Infinitely many weak solutions for p ( x ) -Laplacian-like problems with sign-changing potential

Infinitely many weak solutions for p ( x ) -Laplacian-like problems with sign-changing potential

Qing-Mei Zhou

and Ke-Qi Wang

1Library, Northeast Forestry University, Harbin, 150040, P.R. China

2College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin, 150040, P.R. China

Received 3 October 2019, appeared 1 February 2020 Communicated by Patrizia Pucci

Abstract. This study is concerned with the p(x)-Laplacian-like problems and arising from capillarity phenomena of the following type







−div

1+ √^|∇u|p(x)

1+|∇u|^2p(x)

|∇u|^p(x)−2∇u

=λf(x,u), in Ω, u=0, on∂Ω,

where Ω is a bounded domain inR^N with smooth boundary ∂Ω, p ∈ C(Ω), and the primitive of the nonlinearity f of super-p⁺growth near infinity inuand is also allowed to be sign-changing. Based on a direct sum decomposition of a space W₀^1,p(x)(_Ω)_{, we} establish the existence of infinitely many solutions via variational methods for the above equation. Furthermore, our assumptions are suitable and different from those studied previously.

Keywords: p(x)-Laplacian-like, variational method, multiple solutions, sign-changing potential.

2020 Mathematics Subject Classification: 35D05, 35J60, 35J70.

1 Introduction and main results

The present study is concerned with the existence of infinitely many nontrivial solutions for the nonlinear eigenvalue problems involving the p(x)-Laplacian-like operators, originated from a capillary phenomena,







−div

1+ √^|∇u|p(x)

1+|∇u|^2p(x)

|∇u|^p(x)−2∇u

=λf(x,u), inΩ, u=_{0, on} ∂Ω,

(P)

BCorresponding author. Email: wangkqnefu@163.com

where Ω is a bounded domain in R^N with smooth boundary ∂Ω, p ∈ C(_Ω), λ > 0 is a parameter and f : Ω×_R → _R satisfies a Carathéodory condition and the primitive of the nonlinearity f is allowed to be sign-changing.

Capillarity can be briefly explained by considering the effects of two opposing forces:

adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e., the attractive force between the molecules of the liquid.

The study of capillary phenomena has gained some attention recently. This increasing interest is motivated not only by fascination in naturally-occurring phenomena such as motion of drops, bubbles and waves but also its importance in applied fields ranging from industrial and biomedical and pharmaceutical to microfluidic systems.

Recently, problem (P) has begun to receive more and more attention, see, for example, [2,7,8,11–13,15]. Let us recall some known results on problem (P). When the the primitiveF of f oscillates at infinity, Shokooh and Neirameh [12] showed the existence of infinitely many weak solutions for this problem by using Ricceri’s variational principle. For the case of f isp⁺- superlinear at infinity, Zhou [15] and Ge [7] both obtained the existence of nontrivial solution of problem (P) for every parameterλ >0, under suitable conditions on f. Rodrigues in [11], by using Fountain Theorem, established the existence of sequence of high energy solutions for problem (P), by assuming the following assumptions:

(h₁) f : Ω×_R → _R is a Carathéodory function, that is, t → f(x,t) is continuous for a.e.

x ∈_{Ω, and}x → f(x,t)is Lebesgue measurable for all t∈_R;

(h2) There exists a positive constantCsuch that

|f(_x,t)| ≤C(₁+|t|^r(x)−1)_,

for all x∈_Ωandt∈ _R, wherer ∈C+(_Ω)such that 1< p⁻≤ p⁺ <r⁻≤r(x)< p^∗(x) for allx ∈_Ω, p^∗(x) = ^{N p(x)}

N−p(x) if p(x)< N, p^∗(x) = +_∞if p(x)≥ N;

(h₃)⁰ there existM >0,µ> p⁺ such that for|t| ≥ Mand a.e. x∈ _Ω, 0<µF(x,t)≤t f(x,t),

whereF(x,t) =Rt

0 f(x,s)ds;

(h₄) f(x,−t) =−f(x,t), for all(x,t)∈ _Ω×_R.

Specifically, the author established the following theorem in [11].

Theorem 1.1([11, Theorem 4.7]). Suppose that (h₁), (h₂), (h₃)⁰ and(h₄)hold. Then the problem (P)has an unbounded sequence of weak solutions for every0<λ< ^2r_p+⁺.

Observe that condition (h₃)⁰ plays an important role for showing that any Palais–Smale sequence is bounded in the work. However, there are some functions which do not satisfy condition(h₃)⁰, for example,

f(x,u) =|u|^p+−2uln(1+|u|).

In the present paper, we shall prove the same result as in [11] for problem (P) under more general assumptions on the nonlinearity, which unifies and significantly improves the result of [11]. The underlying idea for proving our main result is motivated by the argument used in [10]. In order to state the main result of this paper, we need the following assumptions:

(h₃) lim

|t|→_∞

|F(x,t)|

|t|^p+ = +_∞uniformly inx, and there existsr₀ >0 such that F(x,t)≥0, ∀(x,t)∈ _Ω×_R, |t| ≥r0;

(h₅) F(x,t) := _p¹+f(x,t)t−F(x,t) ≥ 0, and there exist c₀ > 0 and σ ∈ C+(_Ω) with σ⁻ >

max{1, _p^N−}such that

|F(x,t)|^σ(x) ≤c₀|t|^p−σ(x)F(x,t)_, ∀(x,t)∈_Ω×_R, |t| ≥r₀; (h6) there existµ> p⁺andθ>0 such that

µF(x,t)≤t f(x,t) +θ|t|^p−, ∀(x,t)∈_Ω×_R.

We are now in the position to state our main results.

Theorem 1.2. Suppose that (h₁)–(h₅) hold. Then for each λ ∈ 0,^2r_p+⁺

, problem (P) possesses infinitely many nontrivial solutions.

Theorem 1.3. Suppose that(h₁)–(h₄)hold. and(h₆)hold. Then for eachλ∈ 0,^2r_p+⁺

, problem(P) possesses infinitely many nontrivial solutions.

Remark 1.4. It is easy to see that (h₃) and (h₅) are weaker than(h₃)⁰. In particular, F(x,t) is allowed to be sign-changing in Theorems 1.2 and Theorems 1.3. The role of (h3)⁰ is to ensure the boundedness of the Palais–Smale sequences of the energy functional, it is also significant to construct the variational framework. This is very crucial in applying the critical point theory. However, there are many functions which are superlinear at infinity, but do not satisfy the condition (h₃)⁰ for any µ > p⁺. For example, set f(x,t) = p⁺|t|^p+−2tln(1+t²), then F(x,t) = |t|^p+ln(1+t²)− ^2|t|p

+t

1+t² . It is easy to check that f(x,t)satisfy assumptions(h₃) and(h5).

The rest of this paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, the proof of the main results is given.

2 Preliminaries

In order to discuss problem (P), we need some facts on space W₀^1,p(x)(_Ω) which are called variable exponent Sobolev space. For this reason, we will recall some properties involving the variable exponent Lebesgue–Sobolev spaces, which can be found in [3–6,9] and references therein.

Throughout this paper, we always assume p(x)>1, p ∈C(_Ω). Set C+(_Ω) ={h∈ C(_Ω):h(x)>1 for allx∈ _Ω}. For anyh ∈C+(_Ω), we will denote

h⁻=min

x∈_Ωh(x), h⁺=max

x∈_Ω h(x) and denote by h₁ h₂ the fact that infx∈_Ω(h₂(x)−h₁(x))>_0.

For p(x)∈C+(_Ω), we define the variable exponent Lebesgue space:

L^p(x)(_Ω) =

u:uis a measurable real value function Z

Ω|u(x)|^p(x)dx<+_∞

, with the norm|u|_Lp(x)(_Ω) = |u|_p(x) = inf{λ >0 : R

Ω|^u(x)

λ |^p(x)dx ≤ 1}, and define the variable exponent Sobolev space

W^1,p(x)(_Ω) ={u∈ L^p(x)(_Ω):|∇u| ∈L^p(x)(_Ω)}, with the normkuk=kuk_W1,p(x)(_Ω)=|u|_p(x)+|∇u|_p(x).

We recall that spacesL^p(x)(_Ω)andW^1,p(x)(_Ω)are separable and reflexive Banach spaces.

Denote by L^q(x)(_Ω)the conjugate space of L^p(x)(_Ω)_with ¹

p(x) + ¹

q(x) = 1, then the H ¨older type inequality

Z

Ω|uv|dx≤ 1

p⁻+ ¹ q⁻

|u|_Lp(x)(_Ω)|v|_Lq(x)(_Ω), u∈ L^p(x)(_Ω), v ∈L^q(x)(_Ω) (2.1) holds. Furthermore, if we define the mappingρ:L^p(x)(_Ω)→_Rby

ρ(u) =

Z

Ω|u|^p(x)dx, then the following relations hold

|u|_p(x) <1(=1,>1)⇔ρ(u)<1(=1,>1), (2.2)

|u|_p(x) >1⇒ |u|^p_p⁻_(x) ≤ρ(u)≤ |u|^p_p⁺_(x), (2.3)

|u|_p(x) <1⇒ |u|^p_p⁺_(x) ≤ρ(u)≤ |u|^p_p⁻_(x). (2.4) Next, we denote byW₀^1,p(x)(_Ω)the closure ofC^∞₀ (_Ω)inW^1,p(x)(_Ω). Moreover, we have the following.

Proposition 2.1([6]).

(1) The Poincaré inequality in W₀^1,p(x)(_Ω)holds, that is, there exists a positive constant C such that

|u|_p(x) ≤C|∇u|_p(x), ∀u∈W₀^1,p(x)(_Ω)_.

(2) If q ∈ C(_Ω)and1 < q(x) < p^∗(x)for any x ∈ Ω, then the embedding from W₀^1,p(x)(_Ω)to L^q(x)(_Ω)is compact and continuous, where p^∗(x) = _N^{N p}_−p⁽₍^x_x⁾₎ if p(x) < N or p^∗(x) = +_∞ if p(x)≥ N.

By (1) of Proposition2.1, we know that|∇u|_p(x)andkukare equivalent norms onW₀^1,p(x)(_Ω). We will use|∇u|_p(x) to replacekukin the following discussions.

Proposition 2.2 ([4]). Let p(x) and q(x) be measurable functions such that p(x) ∈ L^∞(_Ω) and 1≤ p(_x)_q(_x)≤_∞almost every where inΩ. Let u∈ L^q(x)(_Ω)_{, u}6=0. Then

|u|_p(x)q(x) ≥1⇒ |u|^p−

p(x)q(x)≤|u|^p(x)

q(x)≤ |u|^p+

p(x)q(x),

|u|_p(x)q(x) ≤₁⇒ |u|^p+

p(x)q(x)≤|u|^p(x)

q(x)≤ |u|^p−

p(x)q(x). In particular, if p(x) = p is a constant, then

|u|^p

q(x)= |u|^p_pq(x).

Consider the following function:

J(u) =

Z

Ω

1 p(x)

|∇u|^p(x)+ q

1+|∇u|^2p(x)

dx, ∀u∈W₀^1,p(x)(_Ω).

We know that J ∈C¹(W₀^1,p(x)(_Ω),R). If we denoteA= J⁰ :W₀^1,p(x)(_Ω)→(W₀^1,p(x)(_Ω))^∗, then hA(u),vi=

Z

Ω



|∇u|^p(x)−2+ |∇u|^2p(x)−2 q

1+|∇u|^2p(x)



(∇u,∇v)_RNdx, for all u,v∈W₀^1,p(x)(_Ω).

Proposition 2.3 ([11]). Set E=W₀^1,p(x)(_Ω), A is as above, then (1) A:E→E^∗ is a convex, bounded and strictly monotone operator;

(2) A:E→E^∗is a mapping of type(S)+, i.e., u_n *u in E andlim sup_n→∞hA(u_n),u_n−ui ≤0, implies u_n→u in E;

(3) A:E→E^∗ is a homeomorphism.

3 Variational setting and proof of the main results

For eachu∈ E, we define ϕ_λ(u) =

Z

Ω

1 p(x)

|∇u|^p(x)+ q

1+|∇u|^2p(x)

dx−λ Z

ΩF(x,u)dx. (3.1) Then we have the following lemma.

Lemma 3.1. If assumptions(h₁)–(h₂)hold, thenϕ∈C¹(E,R)and hϕ⁰_λ(u),vi=

Z

Ω



|∇u|^p(x)−2+ |∇u|^2p(x)−2 q

1+|∇u|^2p(x)



(∇u,∇v)_RNdx−λ Z

Ω f(x,u)vdx (3.2) for all u,v ∈E. Moreover,ψ⁰ :E→ E^∗ is weakly continuous, whereψ(u) =R

ΩF(x,u)dx.

Proof. To prove ϕ_λ ∈C¹(E,R)and (3.2), we only need to show thatψ∈C¹(E,R)and hψ⁰(u),vi=

Z

Ω f(x,u)vdx, ∀u,v∈E.

On the one hand, for anyu,v∈ Eand 0<|t|<1, by condition (h₂), we obtain

|f(x,u+tv)v| ≤C(1+|u+tv|^r(x)−1)|v|

≤C(|v|+2^r+−1|u|^r(x)−1|v|+2^r+−1|v|^r(x)). Note that 1< p(x)<r(x)< p^∗(x), the Hölder inequality implies that

|v|+₂^r+−1|u|^r(x)−1|v|+₂^r+−1|v|^r(x)∈ L¹(_Ω)_.

Consequently, by the mean value theorem and the Lebesgue dominated convergence theorem, there exists 0<λ<1 such that

hψ⁰(u),vi=lim

t→0

Z

Ω

F(x,u+tv)−F(x,u)

t dx

=_lim

t→0

Z

Ω f(x,u+λtv)vdx

=

Z

Ω f(x,u)vdx, for allu,v∈ E. Henceψis Gateaux differentiable.

It remains to prove that ψ⁰ is weakly continuous. Assume thatu_n * u in E. By Proposi- tion2.1, we conclude thatu_n→uinL^r(x)(_Ω)andu_n(x)→u(x)a.e.x ∈_Ω. Recalling

kψ⁰(un)−ψ⁰(u)k_E∗ = sup

kvk≤1

|hψ⁰(un)−ψ⁰(u),vi|

≤ sup

kvk≤1

Z

Ω|f(x,un)− f(x,u)||v|dx.

Set α := limn→+_∞sup_kvk≤1R

Ω|f(x,un)− f(x,u)||v|dx. We claim that α = 0. Suppose, by contradiction, thatα > 0. Hence, there exists a sequence {φ_n} ⊆ E and kφ_nk = 1 such that

R

Ω|f(x,u_n)− f(x,u)||φ_n|dx

> ^α₂ for enough largen. By(h₂), one has

|(f(x,u_n)−f(x,u))φ_n| ≤C(₁+|u_n|^r(x)−1)|φ_n|+C(₁+|u|^r(x)−1)|φ_n|

≤C(2|φ_n|+|u_n|^r(x)−1|φ_n|+|u|^r(x)−1|φ_n|).

Using again Hölder inequality, we get 2|φn|+|un|^r(x)−1|φn|+|u|^r(x)−1|φn| ∈ L¹(_Ω). In view of [14, Lemma A.1], there existw₁ ∈ L¹(_Ω)andξ₁,w₂∈ L^r(x)(_Ω)such that

max{|un(x)|,|u(x)|} ≤ |ξ₁(x)|and|φn(x)| ≤min{|w₁(x)|,|w₂(x)|}. Therefore, it follows from the Lebesgue dominated convergence theorem that

n→+lim_∞ Z

Ω|f(x,un)− f(x,u)||φn|dx=0,

which contradicts with α > _{0. Hence,} kψ⁰(u_n)−ψ⁰(u)k_E∗ → _{0 as} n → +∞. The proof is completed.

Definition 3.2. We say that ϕ_λ ∈ C¹(E,R)_satisfies (C)_c-condition if any sequence {u_n} ⊂ E satisfying

ϕ_λ(u_n)→c and kϕ⁰_λ(u_n)k_E^∗(1+ku_nk)→0 (3.3) contains a convergent subsequence.

Now, we present the following theorem which will play a crucial role in the proof of Main Theorems.

LetXbe a reflexive and separable Banach space. It is well known that there exist{e_n} ⊂X and{e^∗_n} ⊂X^∗ such that

(i) he^∗_n,e_mi=δ_n,m, whereδ_n,m =1 forn=mandδ_n,m =0 forn6=m;

(ii) X=_span{e_n :n∈ N}_andX^∗ =_span{e^∗_n :n∈ N}_.

LetX_i =_Rei, then X=⊕_i≥1X_i. Now, we define

Yn=⊕ⁿ_i=1X_i andZn=⊕_i≥nX_i. (3.4) Then we have the following Fountain Theorem.

Lemma 3.3 ([1,14]). Assume that I ∈ C¹(X,R)satisfies(C)_c-condition for all c> 0and I is even.

If for each sufficiently large n∈ N, there existρ_n> δ_n>0such that the following conditions hold:

(A₁) bn:=inf{I(u):u∈ Zn,kuk=δn} →+_∞as n →+_∞;

(A₂) a_n :=inf{I(u):u∈Y_n,kuk=ρ_n} ≤0.

Then the functional I has an unbounded sequence of critical values, i.e., there exists a sequence{un} ⊂ X such that I⁰(u_n) =0and I(u_n)→+_∞as n→+_∞.

Lemma 3.4. Assume that(h₂),(h₃)and(h₅)hold. Then any(C)_csequence is bounded.

Proof. Let {un} ⊂ E be a (C)_c sequence. To complete our goals, arguing by contradiction, suppose that ku_nk →_{∞, as}n→∞. Observe that fornlarge,

c+1≥ ϕ_λ(un)− ¹

p⁺hϕ⁰_λ(un),uni

=

Z

Ω

1 p(x)

|∇u_n|^p(x)+ q

1+|∇u_n|^2p(x)

dx−λ Z

ΩF(x,u_n)dx

− ¹ p⁺

Z

Ω



|∇u_n|^p(x)+ |∇u_n|^2p(x) q

1+|∇un|^2p(x)



dx+ ^λ p⁺

Z

Ωf(x,u_n)u_ndx

≥

Z

Ω

1 p(_x)− ¹

p⁺

|∇u_n|^p(x)+ q

1+|∇u_n|^2p(x)

dx+λ Z

ΩF(x,u_n)dx

≥λ Z

ΩF(x,u_n)dx.

(3.5)

Sincekunk>1 fornlarge, using (3.3) we have 0= lim

n→_∞

c+o(₁)

kunk^p− = lim

n→_∞

ϕ_λ(u_n) kunk^p−

≥ ¹ p⁺

R

Ω

|∇un|^p(x)+ q

1+|∇un|^2p(x)

dx kunk^p− −λ

Z

Ω

F(x,u_n) kunk^{p− dx}

≥ ² p⁺ −λ

Z

Ω

F(x,un) kunk^{p− dx,} which implies that

2

p⁺λ ≤lim sup

n→_∞ Z

Ω

|F(x,u_n)|

ku_nk^{p− dx. (3.6)}

For 0 ≤ α < β, letΩn(α,β) = {x ∈ _Ω : α≤ |u_n(x)| < β}. Let v_n = _k^un

unk, then kv_nk= 1 and

|v_n|_r(x)≤C₀kv_nk=C₀ for someC₀ >0. Going if necessary to a subsequence, we may assume that v_n*v inEand

v_n→v in L^s(x)(_Ω), 1≤s(x)< p^∗(x) and v_n(x)→v(x) a.e. onΩ. (3.7)

Now, we consider two possible cases: v=0 orv 6=0.

(1) If v = 0, then we have that v_n → 0 in L^s(x)(_Ω) and v_n(x) → 0 a.e. on Ω. Hence, it follows from(h2)that

Z

Ωn(0,r0)

|F(x,u_n)|

ku_nk^{p− dx}≤ ^C(r₀+r₀^r)_meas(_Ω)

ku_nk^p− →0 asn→+_∞, (3.8) wherer=r⁺ifr₀≥1, r=r⁻ifr₀<1.

Setσ⁰(x) = ^σ(x)

σ(x)−1. Sinceσ⁻ >max{1,_p^N−}one sees that 1 < p⁻σ⁰(x)< p^∗(x). So, v_n →0 inL^p−σ0(x)(_Ω)asn→+∞. Hence, we deduce from Proposition2.2,(h5), (3.5) and (3.7) that

Z

Ωn(r0,+_∞)

|F(x,u_n)|

|un|^p− |vn|^p−dx

≤₂

|F(x,u_n)|

|un|^p−

L^σ(x)(_Ωn(r0,+_∞))

|v_n|^p−

L^σ0(x)(_Ωn(r₀,+_∞))

≤_{2 max} Z

Ωn(r0,+_∞)

|F(x,u_n)|^σ(x)

|u_n|^{(p−)σ(x) dx 1}

σ+

, Z

Ωn(r0,+_∞)

|F(x,u_n)|^σ(x)

|u_n|^{(p−)σ(x) dx 1}

σ−

×max (_Z

Ωn(r0,+_∞)

|v_n|^p−σ0(x)dx ₍ ¹

σ0)−

, _Z

Ωn(r0,+_∞)

|v_n|^p−σ0(x)dx ₍ ¹

σ0)+

)

≤2 max (Z

Ωn(r0,+_∞)

F(x,un)dx ¹

σ+

, Z

Ωn(r0,+_∞)

F(x,un)dx ¹

σ−)

×max (_Z

Ωn(r0,+_∞)

|v_n|^p−σ0(x)dx ₍ ¹

σ0)−

, _Z

Ωn(r0,+_∞)

|v_n|^p−σ0(x)dx ₍ ¹

σ0)+

)

≤_{2 max} (

c₀ λ

(c+₁) ¹

σ+

, c₀

λ

(c+₁) ¹

σ−)

×max (_Z

Ωn(r0,+_∞)

|vn|^p−σ0(x)dx ₍ ¹

σ0)−

, _Z

Ωn(r0,+_∞)

|vn|^p−σ0(x)dx ₍ ¹

σ0)+

)

→0, asn→_∞.

(3.9)

Combining (3.8) with (3.9), we get Z

Ω

|F(x,un)|

ku_nk^{p− dx} =

Z

Ωn(0,r0)

|F(x,un)|

ku_nk^{p− dx}+

Z

Ωn(r0,+_∞)

|F(x,un)|

ku_nk^{p− dx}

=

Z

Ωn(0,r0)

|F(x,u_n)|

ku_nk^{p− dx}+

Z

Ωn(r0,+_∞)

|F(x,u_n)|

|u_n|^p− |v_n|^p−dx

→0, asn→_∞,

(3.10)

which contradicts (3.6).

(2) Ifv 6=0, set Ω6= := {x ∈_Ω:v(x)6= 0}, then meas(_Ω6=)> 0. For a.e. x∈ _Ω6=, we have

nlim→_∞|u_n(x)| = +_∞. Hence, Ω6= ⊂ _Ωn(r₀,∞) for large n ∈ N. As the proof of (3.8), we also obtain that

Z

Ωn(0,r₀)

|F(x,u_n)|

kunk^{p+ dx}≤ ^C(r₀+r^r₀)meas(_Ω)

kunk^p+ →0, asn→+_∞. (3.11)

It follows from(h₂),(h₃), (3.11) and Fatou’s Lemma that 0= lim

n→_∞

c+o(1)

ku_nk^p+ = lim

n→_∞

ϕ(un) ku_nk^p+

≤ lim

n→_∞







 1 p⁻

R

Ω

|∇un|^p(x)+ q

1+|∇un|^2p(x)

dx ku_nk^p+ −λ

Z

Ω

F(x,un) ku_nk^{p+ dx}









= lim

n→_∞







 1 p⁻

R

Ω

|∇un|^p(x)+ q

1+|∇un|^2p(x)

dx kunk^p+

−λ Z

Ωn(0,r₀)

F(x,u_n) ku_nk^{p+ dx}−λ

Z

Ωn(r₀,+_∞)

F(x,u_n) ku_nk^{p+ dx}









= lim

n→_∞







 1 p⁻

R

Ω

|∇u_n|^p(x)+ q

1+|∇u_n|^2p(x)

dx ku_nk^p+ −λ

Z

Ωn(r0,+_∞)

F(x,u_n) ku_nk^{p+ dx}









≤ lim

n→_∞



 1 p⁻

R

Ω

|∇u_n|^p(x)+1+|∇u_n|^p(x)dx ku_nk^p+ −λ

Z

Ωn(r0,+_∞)

F(x,u_n) ku_nk^{p+ dx}





= lim

n→_∞

"

2 p⁻

R

Ω|∇un|^p(x)dx ku_nk^p+ −λ

Z

Ωn(r0,+_∞)

F(x,u_n) ku_nk^{p+ dx}

#

≤ lim sup

n→_∞

2 p⁻ −λ

Z

Ωn(r₀,+_∞)

F(x,u_n) ku_nk^{p+ dx}

= ²

p⁻ −lim inf

n→_∞ λ Z

Ωn(r0,+_∞)

F(x,u_n)

|un|^p+ |vn|^p+dx

= ²

p⁻ −lim inf

n→_∞ λ Z

Ω

F(x,un)

|un|^{p+ χΩn(r0,+∞)}(x)|vn|^p+dx

≤ ² p⁻ −λ

Z

Ωlim inf

n→_∞

F(x,un)

|u_n|^{p+ χΩn(r0,+∞)}(x)|v_n|^p+dx

→ −_∞, asn→_∞,

(3.12)

which is a contradiction. Thus{un}is bounded inE. The proof is accomplished.

Lemma 3.5. Suppose that (h₂), (h₃)and(h₅) hold. Then any(C)_c-sequence of ϕhas a convergent subsequence in E.

Proof. Let {u_n} ⊂ E be a (C)_c sequence. In view of the Lemma 3.4, the sequence {u_n} is bounded in E. Then, up to a subsequence we haveu_n*uinE. By Proposition2.2, it follows that

un→uinL^r(x)(_Ω),

{u_n}is bounded inL^r(x)(_Ω).

It is easy to compute directly that Z

Ω|f(x,u_n)− f(x,u)||u_n−u|dx

≤

Z

Ω(|f(x,un)|+|f(x,u)|)|un−u|dx

≤

Z

Ω[C(1+|u_n|^r(x)−1) +C(1+|u|^r(x)−1)]|u_n−u|dx

≤2C Z

Ω|un−u|dx+C Z

Ω|un|^r(x)−1|un−u|dx +

Z

Ω|u|^r(x)−1|u_n−u|dx

≤2C|u_n−u|₁+2C

|u_n|^r(x)−1

r⁰(x)|u_n−u|_r(x) +2C

|u|^r(x)−1

r⁰(x)|u_n−u|_r(x)

≤2C|u_n−u|₁+2Cmaxn

|u_n|^r+−1

r(x) ,|u_n|^r−−1

r(x)

o|u_n−u|_r(x) +2Cmaxn

|u|^r_r⁺_(x⁻₎¹,|u|^r_r⁻_(x⁻₎^1o|u_n−u|_r(x)

→0, asn→_∞,

(3.13)

where _r(¹_x)+ ¹

r⁰(x) =1. Noting that

hA(un)−A(u),un−ui=hϕ⁰(un)−ϕ⁰(u),un−ui +

Z

Ω(f(x,un)− f(x,u))(un−u)dx. (3.14) Moreover, by (3.3), one infers

nlim→_∞hϕ⁰(u_n)−ϕ⁰(u),u_n−ui=0. (3.15) Finally, the combination of (3.13)–(3.15) implies

nlim→_∞hA(u_n)−A(u)_,u_n−ui=_{0. (3.16)} SinceAis of type(S)₊ by Lemma2.3, we obtainu_n→uin E. The proof is complete.

Lemma 3.6. Suppose that (h2), (h3)and(h6)hold. Then any (C)_c-sequence of ϕhas a convergent subsequence in E.

Proof. Similar to the proof of Lemma3.5, we only prove that{u_n}is bounded in E. Suppose by contradiction that kunk → _∞ as n → _{∞. Let} vn = _k^u_uⁿ

nk, then kvnk = 1 and |vn|_r(x) ≤ C₀kv_nk = C₀ for some C₀ > 0. Going if necessary to a subsequence, we may assume that v_n *vinE,

v_n→v inL^r(x)(_Ω), 1≤r(x)< p^∗(x) and v_n(x)→ v(x) a.e. onΩ. (3.17) By (3.1), (3.2) and(h₆)_{, one has}

c+1≥ ϕ_λ(un)− ¹

µhϕ⁰_λ(un),uni

=

Z

Ω

1 p(x)

|∇u_n|^p(x)+ q

1+|∇u_n|^2p(x)

dx−λ Z

ΩF(x,u_n)dx

− ¹ µ

Z

Ω



|∇u_n|^p(x)+ |∇u_n|^2p(x) q

1+|∇un|^2p(x)



dx+ ^λ µ Z

Ω f(x,u_n)u_ndx

≥ 1

p⁺ − ¹ µ

_Z

Ω

|∇u_n|^p(x)+ q

1+|∇u_n|^2p(x)

dx− ^λθ µ

Z

Ω|u_n|^p−dx

≥ ^µ−p⁺

p⁺µ ku_nk^p−− ^λθ µ |u_n|^p_p⁻−, forn∈ N, which implies

1≤ ^λθp

+

µ−p⁺lim sup

n→_∞

|vn|^p_p⁻−. (3.18)

In view of (3.17),v_n → v in L^p−(_Ω). Hence, we deduce from (3.18) that v 6= 0. By a similar reasoning as in the proof of Lemma3.4 step (2), we can conclude a contradiction. Thus,{u_n} is bounded inE. The rest proof is the same as that in Lemma3.5.

Proof of Theorem1.2. Let X = E, Yn and Zn be defined by (3.4). Obviously, ϕ_λ(u) = ϕ_λ(−u) by(h₄), and Lemma3.5 implies that ϕ_λ satisfies the (C)_c condition for any λ> 0. Hence, to prove Theorem1.2, it remains to verify the conditions(A₁)and(A₂)in Lemma3.3.

Verification of(A₁). Setβ_n:=sup_u∈Zn,kuk=1|u|_r(x), where p⁺<r⁻≤r(x)< p^∗(x)andn∈ N. We claim thatβ_n→ _{0 as}n→ ∞. Indeed, it is obvious that β_n ≥ β_n+1 ≥ _{0. so}β_n → β≥₀ asn →∞. For eachn=1, 2, . . . , takingun∈ Zn, kunk=1 such that 0≤βn− |un|_r(x)≤ _n¹. As Eis reflexive,{u_n}has a weakly convergent subsequence, without loss of generality, suppose u_n * uin E. By definition of Z_n, one knows thatu = 0. Proposition2.3implies that u_n →₀ in L^r(x)(_Ω). Thus we have proved thatβ=0.

By the above definition ofβn, foru∈Zn withkuk>1, we have

|u|_r(x)≤ βnkuk. (3.19)

Moreover, we consider the real functionk:R→R, defined by k(t) = ¹

p⁺t^p−−λCβ^r_n⁻t^r+. Choosing δn= (2Cr⁺β^r

− n )

1 p− −r+

forn∈ N, it is clear that k(δ_n) = ¹

p⁺δ^p

−

n −λCβ^r_n⁻δ^r

+ n

= (2Cr⁺β^r

− n )

p− p− −r+

1 p⁺ − ^λ

2r⁺

.

(3.20)

Therefore, sincer⁻ > p⁺,λ< ^2r_p+⁺ andβ_n→0 asn→+∞, we obtain that

δ_n→+_∞, k(δ_n)→+_∞, asn→+_∞. (3.21) It follows from(h₂)that

F(x,t)≤C(|t|+|t|^r(x))≤2C(1+|t|^r(x))

for all(x,t)∈_Ω×R. Then, for anyu∈ Z_n, assume thatkuk=δ_n. It follows from(h₂)_{, (3.19),}