Multiplicity of solutions for

Multiplicity of solutions for

quasilinear elliptic problems involving Φ -Laplacian operator and critical growth

Xuewei Li and Gao Jia

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China Received 27 August 2018, appeared 25 January 2019

Communicated by Dimitri Mugnai

Abstract. In this paper, we study a class of quasilinear elliptic equations with Φ- Laplacian operator and critical growth. Using the symmetric mountain pass theorem and the concentration-compactness principle, we demonstrate that there existsλ_i > 0 such that our problem admitsipairs of nontrivial weak solutions providedλ∈(0,λ_i). Keywords: quasilinear elliptic equation, critical exponent, variational method.

2010 Mathematics Subject Classification: 35J62, 35B33, 35J20.

1 Introduction

In this paper, we discuss the existence of multiple solutions for the quasilinear elliptic problem (−_∆Φu=λ|u|^l∗−2u+ f(x,u), x ∈_Ω,

u=0, x ∈_∂Ω, ^(1.1)

where Ω ⊂ _R^N (N ≥ 2) is a bounded domain with smooth boundary ∂Ω, λ is a positive parameter, l^∗ = _N^lN_−l(1 < l < N) is the critical Sobolev exponent and ∆_Φu denotes the Φ- Laplacian operator, which is defined by∆_Φu=div(φ(|∇u|)∇u). With respect to the function φ:(0,∞)→(0,∞), we assume that it isC¹and satisfies:

(φ₁) φ(t)t →0 ast→0,φ(t)t→_∞ast →_∞; (φ₂) φ(t)t is strictly increasing in(0,∞); (φ₃) 0<l−1 :=inf_t>0(φ(t)t)⁰

φ(t) ≤sup_t>0^(φ_φ⁽₍^t)_t)^t)0 =:m−1< N−1.

Throughout this paper we define Φ(t) =

Z _t

0 φ(s)sds, t≥0,

BCorresponding author. Email: gaojia89@163.com

which is extended as even function,Φ(t) =_Φ(−t), for allt<0. In fact, under the assumptions (φ₁)–(φ₃), the equations like (1.1) may be allowed to possess complicated nonhomogeneous Φ-Laplacian operator. The examples are the following:

(i) p-Laplacian:φ(t) = pt^p−2, for 1< p <N;

(ii) (p,q)-Laplacian:φ(t) = pt^p−2+qt^q−2, for 1< p<q< Nandq∈(p,p^∗)_with p^∗ = _N^pN_−p; (iii) plasticity: φ(t) = pt^p−2(log(1+t))^q+qt^p−1(1+t)⁻¹(log(1+t))^q−1, forp≥1, q>0;

(iv) p(x)-Laplacian: φ(t) = p(x)t^p(x)−2, for p : R^N → _R is Lipschitz continuous and 1 <

p⁻:=inf_RNp(x)≤sup_RN p(x) =: p⁺ < N.

In our discussion, we assume that the nonlinear term f(x,t)∈C(_Ω×_R)satisfies:

(f₁) lim_{|t|→∞ |}^f_t⁽_|l^x,t∗ −⁾1 =0, uniformlyx ∈_Ω;

(f₂) there exist constantsθ ∈(m,l^∗),σ∈[0,l)andC₀,C₁ >0, such that F(x,t)− ¹

θf(x,t)t≤C₀|t|^σ+C₁, forx∈_Ωandt ∈_{R, where} F(x,t) =Rt

0 f(x,s)ds;

(f₃) there exist constantsτ∈(m,l^∗)_andC₂,C₃ >0 such that F(x,t)≤C2|t|^τ+C3

forx∈_Ωandt ∈_R;

(f₄) there exists an open setΩ0 ⊂_Ωwith|_Ω0|>0 such that lim inf

|t|→_∞

F(x,t)

|t|^m = +_∞, uniformlyx ∈_Ω0;

(f₅) f(x, 0) =0 and f(x,−t) =−f(x,t), forx∈ _Ωandt>0.

Remark 1.1. It is easily seen that the following function satisfies hypotheses(f₁)–(f₄): f(x,t) =|t|^r−2t, fort>0 andr∈ (m,l^∗).

The equation (1.1), forΦ(t) =t^p, is well known as the p-Laplacian equation involving critical growth p^∗ = _N^pN_−p. The boundary value problem

(−_∆pu=µ|u|^p∗−2u+ f(x,u), x∈_Ω,

u=_0, x∈∂Ω (1.2)

has been studied by B. Silva and Xavier [11]. The multiplicity of solutions for (1.2) is obtained by the variational method and the minimax critical point theorems. D. Silva improved the variational method and the concentration compactness principle to deal with the problem (cf. [12])

(−_div(|∇u|^p(x)−2∇u) =λ|u|^q(x)−2u+ f(x,u)_, x∈ _Ω,

u=0, x∈ _∂Ω, ^(1.3)

where 0< p(x)≤q(x)≤ p^∗(x) = ^p(x)N

N−p(x),x ∈Ω. Further, one of the main motivations for the study of problem (1.1) is the following problem

−div(φ(|∇u|)∇u) =_b(|u|)u+λf(_x,u)_{, x} ∈_R^N, (1.4) where N ≥ 2, λ > 0 and b(|u|)u possesses critical growth. Fukagai, Ito and Narukawa [5]

proved that problem (1.4) has a positive solution.

As is mentioned in [13], the problem (1.1) has many physical applications, for instance, in nonlinear elasticity, plasticity, generalized Newtonian fluids, etc. We refer the readers to the following related papers (cf. [2,4–6,9]) and references therein.

In this work we will propose a variant symmetric mountain pass theorem for solving the multiplicity of solutions for problem (1.1). This requires the functional associated with the problem (1.1) satisfies the(PS)_c condition below a fixed level. Hence, it will allow us to use a more efficient concentration-compactness type principle than the problem (1.4), which just showed the weak limituis positive in Fukagai, Ito and Narukawa [5].

The main difficulty in dealing with this class of problems is that the associated functional involves the critical growth term so that the embedding ofW₀^1,Φ(_Ω)_into L^l∗(_Ω)is no longer compact. And another difficulty comes from the fact thatΦ-Laplacian operator is nonhomoge- neous, which requires some additional efforts to overcome the estimate. It is worthwhile men- tioning that we exploit the compactness of the embeddingW₀^1,Φ(_Ω),→L_Ψ(_Ω)_{, Φ}≤_{Ψ Φ∗} and the existence of a Schauder basis forW^1,Φ(_Ω)to establish a lower bound for the minimax levels.

Our main result can be stated as follows.

Theorem 1.2. Assume that (φ₁)–(φ₃) and(f₁)–(f₅)hold. Then for any given i ∈ N, there exists λ_i ∈ (0,∞)such that for all λ ∈ (0,λ_i), problem (1.1) possesses at least i pairs of nontrivial weak solutions.

The organization of this paper is as follows. In Section 2, we set up the framework of Orlicz–Sobolev spaces and give some essential results ofΦ-Laplacian. In Section 3, we present the functional associated with the problem (1.1) satisfies the Palais–Smale condition below a given level. Finally, in Section 4, we give some useful lemmas for our main result and the complete proof of the existence of multiple solutions for the problem (1.1).

2 Preliminaries

Due to the nature of the operator∆_Φwe shall work in the framework of Orlicz–Sobolev spaces W^1,Φ(_Ω). For the sake of completeness, we recall some definitions and properties as follows.

The Orlicz space L_Φ(_Ω)_:=

u:Ω→_R: uis measurable and Z

ΩΦ(|u(x)|)dx<_∞

is a Banach space under the usual norm (Luxemburg norm) kuk_Φ =inf

k

k >0

Z

ΩΦ

|u(x)|

k

dx ≤1

.

The Orlicz–Sobolev spaceW^1,Φ(_Ω) is defined as the set of all weakly differentiable u ∈ L_Φ(_Ω) _{such that} D^γu ∈ L_Φ(_Ω) for all multi-indices γ = {γ₁,γ₂, . . . ,γ_N}_with |γ| ≤ _{1. The}

Orlicz–Sobolev norm ofW^1,Φ(_Ω)is defined as

kuk_1,Φ = kuk_Φ+k∇uk_Φ.

We denote by W₀^1,Φ(_Ω) the closure of C^∞₀ (_Ω) with respect to the Orlicz–Sobolev norm of W^1,Φ(_Ω)_.

If

Z ₁

0

Φ⁻¹(s)

s^NN+1 ds<+_∞ and

Z _+∞

1

Φ⁻¹(s)

s^NN+1 ds= +_∞, (2.1) then the Sobolev conjugateN-function functionΦ∗ ofΦis given in [1] by

t∈ (0,∞)7→

Z _t

0

Φ⁻¹(s) s^NN+1 ds.

Notice thatΦis N-function and(φ₃)guarantees (2.1) holds.

The dual(L_Φ(_Ω))^∗ isL

Φe(_Ω)(cf. [6]), whereΦe is called the complement ofΦ, given by Φe(t) =max

s≥0{ts−_Φ(s)}, fort≥0. (2.2) By using of the assumptions(φ₁)and(φ₃), it turns out that Φ,Φ∗ andΦe are N-functions satisfying4₂-condition (cf. [10]), namely there is a constantC₄>0 such that

Φ(_2t)≤C4Φ(_t)_, ∀t>_0.

Meanwhile, the assumptions(φ₃)implies that (φ₃)⁰

1<l:=inf

t>0

φ(t)t²

Φ(t) ≤sup

t>₀

φ(t)t²

Φ(t) =:m< N,

which ensures thatL_Φ(_Ω)andW₀^1,Φ(_Ω)are separable and reflexive Banach spaces (cf. [10]).

Lemma 2.1. Assume that(φ₁)–(φ₃)hold. Then for t≥0, we have

Φe(φ(t)t) =φ(t)t²−_Φ(t)≤_Φ(2t). (2.3) Proof. The convexity ofΦ(t)implies that

Φ(t) +_Φ⁰(t)(s−t)≤_Φ(s), fors,t≥0. By (φ₂)andΦ⁰(t) =φ(t)t, we have

φ(_t)_ts−_Φ(_s)≤φ(_t)_t²−_Φ(_t)_, fors, t≥_{0. Thus by}(_2.2), we obtain

Φe(φ(t)t) =max

s≥0{φ(t)ts−_Φ(s)}

≤φ(t)t²−_Φ(t)

≤φ(t)t²

≤

Z _2t

t φ(z)zdz

≤_Φ(2t), fort ≥0. Hence, this shows (2.3).

Remark 2.2. It is easy to see that(φ₃)⁰ implies that (φ₃)⁰⁰

l≤ ^φ(t)t²

Φ(t) ≤m, t>₀ is verified.

It follows from the Poincaré inequality forΦ-Laplacian operator (cf. [7]) that there exists a constantS₁ >0 such that

kuk_Φ ≤ S₁k∇uk_Φ,

for all u∈W₀^1,Φ(_Ω). As a consequence of this, the normk · k_1,Φ is equivalent to the norm kuk:=k∇uk_Φ

onW₀^1,Φ(_Ω). In this paper, we will usek · kas the norm ofW₀^1,Φ(_Ω).

The embedding results below (cf. [1,3]) are used in this paper. First, we have

W₀^1,Φ(_Ω),→,→ L_Ψ(_Ω), (2.4) ifΦ≤_ΨΦ∗, whereΨ_Φ∗ means that the functionΨessentially grows more slowly than Φ∗. Furthermore,

W₀^1,Φ(_Ω),→L_Φ∗(_Ω). (2.5) Define a constantS₂>0, such that for anyu∈W₀^1,Φ(_Ω),

kuk_Φ∗ ≤S₂kuk. (2.6)

Besides this, it is worth mentioning that if(φ₁)–(φ₂)and(φ₃)⁰⁰are satisfied, we have L_Φ(_Ω),→L^l(_Ω),

L_Φ∗(_Ω),→L^l∗(_Ω). Define a constantS₃>0, such that for anyu∈W₀^1,Φ(_Ω),

kuk_Ll∗(_Ω) ≤S₃kuk_Φ∗. (2.7) Since

W₀^1,Φ(_Ω),→L^l∗(_Ω), (2.8) we can define a constantS₄>0, such that for anyu∈W₀^1,Φ(_Ω),

kuk_Ll∗(_Ω) ≤S₄kuk. (2.9) Lemma 2.3([5]). Assume that(φ₁)–(φ₃)hold. For t≥0, set

η₁(_t) =_min{t^l,t^m}, η₂(_t) =_max{t^l,t^m}. ThenΦsatisfies

η₁(t)_Φ(ρ)≤_Φ(ρt)≤η₂(t)_Φ(ρ), for anyρ, t>0, (2.10) η₁(kuk_Φ)≤

Z

ΩΦ(u)dx≤η₂(kuk_Φ), for u∈L_Φ(_Ω). (2.11)

LetΦe∗ be the complement of Φ∗, we have

Lemma 2.4([5]). Assume that(φ₁)–(φ₃)hold. For t≥0, set

η₃(t) =_min{t^el∗,t^me∗}_, η₄(t) =_max{t^el∗,t^me∗}_, whereel^∗= _l∗^l−^∗1 andme^∗= _m^m∗−^∗1. ThenΦe∗satisfies

me^∗ ≤ ^Φe

0∗(t)t

Φe∗(t) ≤el^∗, for t>0,

η₃(t)Φe∗(ρ)≤ Φe∗(ρt)≤η₄(t)Φe∗(ρ)_, for anyρ, t≥0, (2.12) η3(kuk

Φe∗)≤

Z

ΩΦe∗(u)dx≤η₄(kuk

Φe∗), for u∈ L_Φe∗(_Ω). (2.13) Next, we recall the variational framework for problem (1.1). The functional I_λ:W₀^1,Φ(_Ω)→ Rassociated with our problem is given by

I_λ(u) =

Z

Ω

Φ(|∇u|)− ^λ

l^∗|u|^l∗−F(x,u)

dx, u∈W₀^1,Φ(_Ω)_.

It is easy to verify that I_λ is well-defined and of class C¹ on W₀^1,Φ(_Ω). Hence finding weak solutions for the problem (1.1) is equivalent to find the critical points for the functionalI_λand the Gateaux derivative forI_λ has the following form:

hI_λ⁰(u),ψi=

Z

Ω(φ(|∇u|)∇u∇ψ−λ|u|^l∗−2uψ− f(x,u)ψ)dx, for anyu, ψ∈W₀^1,Φ(_Ω).

Definition 2.5. For given Ea real Banach space and I ∈ C¹(E,R), we say that I satisfies the Palais–Smale condition on the level c ∈ _R, denoted by (PS)_c condition, if every sequence {un} ⊂Esuch thatI(un)→candI⁰(un)→0 asn→∞, possesses a convergent subsequence inE.

In this article we will apply the following version of the symmetric mountain pass theorem (cf. [11]).

Lemma 2.6. Let E = X⊕Y, where E is a real Banach space and X is finite dimensional. Suppose I ∈C¹(E,R)is an even functional, satisfying I(0) =0and

(I₁) there exists a constantρ>0such that I|_∂Bρ∩Y >0;

(I₂) there exist a subspace W of E withdimX <dimW < _∞and M >0such that max_u∈W I(u)<

M;

(I₃) considering M>0given by(I₂), I satisfies(PS)_ccondition, for0<c< M.

Then I possesses at least(_dimW−_dimX)pairs of nontrivial critical points.

3 The Palais–Smale condition

In this section, we will verify that the functionalI_λ satisfies the(PS)_ccondition below a given level whenλ>0 is sufficiently small. In order to do this, we need some preliminary results.

First, we will show the Palais–Smale sequence{un} ⊂W₀^1,Φ(_Ω)is bounded.

Lemma 3.1. Assume that(φ₁)–(φ₃)and(f₁)–(f₂)hold. Then the(PS)_csequence{un} ⊂W₀^1,Φ(_Ω) of I_λ is bounded.

Proof. According to(f₂),(φ₃)⁰⁰ and Hölder’s inequality, it follows that I_λ(u_n)−¹

θhI_λ⁰(u_n),u_ni=

Z

Ω

Φ(|∇u_n|)−¹

θφ(|∇u_n|)|∇u_n|²

dx +λ

1 θ − ¹

l^∗ _Z

Ω|u_n|^l∗dx−

Z

Ω

F(x,u_n)− ¹

θf(x,u_n)u_n

dx

≥ 1

m−¹ θ

Z

Ωφ(|∇un|)|∇un|²dx+λ 1

θ − ¹ l^∗

kunk^l_L^∗_l∗(_Ω)

−C₀ku_nk^σ_Lσ(_Ω)−C₁|_Ω|

≥λ 1

θ − ¹ l^∗

ku_nk^l∗

L^l∗(_Ω)−C₀|_Ω|^1−lσ∗ku_nk^σ

L^l∗(_Ω)−C₁|_Ω|.

(3.1)

Moreover, by Young’s inequality, we have

kunk^σ_Ll∗(_Ω)≤δkunk^l_L^∗_l∗(_Ω)+C_δ, (3.2) whereδ = ^λ(¹_θ⁻_l¹∗)

2C0|_Ω|¹⁻l^σ∗ andC_δ = ^l∗_l⁻∗^σ σ δl^∗

_{l∗ −}^σ

σ.

On the other hand, since{u_n}is a(PS)_c sequence, we have I_λ(un)− ¹

θhI_λ⁰(un),uni ≤I_λ(un) + ¹

θkI_λ⁰(un)k

W₀^1,eΦ(_Ω)kunk

≤ C₅+C₆ku_nk,

(3.3) with some constantsC₅,C₆ >0.

Therefore, from (3.1), (3.2) and (3.3)), there exist constants C₇,C₈>0 such that

kunk^l_L^∗_l∗(_Ω)≤C₇+C₈kunk. (3.4) Now, by(f₁), for given e>0, there exists a constantCe>0 such that

|f(x,t)| ≤C_e+e|t|^l∗−1, forx∈_Ωandt∈_R (3.5) and

|F(x,t)| ≤C_e+ ^e

l^∗|t|^l∗, forx ∈_Ωandt ∈_R. (3.6) Consequently, by (3.4) and (3.6), we have

I_λ(un) =

Z

ΩΦ(|∇un|)dx− ^λ l^∗

Z

Ω|un|^l∗dx−

Z

ΩF(x,un)dx

≥η₁(ku_nk)−^λ+e l^∗ ku_nk^l∗

L^l∗(_Ω)−C_e|_Ω|

≥η₁(ku_nk)−^λ+e

l^∗ C₈ku_nk − ^λ+e

l^∗ C₇−C_e|_Ω|

and

η₁(ku_nk)≤ ^λ+e

l^∗ C₈ku_nk+C(e)_{. (3.7)} This implies that{un}is bounded.

By (2.6), (2.7), (2.11) and Lemma3.1, we obtain

Corollary 3.2. If{un} ⊂W₀^1,Φ(_Ω)is a(PS)_csequence of I_λ, then the sequencesR

ΩΦ(|∇un|)dx andR

Ω|u_n|^l∗dx are bounded.

Next, we use the concentration-compactness type principle which is analogous to Lemma 4.2 of Fukagai, Ito and Narukawa [5]. This will be the keystone that enables us to verify that I_λ satisfies the (PS)_c condition. First, we will recall a measure theory result as follows.

Let {u_n} ⊂ W₀^1,Φ(_Ω) be the (PS)_c sequence. Lemma 3.1 and Corollary 3.2 show that {u_n},{R

ΩΦ(|∇u_n|)dx}and{R

Ω|u_n|^l∗dx}are bounded. Otherwise, we know that L_Φ(_Ω)_and L^l∗(_Ω)are reflexive Banach spaces. Then there exist two nonnegative measuresµ,ν∈ M(_Ω), the space of Radon measures and a subsequence of{u_n}, still denoted by{u_n}, such that

Φ(|∇u_n|)*_{µ, in} M(_Ω)_{, (3.8)}

|u_n|^l∗ *ν, inM(_Ω). (3.9)

Lemma 3.3. Assume that (φ₁)–(φ3) hold. Let {un} o f I_λ be a Palais–Smale sequence such that u_n * u in W₀^1,Φ(_Ω) andΦ(|∇u_n|) * µ, |u_n|^l∗ * ν in M(_Ω), where µ, ν are two nonnegative measures onΩ. Then there exist an at most countable set J and a family{x_j}_j∈J of distinct points in Ωsuch that

(i) ν=|u|^l∗+

∑

j∈J

ν_jδ_xj,

where{ν_j}_j∈J is a family of positive constants andδ_xj is the Dirac measure of mass 1 concentrated at x_j;

(ii) µ≥_Φ(|∇u|) +

∑

j∈J

µ_jδ_xj,

where{µ_j}_j∈J is a family of positive constants, satisfyingν_j ≤max S^l₄^∗µ

l∗ l

j ,S^l₄^∗µ

l∗ m

j for all j∈ J.

Proof. The proof of Lemma3.3 is similar to Lemma 4.2 in Fukagai, Ito and Narukawa [5], we omit the details here.

Lemma 3.4. Assume that (φ₁)–(φ₃) and (f₁)–(f₂) hold. For a given 0 < λ < _∞, let {u_n} ⊂ W₀^1,Φ(_Ω)be a Palais–Smale sequence of I_λ. Considering J given by Lemma3.3, then for each j∈ J, we have eitherν_j =0or

ν_j ≥min





 l λS^l₄

! ^l

∗ l∗ −l

, l

λS₄^{m l}

∗ l∗ −m





 .

Proof. Let us first define ψ ∈ C₀^∞(_R^N)such that ψ(x) = 1 in B(0,¹₂), supp(ψ) ⊂ B(0, 1)and 0≤ψ(x)≤_1, ∀x∈_R^N_{. For each}j∈ J ande>0, let us define

ψ_e(x) =ψ

x−x_j e

, ∀x∈ _R^N.

Then{u_nψ_e(x)} ⊂W₀^1,Φ(_Ω)is bounded inW₀^1,Φ(_Ω). From the fact that I_λ⁰(u_n)→0, it follows that

hI_λ⁰(u_n),u_nψ_ei=o_n(1), i.e.,

Z

Ωφ(|∇un|)∇un∇(unψe) =λ Z

Ω|un|^l∗ψedx+

Z

Ω f(x,un)unψedx+on(1). (3.10) By(φ₃)⁰⁰, we obtain

Z

Ωφ(|∇u_n|)∇u_n∇(u_nψe)dx=

Z

Ωφ(|∇u_n|)|∇u_n|²ψedx+

Z

Ωφ(|∇u_n|)(∇u_n∇ψe)u_ndx

≥l Z

ΩΦ(|∇u_n|)ψ_edx+

Z

Ωφ(|∇u_n|)(∇u_n∇ψ_e)u_ndx.

(3.11)

It is obvious that l

Z

ΩΦ(|∇u_n|)ψ_edx+

Z

Ωφ(|∇u_n|)(∇u_n∇ψ_e)u_ndx

≤λ Z

Ω|un|^l∗ψ_edx+

Z

Ω f(x,un)unψ_edx+on(1).

(3.12)

On the one hand, by Lemma3.1, we know that the Palais–Smale sequence{u_n} ⊂W₀^1,Φ(_Ω) of I_λ is bounded. Taking a subsequence of{un}if necessary, we may suppose that

un*u inW₀^1,Φ(_Ω), (3.13)

un→u in L_Φ(_Ω), (3.14)

u_n→u a.e. inΩ. (3.15)

Moreover, from(_2.3), (2.10) and (2.11) it is easy to see that Z

ΩΦe(φ(|∇un|)∇un)dx≤

Z

ΩΦ(2|∇un|)dx≤η2(2)

Z

ΩΦ(|∇un|)dx ≤η2(2)η2(kunk). Clearly, the sequence{φ(|∇u_n|)∇u_n}is bounded in L

Φe(_Ω). Thus, there exists a subsequence {un}such that for someωe₁∈ L_Φe(_Ω,R^N)

φ(|∇un|)∇un *ω_e1 in L_Φe(_Ω,R^N). (3.16) Therefore, since supp(∇ψe)⊂ B(x_j,e), (3.14) and (3.16), we have

nlim→_∞ Z

Ωφ(|∇un|)(∇un∇ψ_e)undx=

Z

Ω(ω_e1∇ψ_e)udx. (3.17) On the other hand, we will prove

nlim→_∞ Z

Ωf(x,un)unψedx=

Z

Ω f(x,u)uψedx. (3.18) First, we show the following claim.

Claim 1:{f(x,u_n)}is bounded inL_Φe∗(_Ω).

In fact, from (2.12), (3.5), Corollary 3.2, 4₂-condition and the convexity of Φe∗, there exist

constantsC₉,C₁₀>0 such that Z

ΩΦe∗(f(x,un))dx≤C9

Z

ΩΦe∗(|un|^l∗−1)dx+C₁₀ Z

ΩΦe∗(Ce)dx

≤C₉Φe∗(1)

Z

{x∈_Ω;|un|≥1}|u_n|^{(l∗−1)el∗}dx+C₉ Z

{x∈_Ω;|un|<1}

Φe∗(1)dx +C₁₀

Z

ΩΦe∗(C_e)dx

≤C9Φe∗(1)

Z

Ω|un|^l∗dx+C9

Z

ΩΦe∗(1)dx+C₁₀ Z

ΩΦe∗(Ce)dx

< _∞.

Therefore, the claim is proved.

By (3.5), (3.13)–(3.15) and Claim 1, we are now in a position to obtain (3.18).

Now, according to (3.8), (3.9), (3.17), (3.18) and lettingn→_∞in (3.12), it follows that l

Z

Ωψedµ+

Z

Ω(ω_e1∇ψe)udx≤λ Z

Ωψedν+

Z

Ω f(x,u)uψedx. (3.19) Next, we will prove that the second term of the left-hand side converges 0 ase→_0.

By I_λ⁰(un)→0, we have for anyv ∈W₀^1,Φ(_Ω) hI_λ⁰(un),vi=

Z

Ω(φ(|∇un|)∇un∇v−λ|un|^l∗−2unv− f(x,un)v)dx=on(1). (3.20) Moreover, from Claim 1, there is a subsequence{u_n}_{such that}

λ|u_n|^l∗−1+ f(x,u_n)*ω_e2 in L

Φe∗(_Ω), (3.21)

for someωe₂∈ L_Φe∗(_Ω). Hence, by (3.16), (3.20) and (3.21), we conclude Z

Ω(ω_e1∇v−ω_e2v)dx=0, for anyv∈W₀^1,Φ(_Ω). Substitutingv =uψe, we have

Z

Ω(ω_e1∇(uψe)−ω_e2uψe)dx=0, i.e.,

Z

Ω(ω_e1∇ψe)udx=−

Z

Ω(ω_e1∇u−ω_e2u)ψedx.

Notingωe₁∇u−ω_e2u∈ L¹(_Ω), we see that the right-hand side tends to 0 ase→0. Evidently, lim

e→0

Z

Ω(ω_e1∇ψ_e)udx=0. (3.22)

Furthermore, by (3.5) and Lemma3.1, we have Z

Ω|f(x,u)u|dx≤C_e Z

Ω|u|dx+e Z

Ω|u|^l∗dx≤C_ekuk_L1(_Ω)+eS₄kuk^l∗ < _∞. This implies that

lim

e→0

Z

Ω f(x,u)uψedx=0. (3.23)

Consequently, by (3.22) and (3.23), lettinge→0 in (3.19), we obtain for each j∈ J lµ_j ≤ λν_j.

By Lemma3.3, we get

min{S⁻₄^l∗lν_j^l,S⁻₄^l∗mν^m_j } ≤µ^l

∗ j ≤

λ l

l^∗

ν^l

∗ j , i.e.,ν_j =0 or

ν_j ≥min





 l λS₄^l

! ^l

∗ l∗ −_l

, l

λS^m₄ ^l

∗ l∗ −m





 .

Lemma 3.5. Assume that(φ₁)–(φ3)and(f₁)–(f2)hold. Let{un} ⊂W₀^1,Φ(_Ω)be a(PS)_c sequence of I_λ. Then, given M>0, there existsλ^∗ >0such that I_λsatisfies(PS)_ccondition for all0<c< M, provided0<λ<λ^∗.

Proof. Since{un}is a(PS)_c sequence of I_λ and 0<c< M, takingn→_∞in (3.1), we obtain λ

1 θ − ¹

l^∗ _Z

Ωdν≤c+C₁|_Ω|+C₀|_Ω|^1−lσ∗ _Z

Ωdν _l^σ_∗

< M+C₁|_Ω|+C0|_Ω|^1−lσ∗ Z

Ωdν _l^σ∗

.

(3.24)

Therefore, if we choose λ^∗ =min







lS₄^−1l, lS⁻₄ ^m1,

d₁ M+d₂

^l_l^{∗ −}_−σ^l S⁻

l(l∗ −σ) l−σ

4 ,

d₁ M+d₂

^l_m^{∗ −}₋^m_σ S⁻

m(l∗ −σ) m−σ

4





 ,

whered₁= l^{l∗ −l−σσ}(¹_θ −_l¹∗)^{ll∗ −−σl} andd2=C₁|_Ω|+C0|_Ω|^1−lσ∗, then we have from (3.24) Z

Ωdν<min





 l λS^l₄

! ^l

∗ l∗ −l

, l

λS^m₄ ^l

∗ l∗ −m







, (3.25)

for all 0< λ< λ^∗.

As a consequence of this fact and Lemma3.4, we conclude that for each j∈ J,ν_j =0 and

nlim→_∞ Z

Ω|u_n|^l∗dx=

Z

Ω|u|^l∗dx.

Thus, there existsu∈W₀^1,Φ(_Ω)such that, up to subsequence,

u_n→u in L^l∗(_Ω). (3.26)

Next, fromhI_λ⁰(u_n)_,(u_n−u)i=o_n(₁)_{, we have}

nlim→_∞ Z

Ω(φ(|∇un|)∇un∇(un−u)−λ|un|^l∗−2un(un−u)−f(x,un)(un−u))dx=0. (3.27) Hence, we can derive from (3.13)–(3.15), (3.18), (3.26) and (3.27) that

nlim→_∞ Z

Ωφ(|∇u_n|)∇u_n∇(u_n−u)dx=0.

Moreover, by (3.13) and Lemma 5 in [8], we conclude that u_n→u inW₀^1,Φ(_Ω)_.

4 Proof of Theorem 1.2

In order to verify Theorem1.2, we need to prove that Lemma2.6is applicable in our situation, namely the functionalI_λ onW₀^1,Φ(_Ω)satisfies the hypotheses(I₁)and(I₂).

First, since E = W₀^1,Φ(_Ω) is a separable and reflexive Banach space, then there exist a Schauder basis{e_i}_i∈N⊂ Eand{e^∗_j}_j∈N⊂ E^∗ such that

(e_i,e^∗_j) =δ_ij =

(1, i= j, 0, i6= j, and

E=span{e_i|i∈_N}, E^∗ =span{e^∗_j|j∈_N}. Now, fixing a Schauder basis{e_i}_i∈NofW₀^1,Φ(_Ω), we set

X_k :=span{e₁,· · · ,e_k}, Y_k :=

k

\

j=1

Kere^∗_j, (4.1)

in such way thatE=W₀^1,Φ(_Ω) =X_k⊕Y_k, fork∈_N.

Lemma 4.1. Assume that(φ₁)–(φ₃)hold. IfΦ≤_{Ψ Φ∗}, setting

S_k,Ψ :=sup{kuk_LΨ(Ω): kuk=1, u∈Y_k, k∈_N}, thenlim_k→∞S_k,Ψ=0.

Proof. It is clear that 0 ≤ S_k+1,Ψ ≤ S_k,Ψ. Thus we have S_k,Ψ → S_Ψ ≥ _{0, as} k → _{∞. And for} everyk ≥0, there existsu_k ∈Y_k such thatku_kk=1 and

ku_kk_LΨ(Ω)> ^Sk,Ψ

2 . (4.2)

By definition of Y_k, u_k * 0 in W₀^1,Φ(_Ω), as k → ∞. By (2.4), we have u_k → 0 in L_Ψ(_Ω), ask→∞. Using (4.2), we obtainS_k,Ψ →0, ask→∞. Hence we have proved thatS_Ψ=0.

Lemma 4.2. Assume that (φ₁)–(φ3) and (f₁)–(f3) hold. Then there exist constants k, ρ, eλ > 0 andα>0, such that for any u∈Y_k withkuk= ρand0<λ<eλ,

I_λ|_∂B

ρ∩Y_k ≥α.

Proof. From(f3), (2.9), (2.11) and Hölder’s inequality, there exists a constantS₄>0 such that I_λ(u) =

Z

Ω

Φ(|∇u|)− ^λ

l^∗|u|^l∗−F(x,u)

dx

≥η₁(kuk)− ^λ

l^∗S₄^l∗kuk^l∗−C2

Z

Ω|u|^τdx−C3|_Ω|

≥η₁(kuk)− ^λ

l^∗S₄^l∗kuk^l∗−C₂|_Ω|^1−lτ∗kuk^τ

L^l∗(_Ω)−C₃|_Ω|.

(4.3)

By(2.7), Lemma3.5 and Lemma4.1, consideringS_k,Φ∗ to be chosen posteriorly, for allu∈ Y_k andkuk=ρ>1, we have

I_λ(u)≥η₁(kuk)− ^λ

l^∗S₄^l∗kuk^l∗−C₂|_Ω|^1−lτ∗S^τ₃kuk^τ_L

Φ∗(_Ω)−C₃|_Ω|

≥ρ^l− ^λ l^∗S^l₄^∗ρ^l

∗−C₂|_Ω|^1−lτ∗S₃^τS^τ_k,Φ∗ρ^τ−C₃|_Ω|

≥ρ^l(1−C₂|_Ω|^1−lτ∗S₃^τS^τ_k,Φ∗ρ^τ−l)−C₃|_Ω| − ^λ l^∗S^l₄^∗ρ^l

∗.

Now, by Lemma 4.1 again and takingk sufficiently large, there exists sufficiently small S_k,Φ∗ such thatC₂|_Ω|^1−lτ∗S^τ₃S^τ_k,Φ∗ρ^τ−l ≤ ¹₂, ¹₂ρ^l−C₃|_Ω| ≥ ¹₄ρ^l andρ=ρ(S_k,Φ∗)>1.

Consequently, for every u ∈ Y_k with kuk = ρ > 1 and k sufficiently large, there exist sufficiently small eλ>0 and a constant α>0 such that

I_λ(u)≥ ¹ 4ρ^l− ^λ

l^∗S^l₄^∗ρ^l

∗ > α>0 for 0<λ<eλ. Hence, we complete the proof of Lemma4.2.

Lemma 4.3. Assume that(φ₁)–(φ₃)and(f₄)hold. Then for given q∈N, there exist a subspace W of W₀^1,Φ(_Ω)and a constant M_q>0, independent ofλ, such thatdimW = q andmax_u∈W I_λ(u)< M_q. Proof. First, from(f₄)_{, let}x₀∈ _Ω0andr₀>0 be such thatB(x₀,r₀)⊂_Ω0and 0<|B(x₀,r₀)|<

|_Ω0|

2 . We take u₁ ∈ C₀^∞(_Ω) with supp(u₁) = B(x₀,r₀). Considering Ω1 := _Ω0\B(x₀,r₀), we have |_Ω1| > ^|Ω₂^0| > 0. Next, let x₁ ∈ _Ω1 and r₁ > 0 be such that B(x₁,r₁) ⊂ _Ω1 and 0 < |B(x₁,r₁)| < ^|Ω₂^1|. We takeu₂ ∈ C₀^∞(_Ω)with supp(u₂) =B(x₁,r₁). After a finite number of steps, we get u₁,u2, . . . ,uqsuch that supp(u_i)∩supp(u_j) =_∅ and|supp(u_i)|> 0, for alli, j∈ {1, 2, . . . ,q}andi6= j.

LetW =span{u₁,u₂, . . . ,u_q}. For everyu∈W\ {0}, we haveR

Ω0|u|^mdx >0,u=t_uv=tv andv∈∂B(0, 1)∩W. By (2.10) and (2.11), we obtain

max

u∈W\{0}I_λ(u) = max

v∈∂B(0,1)∩W t>₀

Z

Ω

Φ(t|∇v|)− ^λ

l^∗|tv|^l∗−F(x,tv)

dx

≤ max

v∈∂B(0,1)∩W t>0

η₂(t)

Z

ΩΦ(|∇v|)dx−

Z

ΩF(x,tv)dx

≤ max

v∈∂B(_0,1)∩W t>0

η₂(t)η₂(kvk)−

Z

ΩF(x,tv)dx

= max

v∈∂B(0,1)∩W t>0

η₂(t)

1− ¹ η₂(t)

Z

ΩF(x,tv)dx

.

(4.4)

Next, in order to prove the lemma, it suffices to show that

|tlim|→_∞

1

|t|^m

Z

ΩF(x,tv)dx >1 (4.5)

uniformly forv ∈∂B(0, 1)∩W.

In fact, by(f₄), for some positive constantK, there is a constantC_K>0 such that F(x,s)≥K|s|^m−C_K,