OF A p(x)-LAPLACIAN EQUATION IN A BOUNDED DOMAIN

OF A p(x)-LAPLACIAN EQUATION IN A BOUNDED DOMAIN

PETRE SORIN ILIAS¸

We solve the Dirichlet problem

−∆_p(x)u=f(x, u) in Ω

u= 0 on∂Ω,

where p ∈ C+(Ω), Ω is a bounded domain in R^N and f : Ω×R → R is a Carath´eodory function. Especially, the existence of two sequences of weak solu- tions for the problem is obtained.

AMS 2000 Subject Classification: 47H05, 46E35, 35B38.

Key words: p(x)-Laplacian, generalized Lebesgue-Sobolev space, critical point.

1. INTRODUCTION

The aim of this paper is to discuss the existence and multiplicity of weak solutions inW₀^1,p(x)(Ω) of thep(x)-Laplacian equation

(P)

−∆_p(x)u=f(x, u) in Ω

u= 0 on∂Ω

in a bounded domain, Ω⊆R^N wherep: Ω→Ris a continuous function with p(x) >1 for any x∈Ω and f : Ω×R→R is a Carath´eodory function. This kind of problem was studied especially in recent years (see [1], [3]);p(x)-growth conditions can be regarded as a very important class of nonstandard growth conditions. Thep(x)-Laplacian possesses more complicated nonlinearities, for example, it is inhomogeneous, so in the discussion some special techniques will be needed. In this article we will use two variational methods, namely

“Fountain Theorem” and “Dual Fountain Theorem” (see [4]). In this manner we will obtain the existence of two different sequences of weak solutions of problem (P).

REV. ROUMAINE MATH. PURES APPL.,52(2007),6, 639–653

The hypotheses under we work are listed below:

(1.1) p₊= max

x∈Ω p(x)< p^∗₋= inf

x∈Ωp^∗(x);

(1.2) f(x,−t) =−f(x, t) forx ∈Ω, t∈R; (1.3) |f(x, t)| ≤c|t|^α(x)−1+h(x) forx∈Ω, t∈R,

where c ≥ 0 is constant, α ∈ C+(Ω) with α(x) < p^∗(x) for all x ∈ Ω, h ∈ L^β(x)(Ω) and β ∈C₊(Ω) is the conjugate function of α;

(1.4) 0< θF(x, t)≤tf(x, t) forx∈Ω, t∈R with |t| ≥M, whereM >0,θ∈(p₊, p^∗₋), andF(x, t) =_t

−∞f(x, s)dsforx∈Ω, t∈R;

(1.5) α > p₊.

The functionp^∗: Ω→Ris defined by p^∗(x) =

Np(x)

N−p(x) ifp(x)< N +∞ ifp(x)≥N.

The results obtained are generalizations of well-known results for p-Laplacian equations in a bounded domain.

2. PRELIMINAIRES

In this section we will present the basic properties of generalized Lebesgue- Sobolev spaces W₀^1,p(x)(Ω). Write

C+(Ω) =

p∈C(Ω)|p(x)>1 for any x∈Ω , p−= min

x∈Ω p(x), p+= max

x∈Ω p(x) for any p∈C+(Ω), M ={u: Ω→R|u is a measurable real-valued function},

L^p(x)(Ω) =

u∈M |

Ω|u(x)|^p(x)dx <∞

with the norm

u_p = inf

λ >0|

Ω

u(x) λ

^p(x)dx≤1 ,

and

W^1,p(x)(Ω) =

u∈L^p(x)(Ω)| ∃∂u

∂x_i ∈L^p(x)(Ω) for all 1≤i≤N

with the norm

u_W1,p(x) =u_p+|∇u|_p, where|∇u|=

N i=1

∂u

∂xi

₂ .

Also, onL^p(x)(Ω) we introduce the functionϕ_p :L^p(x)(Ω)→Rdefined by ϕp(u) =

Ω|u(x)|^p(x)dx.

The connections between ϕ_p and _p are established by the next result.

Proposition2.1 (Fan and Zhao [2]). a)We have the equivalencesu_p<

(>,=)1 ⇐⇒ ϕ_p(u)<(>,=)1, u_p =α ⇐⇒ ϕ_p(u) =α when α= 0.

b)If u_p >1 then u^p_p⁻ ≤ϕ_p(u)≤ u^p_p⁺. If u_p <1 then u^p_p⁺ ≤ ϕp(u)≤ u^p_p⁻.

c)A⊆L^p(x)(Ω)is bounded if and only if ϕ_p(A)⊆Ris bounded.

d)For a sequence (u_n)_n∈N⊆L^p(x)(Ω) and an element u∈L^p(x)(Ω), the statements below are equivalent:

(1) lim

n→∞un=u in L^p(x)(Ω);

(2) lim

n→∞ϕ_p(u_n−u) = 0;

(3) u_n→u in measure in Ωand lim

n→∞ϕ_p(u_n) =ϕ_p(u);

e) lim

n→∞u_np = +∞ if and only if lim

n→∞ϕ_p(u_n) = +∞.

The generalized Lebesgue-Sobolev space W₀^1,p(x)(Ω) is the closure of C₀^∞(Ω) inW^1,p(x)(Ω). We have

Proposition2.2 (Fan and Zhao [2]). a)The spacesL^p(x)(Ω),W^1,p(x)(Ω) andW₀^1,p(x)(Ω)are separable and reflexive Banach spaces.

b) If q ∈ C+(Ω) and q(x) < p^∗(x) for any x ∈ Ω, then the imbedding fromW^1,p(x)(Ω)into L^q(x)(Ω) is compact.

c)There is a constant C >0 such that

u_p≤C|∇u|_p for all u∈W₀^1,p(x)(Ω).

Using Proposition 2.2 (c), we get that|∇u|_p and u_W1,p(x) are equiv- alent norms on W₀^1,p(x)(Ω). Hence, from now on, we will use the space W₀^1,p(x)(Ω) equipped with the normu_1,p =|∇u|_p.

Remark 2.1. Ifq∈C+(Ω) andq(x)< p^∗(x) for anyx∈Ω, the imbedding fromW₀^1,p(x)(Ω) intoL^q(x)(Ω) is compact.

In order to discuss problem (P), we need also some theory on Nemytskii operators. For a Carath´eodory functionf : Ω×R→ Rand u∈M we know that the functionN_fu: Ω→Rdefined by (N_fu)(x) =f(x, u(x)) forx∈Ω is measurable in Ω. So, with the Carath´eodory functionf we can associate the operator Nf : M → M which is called the Nemytskii operator. Concerning the properties ofNf, we have

Proposition 2.3 (D. Zhao and X.L. Fan [5]). Assume (1.3). Then N_f(L^α(x)(Ω))⊆L^β(x)(Ω)and, moreover, N_f is continuous fromL^α(x)(Ω)into L^β(x)(Ω)and maps bounded sets into bounded sets.

Proposition 2.4. Assume (1.3). Let F : Ω×R → R be the function defined byF(x, t) =_t

−∞f(x, s)dsfor x∈Ω,t∈R. Then

(i) F is a Carath´eodory function and there exist a constant c₁ ≥ 0 and σ∈L¹(Ω)such that

|F(x, t)| ≤c1|t|^α(x)+σ(x) for x∈Ω, t∈R;

(ii) the functional Φ :L^α(x)(Ω)→ R defined by Φ(u) =

ΩF(x, u(x))dx is continuously Fr´echet differentiable and Φ(u) =N_f(u) for all u∈L^α(x)(Ω).

The restrictionα(x)< p^∗(x) for anyx∈Ω from hypothesis (1.3) ensures that the imbeddingW₀^1,p(x)(Ω)→L^α(x)(Ω) is compact. Hence the diagram

W₀^1,p(x)(Ω)→^I L^α(x)(Ω) ^N→^f L^β(x)(Ω) ^I

→

W₀^1,p(x)(Ω) shows thatN_f :W₀^1,p(x)(Ω)→

W₀^1,p(x)(Ω)

is strongly continuous.

Moreover, with the same argument, we get that the functional Φ : W₀^1,p(x)(Ω)→R defined by

Φ(u) =

ΩF(x, u(x))dx

is strongly continuous onW₀^1,p(x)(Ω) and Φ(u) =N_f(u) for allu∈W₀^1,p(x)(Ω).

3. EXISTENCE AND MULTIPLICITY RESULTS For simplicity, throughout this section we denoteX=W₀^1,p(x)(Ω).

As in the case p(x)≡p(a constant), we consider the p(x)-Laplace oper- ator−∆_p(x):X →X defined by

−∆_p(x)u, v

=

Ω|∇u|^p(x)−2∇u· ∇vdx

for allu, v ∈X. The functional Ψ :X→Rdefined by Ψ(u) =

Ω

1

p(x)|∇u|^p(x)dx

is continuously Frechet differentiable and Ψ(u) =−∆_p(x)u for all u∈X. Also, −∆_p(x) is a mapping of type (δ₊), i.e., if u_n u in X and lim sup

n→∞

−∆_p(x)u_n, u_n−u

≤0, then u_n→u inX.

Definition 3.1. A functionu∈Xis said to be a weak solution of problem

(P) if

Ω|∇u|^p(x)−2∇u· ∇vdx=

Ωf(x, u)vdx for allv∈X.

We remark that u ∈X is a weak solution of problem (P) if−∆_p(x)u = N_fuinX^∗. On account of the potentiality of thep(x)-Laplace and Nemytskii operators,u∈Xis a weak solution of problem (P) if and only if Ψ(u) = Φ(u).

Consider the functionalH :X→Rdefined by H(u) =

Ω

1

p(x)|∇u|^p(x)dx−

ΩF(x, u(x))dx.

Consequently, H ∈C¹(X,R) and u∈X is a weak solution of problem (P) if and only ifH(u) = 0 in X^∗.

We shall prove that theC¹-functional H has two different sequences of critical points using two variational methods, namely the “Fountain Theorem”

and the “Dual Fountain Theorem”.

Remark 3.1. As X = W₀^1,p(x)(Ω) is a reflexive and separable Banach space, there exist (e_n)_n∈N∗ ⊆ X and (f_n)_n∈N∗ ⊆ X^∗ such that f_n(e_n) = δnm for all n, m∈N^∗, X = Span{en|n∈N^∗}, X^∗ = Span{fn|n∈N^∗}^w∗.

Fork∈N^∗ denote

X_k= Span{e_k}, Y_k= k

j=1

Xj, Z_k= ∞ j=k

Xj.

Proposition 3.1 (“Fountain Theorem”, see [4]). Let X be a reflex- ive and separable Banach space, H ∈ C¹(X,R) an even functional and the subspaces Xk, Yk, Zk defined in Remark 3.1. If for each k ∈ N^∗ there exist ρk> rk>0 such that

x∈Zkinf,x=rkH(x)→ ∞ as k→ ∞, max

x∈YK,x=ρkH(x)≤0,

andH satisfies the (PS)_c condition for everyc >0, then H has a sequence of critical values to+∞.

Proposition 3.2 (“Dual Fountain Theorem”, see [4]). Let X be a re- flexive and separable Banach space, H∈C¹(X,R) an even functional and the subspaces X_k, Y_k, Z_k defined in Remark3.1. Assume there is a k0∈N^∗ so as to for eachk∈N, k≥k₀, there exists ρ_k> r_k>0 such that

x∈Zkinf,x=ρkH(x)≥0, b_k = max

x∈Yk,x=rkH(x)<0, dk= inf

x∈Zk,x≤ρkH(x)→0 as k→ ∞

and H satisfies the (PS)^∗_c condition for every c ∈ [d_k0,0). Then H has a sequence of negative critical values converging to 0.

Recall thatHis said to satisfy the (PS) condition if any sequence (un)_n∈N

⊆X for which (H(u_n))_n∈N⊆ Ris bounded and H(u_n) → 0 as n→ ∞, has a convergent subsequence. Also, H is said to satisfy the (PS)_c condition for c∈R if any sequence (u_n)_n∈N ⊆X for whichH(u_n) →c and H(u_n)→ 0 as n → ∞, has a convergent subsequence. It is obvious that if H satisfies the (PS) condition, thenH satisfies the (PS)_c condition for all c∈R.

Definition 3.2. Let X be a reflexive and separable Banach space, H ∈ C¹(X,R), c ∈ R and the subspacesYn defined in Remark 3.1. H is said to satisfy the (PS)^∗_c condition if any sequence (un)_n∈N⊆X for whichun∈Ynfor any n∈N^∗,H(un) →c and (H |Yn)(un) →0 as n→ ∞, has a subsequence convergent to a critical point ofH.

We first deal with the (PS) and (PS)^∗_c conditions for H.

Proposition 3.3. Assume (1.4). Then the functional H : X → R de- fined by

H(u) =

Ω

1

p(x)|∇u|^p(x)dx−

ΩF(x, u(x))dx satisfies the(PS) condition.

Proof. Let the sequence (u_n)_n∈N ⊆ X be such that (H(u_n))_n∈N ⊆ R is bounded and H(un) → 0 as n → ∞. Then there exists d ∈ R such that H(un)≤dfor alln∈N. For eachn∈N, denote Ω_n={x∈Ω| |u_n(x)| ≥M}, Ω_n= ΩΩ_n.Without loss of generality, we can suppose that M ≥1.

Ifx∈Ω_n then|u_n(x)|< M and, by Proposition 2.4 (i), F(x, un)≤c1|un(x)|^α(x)+σ(x)≤c1M^α+ +σ(x),

hence (1)

Ω_nF(x, u_n)dx≤

Ω_n(c₁M^α++σ(x))dx≤

Ω(c₁M^α++σ(x))dx=

=c1M^α+meas(Ω) +

Ωσ(x)dx=k1. Ifx∈Ω_n then|u_n(x)| ≥M and, by (1.4),

F(x, u_n)≤ 1

θf(x, u_n(x))u_n(x) which gives

(2)

Ωn

F(x, un)dx≥ 1 θ

Ωn

f(x, un(x))un(x)dx=

= 1 θ

Ωf(x, u_n(x))u_n(x)dx−

Ω_nf(x, u_n(x))u_n(x)dx

. By the growth condition (1.4) again,

Ω_nf(x, u_n(x))u_n(x)dx ≤

Ω_n

c|u_n(x)|^α(x)+h(x)|u_n(x)| dx≤

≤cM^α+meas(Ω_n) +M

Ω_nh(x)dx≤cM^α+meas(Ω) +M

Ωh(x)dx=k2, which yields

(3) −1

θ

Ωn

f(x, un(x))un(x)dx≤ k2

θ. Next, we have

(4)

Ω

1

p(x)|∇u_n|^p(x)dx≤d+

Ωn

F(x, u_n(x))dx+

Ω_nF(x, u_n(x))dx≤

≤d+k1+

Ωn

F(x, un(x))dx.

By (1), (2), (3) and (4), we get (5)

Ω

1

p(x)|∇u_n|^p(x)dx−1 θ

Ωf(x, u_n(x))u_n(x)dx≤k, wherek=d+k1+^k_θ².

On the other hand, becauseH(un)→0 asn→ ∞, there isn0∈Nsuch thatH(u_n) ≤1 for n≥n₀. Consequently, for alln≥n₀ we have

H(un), un

≤ u_n1,p

or

|ϕ_p(|∇u_n|)− N_fun, un| ≤ u_n1,p, hence

(6) −1

θN_fu_n, u_n ≥ −1

θu_n1,p−1

θϕ_p(|∇u_n|). It follows from (5) and (6) that

(7)

1 p₊ −1

θ

ϕ_p(|∇u_n|)−1

θu_n1,p≤k for all n≥n₀. Consider the setsA=

n∈N|n≥n₀ and u_n1,p≤1

and B = n∈ N|n≥n0 and un_1,p >1

. It is obvious that the sequence (un)_n∈A⊆X is bounded. Ifn∈B thenun_1,p >1 and we have the inequality

(8) ϕp(|∇u_n|)≥ u_n^p_1,p⁻. Finally, by (7) and (8) we get

1 p₊ −1

θ

u_n^p_1,p⁻ −1

θu_n1,p≤k for alln∈B.

We know that θ > p− and the last inequality shows that the sequence (u_n)_n∈N ⊆X is bounded. By the Smuljian theorem, we can extract a subse- quence (u_nk)_k∈Nof (u_n)_n∈N weakly convergent to someu∈X. As H(u_nk)→ 0, we have

(9) lim

k→∞

H(u_nk), u_nk−u

= 0.

The Nemytskii operator Nf is strongly continuous, so that lim

k→∞Nf (unk) = N_f(u) in X^∗ and the weak convergence u_nk u inX gives

(10) lim

k→∞N_funk, unk−u= 0. From (9) and (10) we conclude that

k→∞lim

−∆_p(x)u_nk, u_nk−u

= 0

and, since−∆_p(x) is a mapping of type (δ+), we have unk →u inX. Proposition 3.4. Assume (1.3) and (1.4). Then the functional H : X→Rdefined by

H(u) =

Ω

1

p(x)|∇u|^p(x)dx−

ΩF(x, u(x))dx satisfies the(PS)^∗_c condition for any c∈R.

Proof. Let the sequence (un)_n∈N∗ ⊆X and c∈R be such thatun∈Yn

for all n∈N^∗, H(u_n)→c and (H |_Yn)(u_n)→0 as n→ ∞. Similarly to the proof of Proposition 3.3, we get the boundedness of the sequence (u_n)_n∈N∗ ⊆ X. Consequently, by the Smuljian theorem, we can extract a subsequence (unk)_k∈N∗ of (un)_n∈N∗ weakly convergent to some u ∈ X. Next, as X =

n∈N^∗Y_n, we can choose a sequence (v_n)_n∈N∗ ⊆ X such that v_n ∈ Y_n for all n∈N^∗ and lim

n→∞vn=v inX. We then have

(11)

H(unk), unk −u

=

H(unk), unk −vnk

+

H(unk), vnk−u . As

H |_Ynh

(unk)→ 0 ask → ∞, unk−vnk 0 inYnh and vnk →u in X, we deduce that

(12)

H(u_nk), u_nk −v_nk

→0 and

H(u_nk), v_nk−u

→0 as k→ ∞.

From (11) and (12), we conclude that

(13)

H(unk), unk−u

→0 as k→ ∞

We know that under our hypotheses, the Nemytskii operator Nf :X → X^∗ is strongly continuous while the p(x)-Laplace operator is a mapping of type (δ+). So, H : X → X^∗ is a mapping of type (δ+). From (13) and the weak convergenceu_nk →u inX we obtain that lim

k→∞u_nk =u inX. Taking arbitrarilywi∈Yi, notice that when nk≥j we have

H(u), w_i

=

H(u)−H(u_nk), w_i +

H(u_nk), w_i

=

=

H(u)−H(unk), wi +

H|_Ynk

(unk), wi

Lettingk→ ∞ we deduce thatH(u), w_i= 0 for allw_i ∈Y_i, henceH(u) = 0. This shows thatH satisfies the (PS)^∗_c condition for every c∈R.

Before giving our main results we state several lemmas that will be used later.

Lemma3.1. Ifα∈C+(Ω)andα(x)< p^∗(x)for anyx∈Ω, then for each k∈N^∗ there exists βk = sup

u_α |u ∈Zk,u_1,p = 1

<+∞. Moreover,

k→∞lim β_k= 0.

Lemma3.2. Assume thatΨ :X→Ris strongly-continuous and Ψ(0) = 0. Then for each γ >0 and k∈N^∗ there exists

α_k= sup

|Ψ(u)| |u∈Z_k,u_1,p< γ

<+∞. Moreover, lim

k→∞α_k= 0.

Lemma3.3. Assume that the Carath´eodory function f satisfies hypothe- ses (1.3) and (1.4). Then there exist k₁, k₂ > 0, σ ∈ L¹(Ω) and γ ∈ L^∞(Ω) withγ(x)>0 for all x∈Ω such that

F(x, t)≥γ(t)|t|^θ−k1−k2σ(x) for x∈Ω, t∈R.

Now, we are able to prove the main results of this paper.

Theorem3.1. If hypotheses (1.1), (1.2), ( 1.3), (1.4), (1.5)are satisfied, then problem (P) has a sequence of weak solutions (±u_n)_n∈N ⊆ X such that H(±u_n)→+∞ as n→ ∞.

Proof. The C¹-functionalH:X→Rdefined by H(u) =

Ω

1

p(x)|∇u|^p(x)dx−

ΩF(x, u(x))dx

is even and, by Proposition 3.3, it satisfies the (PS) condition. It follows that H satisfies the (PS)_c condition for all c >0.

First, we prove that for each k∈N^∗ there exists r_k>0 such that

u∈Zk,uinf_1,p=rkH(u)→ ∞ ask→ ∞. From the inequalities

H(u)≥ 1

p₊ϕp(|∇u|)−

ΩF(x, u(x))dx,

ΩF(x, u(x))dx ≤

Ω|F(x, u(x))|dx≤

Ω

c₁|u|^α(x)+σ(x)

dx we deduce that

H(u)≥ 1

p+u^p_1,p⁻−c1ϕα(u)−c2

for allu∈X,u_1,p ≥1, where c2=

Ωσ(x)dx.

We know that ϕ_α(u) ≤1 if u_α ≤ 1 and ϕ_α(u) ≤ u^α_α⁺ ifu_α > 1.

From Lemma 3.1 we haveu_α≤β_ku_1,p for all u∈Z_k. Thus, (14) H(u)≥ 1

p+u^p_1,p⁻−c1β_k^α+u^α_1,p⁺ −c2 for all u∈Zk,u_1,p≥1.

Choose the real numbers r_k=

c1α+β_k^α+_p ¹

−−α+ , k∈N^∗. Hypothesis (1.5) ensures that lim

k→∞r_k = +∞. Without loss of generality, we can suppose thatrk≥1 for all k∈N^∗.

Ifu∈Z_k and u_1,p=r_k, inequality (14) gives H(u)≥ α+−p+

α₊p₊

c1α+β_k^α+_p ^p−

−−α+ −c2

for allu∈Z_k, u_1,p=r_k. Consequently,

(15) inf

u∈Zk,u_1,p=rkH(u)≥ α+−p+

α₊p₊

c₁α₊β_k^α+_p ^p−

−−α+ −c₂. From (15) and (1.5), it is obvious that

u∈Zk,uinf_1,p=rkH(u)→+∞ask→ ∞.

Second, we prove that for eachk∈N^∗ there existsρ_k> r_k>0 such that

u∈Yk,umax_1,p=ρkH(u)≤0.

The functional _θ:X→Rdefined by u_θ =

Ωγ(x)|u(x)|^θdx ¹_θ

is a norm on X. The norms _θ and _1,p being equivalent on the finite dimensional subspaceY_k, there is a constant δ >0 such that

(16) u_θ≥δu_1,p for all u∈Yk. By Lemma 3.3 we have

ΩF(x, u(x))dx≥ u^θ_θ−c3, wherec3 =

Ω(k1+k2σ(x)) dx. On the other hand, we have

H(u)≤ 1

p u^p_1,p⁺ −

ΩF(x, u(x))dx for allu∈X,u_1,p ≥1, which yields

(17) H(u)≤ 1

p u^p_1,p⁺ − u^θ_θ+c3 for all u∈Y_k,u_1,p≥1.

By (16) and (17), we conclude that (18) H(u)≤ 1

p u^p_1,p⁺ −δ^θu^θ_1,p+c₃ for all u∈Y_k,u_1,p≥1.

Becauseθ > p+, we have

t→+∞lim 1

p t^p+−δ^θt^θ+c₃

=−∞

and we can chooseρ_k> r_k>0 such that (19) 1

p u^p_1,p⁺ −δ^θu^θ_1,p+c3 ≤0 ₃ for all u∈Y_k,u_1,p=ρ_k. It is obvious from (18) and (19) that

u∈Yk,umax_1,p=ρkH(u)≤0.

We now apply Proposition 3.1 to theC¹-functional H and obtain a se- quence of critical values of H converging to +∞. Consequently, there is a sequence (±un)_n∈N of critical points for H such that H(±un) → +∞ as n→ ∞, and the proof is complete.

Theorem3.2. If hypotheses (1.1), (1.2), (1.3), (1.4), (1.5) are satisfied, then problem (P) has a sequence of weak solutions (±v_n)_n∈N ⊆ X such that H(±v_n)≤0 for alln∈N and H(±v_n)→0 as n→ ∞.

Proof. The functionfbeing odd in its second argument, theC¹-functional H is even. By Proposition 3.4, theC¹-functionalH satisfies the (PS)^∗_c condi- tion for allc∈R, in particular for allc∈[d_k0,0).

First, we prove that for each k∈N^∗ there exists rk>0 such that

u∈Yk,maxu_1,p=rkH(u)<0.

The functional _θ:X→Rdefined by u_θ =

Ωγ(x)|u(x)|^θdx ¹

θ

is a norm on X. The norms _θ and _1,p being equivalent on the fi- nite dimensional subspace Y_k, there is a constant δ > 0 such that u_θ ≥ δu_1,p for all u∈Yk. Similarly to Theorem 3.1, we have the inequality

H(u)≤ 1

p u^p_1,p⁺ −δ^θu^θ_1,p+c3 for all u∈Yk,u_1,p≥1.

Butθ > p+ and it is obvious that

t→+∞lim 1

p t^p+ −δ^θt^θ+c₃

=−∞. So, there existst0∈(1,+∞) such that

1

p t^p+−δ^θt^θ+c₃ <−1 for all t∈[t₀,+∞).

The last two inequalities show that H(u) < −1 for all u ∈ Y_k, u_1,p = t0. Choosingr_k =t₀ for all k∈N^∗, we have

u∈Yk,umax_1,p=rkH(u)≤ −1<0.

Second, we prove that there existsk₀∈N^∗such that for eachk∈N^∗, k≥ k0, there exists ρk> rk>0 for which

u∈Zk,uinf_1,p=ρkH(u)≥0.

Similarly to Theorem 3.1 we have the inequality H(u)≥ 1

p+u^p_1,p⁻ −c1β_k^α+u^α_1,p⁺−c2 for all u∈Zk,u_1,p ≥1.

We also have

k→∞lim

α₊−p₊ α+p+

c₁α₊β_k^α+_p ^p−

−−α+ = +∞. There existsk0∈N^∗ such that

α₊−p₊ α₊p₊

c1α+β_k^α+_p ^p−

−−α+ −c2 ≥0

and

c1α+β_k^α+_p ¹

−−α+ > t0

for all k≥ k0. Choose ρ_k =

c1α+β_k^α₀⁺_p ¹

−−α+ for k ≥k0. It is obvious that ρk > rk >0 for all k∈N^∗, k≥k0. Ifu ∈Zk with u_1,p =ρk, from (14) we obtain that

H(u)≥ α+−p+

α+p+

c1α+β_k^α₀⁺_p ^p−

−−α+ −c2≥0 which gives

u∈Zk,uinf_1,p=ρkH(u)≥0.

Denote

bk= max

u∈Yk,u_1,p=rkH(u), dk= inf

u∈Zk,u_1,p≤ρkH(u).

BecauseZ_k∩Y_k=and 0< r_k < ρ_k, we have (20) dk ≤bk<0 for all k≥k0. Hypothesis (1.3) helps us to prove that

|F(x, t)| ≤ c

α(x)|t|^α(x)+|t|h(x)

forx∈Ω, t∈R. Let Ψ₁ :X→R, Ψ₂ :X→Rbe the functionals defined by Ψ₁(u) =

Ω

c

α(x)|u(x)|^α(x)dx, Ψ₂(u) =

Ω|u(x)|h(x)dx.

Obviously, Ψ₁(0) = Ψ₂(0) = 0.

The imbedding from X into L^α(x)(Ω) is compact. A straightforward computation shows that the functionals Ψ₁ :X→R,Ψ₂ :X→Rare strongly continuous. Denote

γ_k = sup

|Ψ₁(u)| |u∈Z_k,u_1,p≤1

, ε_k= sup

|Ψ₂(u)| |u∈Z_k,u_1,p≤1

. Lemma 3.2 implies that lim

k→∞γ_k= lim

k→∞ε_k= 0. Considerv∈Z_k withv_1,p = 1 and 0< t < ρ_k. Then

H(tv) =

Ω

1

p(x)|∇(tv)|^p(x)dx−

ΩF(x, tv(x))dx≥

≥ −

ΩF(x, tv(x))dx≥ −Ψ₁(tv)−Ψ₂(tv).

But

Ψ₂(tv) =

Ω|tv(x)|h(x)dx=tΨ₂(v) and

Ψ₁(tv) =

Ω

c

α(x) |tv(x)|^α(x)dx≤t^α+Ψ₁(v).

Combining the last three assertions yields

H(tv)≥ −ρ^α_k⁺Ψ₁(v)−ρ_kΨ₂(v)≥ −ρ^α_k⁺γ_k−ρ_kε_k for allt∈(0, ρ_k) and v∈Z_k withv_1,p= 1.

It is now obvious that

(21) d_k≥ −ρ^α_k⁺γ_k−ρ_kε_k. Recall thatρ_k=

c1α+β^α_k0⁺_p ¹

−−α+ is constant and lim

k→∞γ_k= lim

k→∞ε_k= 0.

Lettingk→ ∞in (21) and taking into account that d_k <0 for allk≥k₀, we get lim

k→∞d_k = 0.

The C¹-functional H satisfies the hypotheses of Proposition 3.2. We conclude that there exists a sequence of negative critical values converging to 0. Consequently, there is a sequence (±vn)_n∈N ⊆ X of critical points for H such thatH(±vn)≤0 for all n∈Nand lim

n→∞H(±vn) = 0.

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Received 5 October 2006 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania ilias@fmi.unibuc.ro