iii)G⁰ :V →V^∗ is the dierential of the functional G:V →R.

As usual, X^∗ (resp. V^∗) denotes the dual space of X (resp. V) and h·,·i_X,X∗ (resp. h·,·i_V,V∗) denotes the duality pairing between X^∗ and X

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 419437

(resp. V^∗ and V). Often, we shall omit to indicate the spaces in duality and, simply, we shall write h·,·i.

Our approach is a variational one, the Z2-version of the Mountain Pass Theorem due to Rabinowitz ([17]) being the basic ingredient which is used.

Equations of the form (1.1) withJϕ a duality mapping on Orlicz-Sobolev spaces, are considered as applications. As particular cases of these results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Moreover, these results apply to many dierential operators which in fact are duality mappings on some appropriate spaces of functions (for example, if

∆p,1< p <∞, is the so calledp-Laplacian, then−∆_p is the duality mapping on W₀^1,p(Ω)corresponding to the gauge functionϕ(t) =t^p−1,t≥0).

2. THE MAIN RESULT

Theorem 1. Let X be a real reexive smooth Banach space having the Kade£-Klee property and compactly imbedded in a real Banach space V. Let H ∈ C¹(X,R) be an even functional of the form

(2.1) H = Ψ−G,

where

(i) Ψ(u) = Φ(kuk) at any u∈X with

(2.2) Φ(t) =

Z _t

0

ϕ(ξ) dξ, ∀t≥0,

ϕ:R+→R+ being a gauge function which satisesp^∗= sup

t>0 tϕ(t) Φ(t) <∞; (ii)G:V →R satises:

(ii)₀ G(0) = 0;

(ii)₁ G⁰ :V →V^∗ is demicontinuous;

(ii)₂ there is a constant θ > p^∗ such that, at any y∈V,

(2.3)

G⁰(y), y

V,V^∗−θG(y)≥C=const.;

(iii) there exists c0>0 such that for any u∈X with kuk_X < c0 one has (2.4) H(u)> c1kuk^p_X −c2ki(u)k^q_V,

whereistands for the compact injection ofX inV while0< p < qandc₁ >0, c2>0;

(iv) for any nite dimensional subspaceX₁⊂X, there exist real constants d0 >0, d1, d2>0,d3, s >0 andr < s (generally depending onX1) such that (2.5) H(u)≤d1kuk^r_X −d2kuk^s_X +d3,

for any u∈X1 with kuk_X > d0.

Then the functionalHpossesses an unbounded sequence of critical values.

Before proceeding to the proof of Theorem 1, we list some results we need.

First, we recall that a real Banach spaceX is said to be smooth if it has the following property: for anyx∈X,x6= 0, there exists a uniqueu^∗(x)∈X^∗ such that hu^∗(x), xi = kxk_X and ku^∗(x)k_X^∗ = 1. It is well known (see, for instance, Diestel [8], Zeidler [22] ) that the smoothness of X is equivalent to the Gâteaux dierentiability of the norm. Consequently, if (X,k · k_X) is smooth, then for any x ∈ X, x 6= 0, the only element u^∗(x) ∈ X^∗ with the properties hu^∗(x), xi =kxk_X and ku^∗(x)k_X^∗ = 1 is u^∗(x) =k · k⁰_X(x) (where k · k⁰_X(x) denotes the Gâteaux gradient of the k · k_X-norm at x).

A functionϕ:R+→R+is said to be a gauge function ifϕis continuous, strictly increasing, ϕ(0) = 0, and ϕ(t)→ ∞ast→ ∞.

Denition 1. If X is a real smooth Banach space andϕ:R+→ R+ is a gauge function, the duality mapping on X corresponding toϕ is the mapping J_ϕ:X →X^∗ dened by

(2.6) J_ϕ0 = 0, J_ϕx=ϕ(kxk_X)k · k⁰_X(x) if x6= 0.

The following metric properties are consequences of Denition 1:

(2.7) kJ_ϕxk_X^∗ =ϕ(kxk_X), hJ_ϕx, xi=ϕ(kxk_X)kxk_X, ∀x∈X.

For the main properties of duality mappings, see [6], [9], [22].

In order to state the next result, we recall that ifXis a real Banach space and H ∈ C¹(X,R), we say that H satises the Palais-Smale condition onX ((P S)-condition, for short) if any sequence (u_n) ⊂X with (H(u_n)) bounded and H⁰(u_n)→0 asn→ ∞, possesses a convergent subsequence.

The basic result we need for proving Theorem 1 is theZ2-version of the Mountain Pass Theorem due to Rabinowitz:

Theorem 2 (Rabinowitz [17, Theorem 9.12]). Let X be an innite di- mensional real Banach space. Assume H ∈ C¹(X,R) is even, satises the (P S)-condition and H(0) = 0. If

(G₁) there exist ρ >0 andr >0 such that H(u)≥r forkuk=ρ; (G2) for each nite dimensional subspace X1 of X the set {u ∈ X1 | H(u) ≥ 0} is bounded, then H possesses an unbounded sequence of critical values.

Now, we are able to give the proof of Theorem 1. The idea is that the hypotheses of Theorem 1 entail those of Theorem 2.

Indeed, the following result holds (see [12, Corollary 2]):

Proposition 1. Let X be a real reexive smooth Banach space having the Kade£-Klee property and compactly imbedded in a real Banach space V. Let H ∈ C¹(X,R) be a functional of the form

(2.8) H = Ψ−G,

where

(i) Ψ(u) = Φ(kuk) at any u∈X with Φ(t) =

Z t 0

ϕ(s) ds, ∀t≥0,

and ϕ:R+→R+ being a gauge function which satises sup

t>0

tϕ(t)

Φ(t) =p^∗ <∞;

(ii)G:V →R satises:

(ii)₁ G⁰ :V →V^∗ is demicontinuous;

(ii)₂ there is a constant θ > p^∗ such that

(2.9)

G⁰(y), y

Y,Y^∗−θG(y)≥C =const. ∀y∈V. ThenH satises the (P S)-condition.

Proof. It is now clear that, under the hypotheses of Theorem 1, the fact that H satises the(P S)-condition is a direct consequence of Proposition 1.

We will show that hypothesis(G)₁ of Theorem 2 is satised.

Indeed, taking into account that ki(u)k_V ≤ ckuk_X, ∀x ∈ X, it follows from (2.4) that

H(u)>kuk^p_X ·

c1−c2c^qkuk^q−p_X for all u∈X withkuk_X < c0. So, for

kuk_X =ρ≤min c0, c1

2c^qc2

_q−p¹ ! , we have

H(u)> C

2ρ^p >0,

that is, hypothesis (G1) of Theorem 2 is fullled withr = ^C₂ρ^p.

On the other hand, let X₁ be a nite dimensional subspace of X. We shall show that the set S = {u∈X1|H(u)≥0} is bounded. Indeed, taking into account (2.5), if u∈S,kuk_X > d₀, then

(2.10) d1kuk^r_X−d2kuk^s_X +d3≥0.

Sinces > r, we conclude thatSis bounded, thus hypothesis(G2)of Theorem 2 is fullled.

The conclusion follows from Theorem 2.

3. APPLICATIONS TO ORLICZ-SOBOLEV SPACES

Throughout this sectionΩdenotes a bounded open subset ofR^N,N ≥2. Leta:R→Rbe a strictly increasing odd continuous function with lim

t→+∞a(t) = +∞. For m∈N^∗ let us denote by W₀^mEA(Ω)the Orlicz-Sobolev space gene- rated by the N-functionA, dened by

(3.1) A(t) =

Z t 0

a(s)ds. We shall always suppose that

(3.2) lim

t→0

Z 1 t

A⁻¹(τ)

τ^N+1N dτ <∞,

replacing, if necessary, A by anotherN-function equivalent to A near innity (which determines the same Orlicz space).

Suppose also that

(3.3) lim

t→∞

Z t 1

A⁻¹(τ)

τ^N+1N dτ =∞.

With (3.3) satised, we dene the Sobolev conjugate A∗ ofA by setting

(3.4) A⁻¹_∗ (t) =

Z t 0

A⁻¹(τ)

τ^N+1N dτ, t≥0.

The existence and multiplicity of weak solutions to the boundary value problem

(3.5) Jau= X

|α|<m

(−1)^|α|D^αgα(x, D^αu) inΩ,

(3.6) D^αu= 0 on∂Ω, |α| ≤m−1,

is studied in this section in the following functional framework:

•T[u, v]is a nonnegative symmetric bilinear form on the Orlicz-Sobolev space W₀^mE_A(Ω), only involving the generalized derivatives of order m of the functions u, v∈W₀^mEA(Ω), satisfying

(3.7) c₁ X

|α|=m

(D^αu)² ≤T[u, u]≤c₂ X

|α|=m

(D^αu)², ∀u∈W₀^mL_A(Ω),

with c1,c2 positive constants;

• kuk_m,A =kp

T[u, u]k_(A) is a norm on W₀^mE_A(Ω), k · k_(A) designating the Luxemburg norm on the Orlicz space LA(Ω);

•J_a: (W₀^mE_A(Ω),k · k_m,A)→(W₀^mE_A(Ω),k · k_m,A)^∗ is the duality map- ping on (W₀^mEA(Ω),k · k_m,A)subordinated to the gauge function a;

• g_α : Ω×R → R, |α| < m, are Carathéodory functions satisfying hypotheses (H)₁ and(H)₂ below:

(H)₁ there exist N-functions Mα, |α| < m, which increase essentially more slowly than A∗ near innity and satisfy the ∆₂-condition, such that (3.8) |g_α(x, s)| ≤c_α(x) +d_αM⁻¹_α (M_α(s)), x∈Ω,s∈R, |α|< m, where M_α are the complementary N-functions to M_α,c_α ∈K_M

α and d_α are positive constants;

(H)₂ for any α with|α|< m, there existsα >0 and θα > p^∗ = sup

t>0 ta(t) A(t)

such that

(3.9) 0< θ_αG_α(x, s)≤sg_α(x, s) for a.e. x∈Ω and allswith|s| ≥sα, where

(3.10) Gα(x, s) =

Z s 0

gα(x, τ)dτ. Assume also that

(H)₃ the function ^a(t)_t is nondecreasing on (0,∞), (3.2) and (3.3) being fullled as well (see the beginning of this section);

By (weak) solution to problem (3.5), (3.6) we understand a solution to the equation

(3.11) Jau=G⁰(u),

in the following functional framework:

(i)X=W₀^mE_A(Ω)normed withk·k_m,A,V = T

|β|<m

W^m−1L_Mβ(Ω)normed with the norm

kuk_V = X

|β|<m

kuk_W^m−1_L

Mβ(Ω);

(ii)Ja is the duality mapping on (W₀^mEA(Ω),k · k_m,A) corresponding to the gauge function a;

(iii)G⁰ :V →V^∗ is the dierential of the functional G:V →R, G(u) = X

|α|<m

Z

Ω

G_α(x, D^αu(x)) dx, u∈V.

According to [11, Proposition 6.2],X is compactly imbedded in V. To apply Theorem 1 some prerequisites are necessary.

We begin with the following result ([10, Proposition 4]):

Proposition 2. LetXbe a real reexive Banach space, compactly imbed- ded in a real Banach space Z. Denote by ithe compact injection of X into Z and, for any p∈[1,∞), dene

λ_1,p= inf

kuk^p_X

ki(u)k^p_Z |u∈X\ {0_X}

. Thenλ1,p is attained and λ⁻

1 p

1,p is the best constantcZ in the imbedding of X into Z, namely,

ki(u)k_Z ≤c_Zkuk_X for all u∈X.

Now, let us consider the Banach spaceZ =W₀^m−1E_A(Ω), with the norm kuk_Wm−1

0 EA(Ω)= X

|α|<m

kD^αuk²_(A)

!1/2

.

According to ([20, Theorem 2.7]), one has

Proposition 3. LetΩ be any domain inR^N. LetA(u) =R|u|

0 a(t) dt be an N-function and let m∈N^∗ be given. Then the imbedding

(3.12) W₀^mE_A(Ω),→ⁱ W₀^m−1E_A(Ω) exists and is compact.

On account of Propositions 2 and 3, we also have

Corollary 1. Under the hypotheses of Proposition 3, we further assume that the N-functions A and A satisfy the ∆₂-condition (that is, W₀^mE_A(Ω) is a reexive space). For any p∈[1,∞) dene

(3.13) λ1,p:= inf







kuk^p_m,A ki(u)k^p

W₀^m−1EA(Ω)

|u∈W₀^mEA(Ω)\ {0_W^m

0 EA(Ω)}





 ,

where i is the compact injection (3.12). Then λ1,p is attained and λ⁻

1 p

1,p is the best constant c in the imbedding of W₀^mE_A(Ω)into W₀^m−1E_A(Ω), namely,

ki(u)k_Wm−1

0 EA(Ω) ≤ckuk_m,A for all u∈W₀^mE_A(Ω). We need the following technical result.

Lemma 1. Let A(u) = R|u|

0 a(t) dt be an N-function. If A satises the

∆2-condition and

p₀= inf

t>0

ta(t) A(t),

then ∞> p₀ ≥1 and for anyu∈L_A(Ω) with kuk_(A)<1 one has

(3.14) Z

Ω

A(u(x)) dx≤ kuk^p_(A)⁰ .

Proof. First, we remark that from Young's equality we have ta(t)

A(t) >1 for any t >0,

therefore, p₀≥1. SinceAsatises the∆₂-condition, we havekA(t)≥A(2t)>

ta(t), therefore

ta(t)

A(t) < k with k >2. Thus, p₀ is nite and

a(τ) A(τ) ≥ p₀

τ , τ >0.

Now, let u be such that kuk_(A) < 1. Integrating over the interval h|u(x)|,_kuk^|u(x)|

(A)

i, we obtain

A(u(x))≤A

|u(x)|

kuk_(A)

kuk^p_(A)⁰ . Integrating over Ω and taking into account that

Z

Ω

A

|u(x)|

kuk_(A)

dx= 1, inequality (3.14) follows.

We will apply Theorem 1 to the functionalF :W₀^mEA(Ω)→Rgiven by (3.15) F(u) =A(kuk_m,A)− X

|α|<m

Z

Ω

Gα(x, D^αu(x)) dx,

whereA and G_α,|α|< m, are given by (3.1) and (3.10), respectively.

The following result is useful.

Proposition 4 (see [12, Proposition 4]). Let A : R → R+ be the N- function given by (3.1). Furthermore, assume that A satises (3.2) and (3.3), the ∆₂-condition being also satised byAandA. Let g_α: Ω×R→R,|α|< m be Carathéodory functions satisfying condition (H)1.

Then the functionalH :W₀^mEA(Ω)→Rdened by

(3.16) H(u) = Ψ(u)−G(u),

with

Ψ(u) =A(kuk_m,A), G(u) = X

|α|<m

Z

Ω

Gα(x, D^αu(x)) dx,

for all u∈W₀^mE_A(Ω)is well-dened and C¹ on W₀^mE_A(Ω), with H⁰(u) =J_au− X

|α|<m

(−1)^|α|D^αg_α(x, D^αu). The main result of this section is as follows.

Theorem 3. Let A : R → R+ be the N-function given by (3.1), ful- lling (3.2), (3.3) and hypothesis (H)₃. Let g_α : Ω×R → R, |α| < m, be Carathéodory functions satisfying (H)1, (H)2 and being odd in the second ar- gument: gα(x,−s) =−g_α(x, s). Suppose that the N-functionsA, A and Mα,

|α|< m, satisfy the ∆₂-condition. With (3.17) p₀ = inf

t>0

ta(t)

A(t), p^∗ = sup

t>0

ta(t)

A(t) <∞, p∗ = lim inf

t→∞

tA⁰_∗(t) A∗(t) , A∗ being the Sobolev conjugate of A, we further assume:

(H)₄ there exists a positive constant C >0 such that A(t)≥C·t^p0, ∀t∈(0,1);

(H)5

lim sup

s→0

g_α(x, s)

a(s) < Cλ_1,p0

2N0 , |α|< m,

uniformly with respect to almost all x ∈ Ω, where λ1,p0 are given by (3.13) and N₀ = P

|α|<m

1; (H)6 p0< p∗.

Then the functional (3.16) has an unbounded sequence of critical values.

Proof. Theorem 1 applies. Indeed, since ^a(t)_t is nondecreasing on(0,∞), W₀^mEA(Ω)is uniformly convex ([11, Theorem 3.14]). Consequently,W₀^mEA(Ω) is reexive and has the Kade£-Klee property. The same space is smooth ([11, Theorem 3.6]) and compactly imbedded in T

|β|<m

W^m−1L_Mβ(Ω)([11, Proposi- tion 6.2]).

The functionalH ∈ C¹(X,R) (Proposition 4) is even (sinceg_α are odd in the second argument) and satises hypotheses (i)(iv) of Theorem 1.

Since (3.17) holds, hypothesis (i) is obviously satised withϕ=a. Since G⁰ :V →V^∗ is continuous ([11, Proposition 6.3]),(ii)₁ is obviously satised.

By [11, Lemma 7.7], there exists a positive constantC such that (3.18) X

|α|<m

Z

Ω

1

θgα(x, D^αun(x))D^αun(x)−Gα(x, D^αun(x))

dx≥ −C, whereθ= min

|α|<mθ_α. We remark that (3.18) can be rewritten as 1

θ

G⁰(u_n), u_n

−G(u)≥ −C, therefore (ii)₂ in Theorem 1 is fullled.

Let us prove that hypothesis (iii) of Theorem 1 is fullled. For the rst term in (3.16), according to (H)₄, we have

(3.19) A(kuk_m,A)≥Ckuk^p_m,A⁰ for all u∈W₀^mE_A(Ω)withkuk_m,A <1.

We shall now handle the estimations for the second term in (3.16). As in [11, Lemma 7.6], from (H)5 we deduce that, for anyα with |α|< m, there exist µ_α∈

0,^Cλ_2N^1,p0

0

ands_α>0such that (3.20) G_α(x, s)< µ_αA(s)

for x∈ Ω,0 <|s|< s_α. We denote Ω_α ={x ∈Ω| |D^αu(x)| ≥s_α},|α|< m. It is shown in [11, Lemma 7.6], that there exist positive constants c ≤1 and D such that, ifkuk_m,A < cthen we have

(3.21) X

|α|<m

Z

Ωα

Gα(x, D^αu(x)) dx≤D· kuk^p_m,A^∗−µ.

On the other hand, by the denition of λ_1,p0, for any α with|α|< mwe have (3.22) λ_1,p0 ≤ kuk^p_m,A⁰

kuk^p0

W₀^m−1EA(Ω)

≤ kuk^p_m,A⁰ kD^αuk^p_(A)⁰ .

Therefore, for anyαwith|α|< m, from (3.20), Lemma 1 and (3.22), we deduce Z

Ω\Ωα

G_α(x, D^αu(x))dx < µ_α Z

Ω

A(D^αu(x))dx≤

≤µαkD^αuk^p_(A)⁰ ≤ µ_α λ1,p0

kuk^p_m,A⁰ < C 2N0

kuk^p_m,A⁰ .

Consequently, summing over α, we have

(3.23) X

|α|<m

Z

Ω\Ωα

G_α(x, D^αu(x))dx < C

2kuk^p_m,A⁰ . Then, from (3.19), (3.21), (3.23) we obtain

F(u)> Ckuk^p_m,A⁰ − C

2kuk^p_m,A⁰ −D· kuk^p_m,A^∗−µ= C

2kuk^p_m,A⁰ −Dkuk^p_m,A^∗−µ, that is (2.4).

Now, we will prove that hypothesis (iv) of Theorem 1 is fullled. LetX1

be a nite dimensional subspace ofW₀^mE_A(Ω). According to [11, Lemma 7.6, inequality (7.46)], for any α with|α|< m, one has

G_α(x, s)≥γ_α(x)|s|^θα for a.e. x∈Ω and|s| ≥s_α,whereγ_α∈L^∞(Ω).

Forα with|α|< mand v∈W₀^mEA(Ω), dene

Ω^α_≥={x∈Ω| |D^αv(x)| ≥s_α}, Ω^α_<= Ω\Ω^α_≥. Then

Z

Ω

Gα(x, D^αv(x) dx≥ Z

Ω^α_≥

γα(x)|D^αv(x)|^θαdx+ Z

Ω^α_<

Gα(x, D^αv(x) dx. But

Z

Ω^α_≥

γα(x)|D^αv(x)|^θαdx= Z

Ω

γα(x)|D^αv(x)|^θαdx− Z

Ω^α_<

γα(x)|D^αv(x)|^θαdx.

Since Z

Ω^α_<

γα(x)|D^αv(x)|^θαdx≤ kγ_αk_∞s^θ_α^αvol(Ω), we have

Z

Ω

Gα(x, D^αv(x) dx≥ Z

Ω

γα(x)|D^αv(x)|^θαdx+ Z

Ω^α_<

Gα(x, D^αv(x) dx−kα, wherekα=kγ_αk_∞s^θ_α^αvol(Ω). On the other hand, it follows from (3.8) that

Z

Ω^α_<

Gα(x, D^αv(x) dx≤ kc_αk_L1(Ω)sα+ 2dαMα(sα)vol(Ω).

Therefore, Z

Ω

Gα(x, D^αv(x)dx≥ Z

Ω

γα(x)|D^αv(x)|^θαdx−Kα, whereK_α =k_α+kc_αk_L1(Ω)s_α+ 2d_αM_α(s_α)vol(Ω).

Consequently,

F(v)≤A(kvk_m,A)− X

|α|<m

Z

Ω

γα(x)|D^αv(x)|^θαdx+K,

whereK is a positive constant andθ_α are given by(H)₂. By the denition of p^∗, for kvk_m,A >1 we have

(3.24) F(v)≤A(1)kvk^p_m,A^∗ − X

|α|<m

Z

Ω

γ_α(x)|D^αv(x)|^θαdx+K. Now, the functionalk · k_γ:W₀^mEA(Ω)→R dened by

kuk_γ = X

|α|<m

Z

Ω

γ_α(x)|D^αu(x)|^θαdx 1/θα

is a norm onW₀^mE_A(Ω). Dening kD^αuk_θα =

Z

Ω

γ_α(x)|D^αu(x)|^θαdx 1/θα

, one has

kuk_γ = X

|α|<m

kD^αuk_θα. Let α be a multiindex satisfying

kD^αuk_θα = max

|α|<mkD^αuk_θα. Then

kuk_γ≤N0kD^αuk_θα, whereN0= P

|α|<m

1. Therefore X

|α|<m

Z

Ω

γ_α(x)|D^αu(x)|^θαdx≥ Z

Ω

γ_α(x)|D^αu(x)|^θα dx (3.25)

=kD^αuk^θ_θ^α

α ≥ 1 N0

kuk^θ_γ^α.

Since the k · k_m,A and k · k_γ-norms are equivalent on the nite dimensional subspaceX1, there is a constantδ =δ(X1)>0 such that

(3.26) kuk_m,A ≤δkuk_γ.

Therefore,

F(v)≤A(1)kvk^p_m,A^∗ − 1

N0δ^θαkvk^θ_m,A^α +K if v∈X1,kvk_m,A >1.

By Theorem 1, the functionalF possesses a sequence of critical positive values. By Proposition 4, equation (3.11) possesses a sequence of solutions in W₀^mEA(Ω)or, equivalently, problem (3.5), (3.6) possesses a sequence of weak solutions inW₀^mE_A(Ω).

4. EXAMPLES

Example 1. Consider problem (3.5), (3.6), under the following hypothe- ses:

(i) the function a : R → R is dened by a(t) =

n

P

i=1

a_i|t|^pi−2t, where a_i >0,1≤i≤n,p_i+1> p_i ≥2,1≤i≤n−1,p_n< N;

(ii) the Carathéodory functionsgα : Ω×R→R,|α|< m, are odd in the second argument, that is,g_α(x,−s) =−g_α(x, s), and satisfy the conditions

(4.1) lim sup

s→0

gα(x, s)

a(s) < a1λ1,p1

2p₁N₀, |α|< m,

uniformly with respect to almost all x ∈ Ω, where λ1,p1 is dened by (3.13) and N₀ = P

|α|<m

1;

(iii) there existqα,1< qα< _N^{N p}_−pⁿ

n,|α|< m, such that

(4.2) |g_α(x, s)| ≤a_α+b_α|s|^qα−1,x∈Ω,s∈R,a_α,b_α positive constants;

(iv) if G_α, |α| < m, are given by (3.10), then there exist s_α > 0 and θα > pn such that

(4.3) 0< θ_αG_α(x, s)≤sg_α(x, s) for a.e. x∈Ω and allswith |s| ≥s_α. Under these assumptions problem (3.5), (3.6) has a sequence of weak solutions.

Proof. The idea of the proof is as follows: the assumptions made entail the hypotheses of Theorem 3.

First, we prove that hypothesis (H)₃ is satised. Since ^a(t)_t =

n

P

i=1

ait^pi−2 for allt >0, the function^a(t)_t is nondecreasing on(0,∞). In order to prove that (3.2) and (3.3) are satised, the result below is needed. See [11, Lemma 8.1 and (ii), Lemma 8.2].

Lemma 2. Let A : R → R+, A(t) = R|t|

0 a(s) ds, be an N-function.

Assume that

p^∗ = sup

t>0

ta(t) A(t) < N

and there are constants 0< γ < N and δ >0 such that (4.4) A(t)≥Ct^γ, ∀t∈(0, A⁻¹(δ)).

Then (3.2) and (3.3) are satised (consequently, the Sobolev conjugate A∗ of A, can be dened).

If lim

t→∞

ta(t)

A(t) =l <∞ then

(4.5) lim

t→∞

tA⁰_∗(t)

A∗(t) = N l N −l. In our case,p^∗ =p_n and p_n< N (by (i)). Since

(4.6) A(t) =

n

X

i=1

ai

pi

t^pi ≥ a1

p1

t^p1, ∀t >0, condition (4.4) is satised with C= ^a_p¹

1,γ =p₁ and any δ >0. Therefore, A∗

exists and we can compute p∗. We obtain p∗ = lim inf

t→∞

tA⁰_∗(t)

A∗(t) = N pn

N −pn.

Second, we prove that hypothesis (H)₁ is satised. By setting Mα(s) = |s|^qα

qα , |α|< m,s∈R, (4.2) can be written as

|g_α(x, s)| ≤aα+bα(qα−1)

1 q0

α M⁻¹_α (Mα(s)), x∈Ω,s∈R, |α|< m, showing that (3.8) is satised.

What remains to be proved is thatM_α,|α|< m, satisfy the∆₂-condition and increase essentially more slowly than A∗ near innity. It is easy to check (by denition) that M_α,|α|< m, satisfy the ∆₂-condition.

By using l'Hôspital rule, we also have (4.7) lim

t→∞

A⁻¹_∗ (t)

M_α⁻¹(t) = lim

t→∞c_αA⁻¹(t) t^{qα1 +N1}

= lim

s→∞c_α s

(A(s))^{qα1 +N1}

= 0, c_α=q^(q_α^α−1)/qα,

since, by (iii), the degree of denominator is p_n

1

qα +_N¹

> 1. Thus, M_α,

|α|< m, increase essentially more slowly thanA∗.

Hypothesis (H)₂ is covered by (iv) (with g_α odd functions in the second argument, according to (ii).

In order to prove thatAandAsatisfy the∆₂-condition, the result below is needed (see [11, Lemma 8.1, (i)]).

Lemma 3. LetA:R→R+,A(t) =R|t|

0 a(s) ds, be anN-function and A the complementary N-function to A. Assume that

p^∗= sup

t>0

ta(t)

A(t) <∞ and p₀ = inf

t>0

ta(t) A(t) >1. Then bothA andA satisfy the ∆2-condition.

In our case, as one has already seen, p^∗ = pn < N and p0 = p1 > 1 (according to (i)). SinceM_α(s) = ^|s|q

0α

q⁰_α , _q¹_α +_q¹0

α = 1,|α|< m,s∈R, it is easy to check (by denition) that Mα,|α|< m, satisfy the ∆2-condition.

On the other hand, since p₀ =p₁, (4.6) says that (H)₄ in Theorem 3 is satised.

Finally, sincep₀ =p₁< p_n< _N−p^{N pn}

n =p∗,(H)₆ is satised too.

The result now follows by Theorem 3.

Example 2. Consider problem (3.5), (3.6), under the following hypothe- ses: (i) the function a:R→ Ris dened by a(t) = |t|^p−2t√

t²+ 1,2 ≤p <

N −1;

(ii) the Carathéodory functionsg_α : Ω×R→R,|α|< m, are odd in the second argument, that is,gα(x,−s) =−g_α(x, s), and satisfy the conditions

lim sup

s→0

gα(x, s)

a(s) < λ1,p

2pN₀, |α|< m,

uniformly with respect to almost allx∈Ω, where λ1,pare given by (3.13) and N₀= P

|α|<m

1;

(iii) there exist qα, 1 < qα < ^N_N^(p+1)_−p−1, |α| < m, such that the growth conditions (4.2) hold;

(iv) there exists_α>0 and θ_α > p+ 1such that conditions (4.3) hold.

Under these assumptions, problem (3.5), (3.6) has a sequence of weak solutions.

Proof. The idea of the proof is that used in Example 1, namely, we shall show that the assumptions made entail those of Theorem 3.

First, we prove that hypothesis(H)₃is satised. Since ^a(t)_t =t^p−2√ t²+ 1 for all t >0, the function ^a(t)_t is nondecreasing on (0,∞). In order to prove that (3.2) and (3.3) are satised, we shall use Lemma 2. In our case,p^∗=p+ 1 ([11, Example 8.6]) and p+ 1< N (by (i)). Since a(t)≥t^p−1,t >0, one has

(4.8) A(t)≥ 1

pt^p, ∀t >0.

Therefore, (4.4) is satised with C = ¹_p, γ =p < N and any δ > 0. So, A∗

exists and we can compute p∗. We obtain p∗ = lim inf

t→∞

tA⁰_∗(t)

A∗(t) = N(p+ 1) N −p−1.

Second, the hypothesis (H)₁ in Theorem 3 is satised with M_α(s) =

|s|^qα

qα ,|α|< m, which, obviously, satisfy the ∆₂-condition. Also, M_α,|α|< m, increase essentially more slowly than A∗ near innity. Indeed, as in (4.7),

t→∞lim

A⁻¹_∗ (t)

Mα⁻¹(t) = lim

s→∞c_α s

(A(s))^{qα1 +N1} . It is sucient to show that

s→∞lim

s (A(s))^{qα1 +N1}

= 0.

Since a(t)≥t^p,∀t≥0, we haveA(t)≥ ^t_p+1^p+1,∀t≥0. Consequently,

s→∞lim

s (A(s))^{qα1 +N1}

≤ lim

s→∞

s (p+ 1)^{qα1 +N1} ·s

1 qα+_N¹

(p+1) = 0, as by (iii), the degree of denominator is (p+ 1)

1 qα + _N¹

>1.

The hypothesis (H)₂ is covered by (iv) (with gα odd functions in the second argument, according to (ii).

In order to prove that A and A satisfy the ∆2-condition, we shall use Lemma 3. In our case, as one has already seen,p^∗=p+ 1< N andp₀ =p >1 (according to (i)). Also, the functions M_α(s) = ^|s|q

0α

q_α⁰ , _q¹_α +_q¹0

α = 1, |α|< m, s∈R, satisfy the ∆2-condition.

On the other hand, sincep₀ =p (see [11, Example 8.6]), (4.8) says that (H)₄ in Theorem 3 is satised.

Finally, sincep0 =p < p+ 1< ^N_N^(p+1)_−p−1 =p∗,(H)₅ is satised, too.

The result now follows by Theorem 3.

Example 3. Consider problem (3.5), (3.6), under the following hypothe- ses: (i) the function a : R → R is dened by a(t) = |t|^p−2tln (1 +α+|t|), 2≤p≤N−1,α >0;

(ii) the Carathéodory functionsgα : Ω×R→R,|α|< m, are odd in the second argument, that is,gα(x,−s) =−g_α(x, s), and satisfy the conditions:

lim sup

s→0

g_α(x, s)

a(s) < ln(1 +α)λ_1,p

2pN₀ , |α|< m,

uniformly with respect to almost allx∈Ω, where λ1,pare given by (3.13) and N₀= P

|α|<m

1;

(iii) there exist qα, 1 < qα < _N^{N p}_−p, |α| < m, such that the growth conditions (4.2) hold;

(iv) there existsα>0 and θα > p+ 1such that conditions (4.3) hold.

Under these asumptions, problem (3.5), (3.6) has a sequence of weak solutions.

Proof. The idea of the proof is that used in Example 1, namely, we shall show that the assumptions made entail those of Theorem 3.

First, we prove that hypothesis(H)₃ is satised. Since ^a(t)_t =t^p−2ln(1 + α+t) for all t >0, the function ^a(t)_t is nondecreasing on (0,∞). In order to prove that (3.2) and (3.3) are satised, we shall use Lemma 2. In our case, p^∗ =p+C₀ < p+ 1 ([11, Example 8.10]) and p+ 1≤N (by (i)). Since (see [11, Example 8.10, inequality (8.23)])

(4.9) A(t)≥ ln (1 +α)

p t^p, ∀t≥0,

it follows that (4.4) is satised with C = ^ln(1+α)_p , γ =p < N and any δ > 0. Therefore, A∗ exists and we can compute p∗. We obtain

p∗ = lim inf

t→∞

tA⁰_∗(t)

A∗(t) = N p N −p.

Second, hypothesis (H)1 in Theorem 3 is satised with Mα(s) = ^|s|_q^qα

α ,

|α| < m, which, obviously, satisfy the ∆2-condition. Also, Mα, |α| < m, increase essentially more slowly thanA∗ near innity. As in the preceding two examples, this amounts to show that

s→∞lim

s (A(s))^{qα1 +N1}

= 0.

This last equality is true since (4.9) holds. Therefore,

s→∞lim

s (A(s))^{qα1 +N1}

≤ lim

s→∞

s ln(1+α)

p

¹

qα+_N¹

·s

1 qα+_N¹

p

= 0,

since, by (iii), the degree of denominator is p 1

qα +_N¹

>1.

The necessary arguments in order to prove that hypothesis(H)₂of Theo- rem 3 is satised are those used in the preceding two examples.

In order to prove that A and A satisfy the ∆₂-condition, we shall use Lemma 3. In our case, as one has already seen, p^∗ = p+C0 < N and

p0=p >1(according to (i)). Also, the functionsMα(s) = ^|s|q

0 α

q_α⁰ , _q¹_α +_q¹0 α = 1,

|α|< m,s∈R, satisfy the∆₂-condition.

On the other hand, sincep0 =p (see [11, Example 8.6]), (4.9) says that (H)₄ in Theorem 3 is satised.

Finally, sincep₀ =p < _N−p^{N p} =p∗,(H)₅ is satised, too.

The result now follows by Theorem 3.

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Received 19 May 2008 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania Romanian Academy

Gheorghe MihocCaius Iacob Institute of Mathematics Statistics and

Applied Mathematics Calea 13 Septembrie nr. 13

Bucharest, Romania dinca@fmi.unibuc.ro

and

Technical University of Civil Engineering Department of Mathematics

Bd. Lacul Tei nr. 124 020396 Bucharest, Romania

pavel.matei@gmail.com