On a Class of Degenerate Nonlocal Problems with Sign-Changing Nonlinearities

MALAYSIANMATHEMATICAL

SCIENCESSOCIETY http://math.usm.my/bulletin

On a Class of Degenerate Nonlocal Problems with Sign-Changing Nonlinearities

1NGUYENTHANHCHUNG AND2HOANGQUOCTOAN

1Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam

2Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

1[email protected],²hq [email protected]

Abstract. Using variational techniques, we study the nonexistence and multiplicity of so- lutions for the degenerate nonlocal problem

( −M(^R_Ω|x|^−ap|∇u|^pdx)div |x|^−ap|∇u|^p−2∇u

=λ|x|^−p(a+1)+cf(x,u) inΩ,

u =0 on∂Ω,

whereΩ⊂R^N(N≥3) is a smooth bounded domain, 0∈Ω, 0≤a<^N−p_p , 1<p<N,c>0, M:R⁺→R⁺is a continuous function that may be degenerate at zero,f:Ω×R→Ris a sign-changing Carath´eodory function andλis a parameter.

2010 Mathematics Subject Classification: 35D35, 35J35, 35J40, 35J62

Keywords and phrases: Degenerate nonlocal problems, nonexistence, multiplicity, varia- tional methods.

1. Introduction and preliminaries

In this paper, we are concerned with the problem (1.1)

( −M ^R_Ω|x|^−ap|∇u|^pdx

div |x|^−ap|∇u|^p−2∇u

=λ|x|^−p(a+1)+cf(x,u) inΩ,

u =0 in∂Ω,

whereΩ⊂R^N(N≥3) is a smooth bounded domain, 0∈Ω, 0≤a<^N−p_p , 1<p<N,c>0, M:R⁺→R⁺is a continuous function, f :Ω×R→Ris a sign-changing Carath´eodory function, andλ is a parameter. It should be noticed that ifa=0 andc=pthen problem (1.1) becomes

(1.2)

( −M(^R_Ω|∇u|^pdx)∆_pu =λf(x,u) inΩ,

u =0 on∂Ω.

Since the first equation in (1.2) contains an integral overΩ, it is no longer a pointwise identity; therefore it is often called nonlocal problem. This problem models several physical

Communicated byNorhashidah Mohd. Ali.

Received:April 13, 2012;Revised:September 13, 2012.

and biological systems, whereudescribes a process which depends on the average of itself, such as the population density, see [5]. Moreover, problem (1.2) is related to the stationary version of the Kirchhoff equation

(1.3) ρ∂²u

∂t² − P0

h + E 2L

Z _L 0

∂u

∂x

²dx∂²u

∂x² =0

presented by Kirchhoff in 1883, see [11]. This equation is an extension of the classical d’Alembert’s wave equation by considering the effects of the changes in the length of the string during the vibrations. The parameters in (1.3) have the following meanings: Lis the length of the string, his the area of the cross-section,E is the Young modulus of the material,ρ is the mass density, andP0is the initial tension.

In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to [2, 7, 8, 10, 12, 13, 16, 17, 19], in which the authors have used different methods to get the existence of solutions for (1.2). In [15, 21], Z. Zhanget al. studied the existence of nontrivial solutions and sign-changing solutions for (1.2). One of the important hypotheses in these papers is that the Kirchhoff functionMis non-degenerate, i.e.,

(1.4) M(t)≥m₀>0 for allt∈R⁺.

Motivated by the ideas introduced in [6, 9, 14, 20], the goal of this paper is to study the existence of solutions for problem (1.1) without condition (1.4). More exactly, we con- sider problem (1.1) in the case when f is a sign-changing Carath´eodory function and the Kirchhoff functionMis allowed to take the value 0 at 0. Using the minimum principle com- bined with the mountain pass theorem, we show that problem (1.1) has at least two distinct, non-negative nontrivial weak solutions for λ large enough. We also prove that (1.1) has no nontrivial solution ifλis small enough. Our results supplement the previous ones in the non-degenerate case. Moreover, we consider problem (1.1) in the general case 0≤a<^N−p_p , 1<p<N,c>0. To our best knowledge, the present paper is the first contribution related to a Kirchhoff equation in this direction.

In order to state the main results, let us introduce the following conditions:

(M₀) M:R⁺→R⁺is a continuous function and satisfies M(t)≥m0t^α−1for allt∈R⁺, wherem₀>0 and 1<α<minn

N

N−p,^N−p(a+1)+c_N−p(a+1) o

; (F₁) f:Ω×[0,+∞)→Ris a Carath´eodory function, such that

|f(x,t)| ≤Ct^αp−1for allt∈[0,+∞)andx∈Ω, whereαis given in(M₀);

(F₂) There existt₀,t₁>0 such thatF(x,t)≤0 for all 0≤t≤t₀andF(x,t₁)>0 for all x∈Ω, whereF(x,t) =^R₀^tf(x,s)ds;

(F₃) It holds that

lim sup

t→∞

F(x,t)

t^αp ≤0 uniformly inx∈Ω.

We point out that ifa=0, c=p andM(t)≡1, problem (1.1) has been studied by K.

Perera [14]. We emphasize that the main difference between the local case (M≡1) and the present paper (M 6≡1) is that the operator appears in problem (1.1) is not homogeneous.

Moreover, from the physical point of view, nonlocal coefficientM(^R_Ω|x|^−ap|∇u|^pdx)of the

divergence term in (1.1) is a function (may be degenerate at zero) depending on the average of the kinetic energy. It should be noticed that since 0≤a<^N−p_p , 1<p<N,c>0, our results are better than those in [14] even in the caseM≡1. Finally, with the same arguments used in this work, we can deal with the caseα=1. Thus, our paper is a natural extension from [14] and recent results onp-Kirchhoff type problems.

We start by recalling some useful results in [3, 4, 20]. We have known that for allu∈ C₀^∞(R^N), there exists a constantC_a,b>0 such that

(1.5)

Z

R^N

|x|^−bq|u|^qdx _q^p

≤C_a,b Z

R^N

|x|^−ap|∇u|^pdx, where−∞<a<^N−p_p ,a≤b≤a+1,q=p^∗(a,b) =_{N−d p}^{N p} ,d=1+a−b.

LetW₀^1,p(Ω,|x|^−ap)be the completion ofC₀^∞(Ω)with respect to the norm kuk_a,p=

Z

Ω

|x|^−ap|∇u|^pdx ¹_p

.

ThenW₀^1,p(Ω,|x|^−ap)is a reflexive Banach space. From the boundedness ofΩand the stan- dard approximation argument, it is easy to see that (1.5) holds for anyu∈W₀^1,p(Ω,|x|^−ap) in the sense that

(1.6)

_Z

R^N

|x|^−α|u|^rdx ^p_r

≤C_a,b Z

R^N

|x|^−ap|∇u|^pdx, for 1≤r≤p^∗=_N−p^{N p} ,α≤(1+a)r+N

1−^r_p

, that is, the embeddingW₀^1,p(Ω,|x|^−ap),→ L^r(Ω,|x|^−α)is continuous, whereL^r(Ω,|x|^−α)is the weightedL^r(Ω)space with the norm

|u|_r,α:=|u|_Lr(Ω,|x|^−α)= Z

Ω

|x|^−α|u|^rdx ¹_r

.

In fact, we have the following compact embedding result which is an extension of the clas- sical Rellich-Kondrachov compactness theorem (see [20]).

Lemma 1.1 (Compact embedding theorem). Suppose thatΩ⊂R^N is an open bounded domain with C¹boundary and that0∈Ω, and1<p<N,−∞<a<^N−p_p ,1≤r<_N^{N p}_−p and α <(1+a)r+N

1−^r_p

. Then the embedding W₀^1,p(Ω,|x|^−ap),→L^r(Ω,|x|^−α)is compact.

From Lemma 1.1, B. Xuan proved in [20] that the first eigenvalue λ1 of the singular quasilinear equation

( −div |x|^−ap|∇u|^p−2∇u

=λ|x|^−p(a+1)+c|u|^p−2u inΩ,

u =0 in∂Ω,

is isolated, unique (up to a multiplicative constant), that is, the first eigenvalue is simple and it is given by

λ1= inf

u∈W^1,p

0 (Ω,|x|^−ap)\{0}

R

Ω|x|^−ap|∇u|^pdx R

Ω|x|^−p(a+1)+c|u|^pdx>0.

This is a natural extension from the previous results on the casea=0 andc=prelying esstentially on the Caffarelli-Kohn-Nirenberg inequalities.

Definition 1.1. We say that u∈X=W₀^1,p(Ω,|x|^−ap)is a weak solution of problem (1.1) if for allϕ∈X , it holds that

M Z

Ω

|x|^−ap|∇u|^pdx Z

Ω

|x|^−ap|∇u|^p−2∇u·∇ϕdx−λ Z

Ω

|x|^−p(a+1)+cf(x,u)ϕdx=0.

Our main results of this paper can be described as follows.

Theorem 1.1. Assume that the conditions(M₀)and(F₁)hold. Then there exists a positive constantλ∗such that for anyλ<λ∗, problem (1.1) has no nontrivial weak solution.

Theorem 1.2. Assume that the conditions(M₀)and(F₁)-(F₃)hold. Then there exists a positive constantλ^∗such that for anyλ ≥λ^∗, problem (1.1) has at least two distinct non- negative, nontrivial weak solutions.

2. Proof of the main results

For simplicity, we denoteX=W₀^1,p(Ω,|x|^−ap). In the following, when there is no misun- derstanding, we always useC_ito denote positive constants.

Proof of Theorem 1.1. First, since 1<α <minn

N

N−p,^N−p(a+1)+c_N−p(a+1) o

, the embeddingX ,→ L^αp(Ω,|x|^−p(a+1)+c)is compact, see Lemma 1.1. Then there existsC₁>0 such that

C₁kuk_Lαp(Ω,|x|^−p(a+1)+c)≤ kuk_a,pfor allu∈X or

C₁^αp Z

Ω

|x|^−p(a+1)+c|u|^αpdx≤ _Z

Ω

|x|^−ap|∇u|^pdx α

for allu∈X.

It follows that the number

(2.1) λα:= inf

u∈X\{0}

(^R_Ω|x|^−ap|∇u|^pdx)^α R

Ω|x|^−p(a+1)+c|u|^αpdx>0.

Ifu∈X is a nontrivial weak solution, then multiplying (1.1) byu, integrating by parts and using(M₀),(F₁)gives

m₀ Z

Ω

|x|^−ap|∇u|^pdx α

≤M Z

Ω

|x|^−ap|∇u|^pdx Z

Ω

|x|^−ap|∇u|^pdx

=λ Z

Ω

|x|^−p(a+1)+cf(x,u)udx

≤Cλ Z

Ω

|x|^−p(a+1)+c|u|^αpdx.

(2.2)

From (2.2), choosing λ∗=^λαm0

C , where λα is given by (2.1), we conclude the proof of Theorem 1.1.

We will prove Theorem 1.2 using critical point theory. Set f(x,t) =0 fort<0. For all λ∈R, we consider the functionalT_λ:X→Rgiven by

T_λ(ω) =1 pMb

Z

Ω

|x|^−ap|∇u|^pdx

−λ Z

Ω

|x|^−p(a+1)+cF(x,u)dx (2.3)

=J(u)−λI(u),

where

J(u) =1 pMb

_Z

Ω

|x|^−ap|∇u|^pdx

, (2.4)

I(u) = Z

Ω

|x|^−p(a+1)+cF(x,u)dx, u∈X.

By Lemma 1.1 and the condition(F₁), a simple computation implies thatT_λ is well-defined and ofC¹class inX. Thus, weak solutions of problem (1.1) correspond to the critical points of the functionalT_λ.

Lemma 2.1. The functional T_λ given by (2.3) is weakly lower semicontinuous X .

Proof. Let{u_m}be a sequence that converges weakly touinX. Then, by the continuity of norm, we have

lim inf

m→∞

Z

Ω

|x|^−ap|∇u_m|^pdx≥ Z

Ω

|x|^−ap|∇u|^pdx.

Combining this with the continuity and monotonicity of the function ψ:R⁺→R, t7→

ψ(t) =¹

pM(t), we getb lim inf

m→∞ J(u_m) =lim inf

m→∞

1 pMb

Z

Ω

|x|^−ap|∇u_m|^pdx

=lim inf

m→∞ ψ Z

Ω

|x|^−ap|∇u_m|^pdx

≥ψ

lim inf

m→∞

Z

Ω

|x|^−ap|∇u_m|^pdx

≥ψ _Z

Ω

|x|^−ap|∇u|^pdx

= 1 pMb

_Z

Ω

|x|^−ap|∇u|^pdx

=J(u).

(2.5)

We shall show that

(2.6) lim

m→∞

Z

Ω

F(x,u_m)dx= Z

Ω

F(x,u)dx.

Using(F₁)and H¨older’s inequality, it follows that

Z

Ω

|x|^−p(a+1)+c[F(x,u_m)−F(x,u)]dx

≤ Z

Ω

|x|^−p(a+1)+c|f(x,u+θ_m(u_m−u))||u_m−u|dx

≤C Z

Ω

|x|^−p(a+1)+c|u+θm(u_m−u)|^αp−1|u_m−u|dx

≤Cku+θ_m(u_m−u)k^αp−1

L^αp(Ω,|x|^−p(a+1)+c)ku_m−uk_Lαp(Ω,|x|^−p(a+1)+c), (2.7)

where 0≤θm(x)≤1 for allx∈Ω.

On the other hand, since 1<α<minn

N

N−p,^N−p(a+1)+c_N−p(a+1) o

,X,→L^αp(Ω,|x|^−p(a+1)+c)is compact, the sequence{u_m}converges strongly touin the spaceL^αp(Ω,|x|^−p(a+1)+c). It is easy to see that the sequence{ku+θ_m(u_m−u)k_Lαp(Ω,|x|^−p(a+1)+c)}is bounded. Thus, it follows from (2.7) that relation (2.6) holds true. The proof of Lemma 2.1 is proved.

Lemma 2.2. The functional T_λ is coercive and bounded from below.

Proof. By the conditions(F₁)and(F₃), there existsC_λ >0 such that for allt∈Rand a.e.

x∈Ω, one has

(2.8) λF(x,t)≤m₀λα

2αp |t|^αp+C_λ, whereλα is given by (2.1). Hence, using(M₀)and the fact that

0<

Z

Ω

|x|^−p(a+1)+cdx<∞ we get

T_λ(u)≥m0

αp _Z

Ω

|x|^−ap|∇u|^pdx α

−λ Z

Ω

|x|^−p(a+1)+cF(x,u)dx

≥m₀ αp

_Z

Ω

|x|^−ap|∇u|^pdx α

− Z

Ω

|x|^−p(a+1)+c m₀λα

2αp |u|^αp+C_λ

dx

≥ m0

2αpkuk^α_a,p^p−C_λ, (2.9)

whereC_λ>0 is a constant. So,T_λ is coercive and bounded from below.

Lemma 2.3. If u∈X is a weak solution of problem (1.1) then u≥0inΩ.

Proof. Indeed, ifu∈X is a weak solution of problem (1.1), then we have 0=hT_λ⁰(u),u⁻i

=M _Z

Ω

|x|^−ap|∇u|^pdx _Z

Ω

|x|^−ap|∇u|^p−2∇u·∇u⁻dx−λ Z

Ω

|x|^−p(a+1)+cf(x,u)u⁻dx

≥m_{0 Z}

Ω

|x|^−ap|∇u⁻|^pdx α

,

whereu⁻=min{u(x),0}is the negative part ofu. It follows thatu≥0 inΩ.

By Lemmas 2.1-2.3, applying the minimum principle (see [18, p. 4, Theorem 1.2]), the functionalT_λ has a global minimum and thus problem (1.1) admits a non-negative weak solutionu₁∈X. The following lemma shows that the solutionu₁is not trivial provided that λ is large enough.

Lemma 2.4. There existsλ^∗>0such that for allλ ≥λ^∗,infu∈XT_λ(u)<0and hence the solution u₁6≡0.

Proof. Indeed, letΩ⁰be a sufficiently large compact subset ofΩand a functionu₀∈C₀^∞(Ω), such thatu0(x) =t0onΩ⁰, 0≤u0(x)≤t0onΩ\Ω⁰, wheret0is as in(F₂). Then we have

Z

Ω

|x|^−p(a+1)+cF(x,u₀)dx= Z

Ω⁰

|x|^−p(a+1)+cF(x,u₀)dx+ Z

Ω\Ω⁰

|x|^−p(a+1)+cF(x,u₀)dx

≥ Z

Ω⁰

|x|^−p(a+1)+cF(x,t₀)dx−C Z

Ω\Ω⁰

|x|^−p(a+1)+c|u₀|^pdx

≥ Z

Ω⁰

|x|^−p(a+1)+cF(x,t0)dx−Ct₀^p Z

Ω\Ω⁰

|x|^−p(a+1)+cdx>0,

provided that|Ω\Ω⁰|>0 is small enough. So, we deduce that T_λ(u₀) =1

pMb Z

Ω

|x|^−ap|∇u₀|^pdx

−λ Z

Ω

|x|^−p(a+1)+cF(x,u0)dx

≤ 1 pMb

Z

Ω

|x|^−ap|∇u₀|^pdx

−λ Z

Ω⁰

|x|^−p(a+1)+cF(x,t₀)dx−Ct₀^p Z

Ω\Ω⁰

|x|^−p(a+1)+cdx

. Hence, ifΩ⁰ is large enough, there existsλ^∗such that for allλ ≥λ^∗we haveT_λ(u₀)<0 and thusu16≡0. Moreover,T_λ(u₁)<0 for allλ ≥λ^∗.

Our idea is to obtain the second weak solutionu2∈X by applying the mountain pass theorem in [1]. To this purpose, we first show that for allλ≥λ^∗, the functionalT_λ has the geometry of the mountain pass theorem.

Lemma 2.5. There exist a constantρ∈(0,ku₁k_a,p)and a constant r>0such that T_λ(u)≥r for all u∈X withkuk_a,p=ρ.

Proof. For eachu∈X, we set

(2.10) Ωu:={x∈Ω: u(x)>t₀},

wheret₀is given by(F₂). Then, we haveF(x,u(x))≤0 onΩ\Ω_u, so T_λ(u)≥ m0

αp ^Z

Ω

|x|^−ap|∇u|^pdx α

− Z

Ωu

F(x,u)dx

= m0

αpkuk^αp_a,p− Z

Ωu

F(x,u)dx.

(2.11)

Using the H¨older inequality and Lemma 1.1, we get Z

Ω_u

|x|^−p(a+1)+cF(x,u)dx≤C Z

Ω_u

|x|^−p(a+1)+c|u|^αpdx

≤C Z

Ωu

|x|^−p(a+1)+c|u|^qdx ^αp_q Z

Ωu

|x|^−p(a+1)+cdx 1−^αp_q

≤C₂kuk^α_a,p^p _Z

Ωu

|x|^−p(a+1)+cdx 1−^αp_q

, (2.12)

whereαp<q<min n N p

N−p,p(N−p(a+1)+c) N−p(a+1)

o . From (2.11) and (2.12), it implies that (2.13) T_λ(u)≥ kuk^α_a,p^p

"

m0

αp−C₂ Z

Ωu

|x|^−p(a+1)+cdx 1−^αp_q #

.

From (2.13), in order to prove Lemma 2.5, it is enough to show that Z

Ω_u

|x|^−p(a+1)+cdx→0 askuk_a,p→0.

Givenε>0, take a compact subsetΩε ofΩsuch that Z

Ω\Ω_ε

|x|^−p(a+1)+cdx<ε

and letΩu,ε=Ω_u∩Ω_ε. Then (2.14)

Z

Ω

|x|^−ap|∇u|^p≥C₃ Z

Ωu,ε

|x|^−p(a+1)+c|u|^pdx≥C₃t₀^p Z

Ωu,ε

|x|^−p(a+1)+cdx, so

Z

Ωu,ε

|x|^−p(a+1)+cdx→0 askuk_a,p→0.

But sinceΩ_u⊂Ωu,ε∪(Ω\Ωε), we have Z

Ωu

|x|^−p(a+1)+cdx<

Z

Ωu,ε

|x|^−p(a+1)+cdx+ε, andεis arbitrary. This shows that

Z

Ωu

|x|^−p(a+1)+cdx→0 askuk_a,p→0 and thus, Lemma 2.5 is proved.

Lemma 2.6. The functional T_λ satisfies the Palais-Smale condition in X .

Proof. By Lemma 2.2, we deduce thatT_λ is coercive onX. Let{u_m}be a sequence such that

(2.15) T_λ(u_m)→c<∞, T_λ⁰(u_m)→0 inX^∗asm→∞, whereX^∗is the dual space ofX.

SinceT_λ is coercive onX, relation (2.15) implies that the sequence{u_m}is bounded in X. SinceXis reflexive, there existsu∈Xsuch that, passing to a subsequence, still denoted by{u_m}, it converges weakly touinX. Hence,{ku_m−uk}is bounded. This and (2.15) imply thatT⁰

λ(u_m)(u_m−u)converges to 0 asm→∞. Using the condition(F₁)combined with H¨older’s inequality, we conclude that

Z

Ω

|x|^−p(a+1)+c|f(x,u_m)||u_m−u|dx≤C Z

Ω

|x|^−p(a+1)+c|u_m|^αp|u_m−u|dx

≤C₄ku_mk^αp

L^αp(Ω,|x|^−p(a+1)+c)ku_m−uk_Lαp(Ω,|x|^−p(a+1)+c), which shows that

(2.16) lim

m→∞

I⁰(u_m),u_m−u

=0.

Combining this with (2.15) and the fact that J⁰(u_m),u_m−u

=

T_λ⁰(u_m),u_m−u +λ

I⁰(u_m),u_m−u imply that

(2.17) lim

m→∞M Z

Ω

|x|^−ap|∇u_m|^pdx Z

Ω

|x|^−ap|∇u_m|^p−2∇um·(∇u_m−∇u)dx=0.

Since{u_m}is bounded inX, passing to a subsequence, if necessary, we may assume that Z

Ω

|x|^−ap|∇u_m|^pdx→t₀≥0 asm→∞.

Ift₀=0 then{u_m}converges strongly tou=0 inX and the proof is finished. Ift₀>0 then by(M₀)and the continuity ofM, we get

M _Z

Ω

|x|^−ap|∇u_m|^pdx

→M(t₀)>0asm→∞.

Thus, formsufficiently large, we have

(2.18) 0<C₅≤M

Z

Ω

|x|^−ap|∇u_m|^pdx

≤C₆. From (2.17) and (2.18), we have

(2.19) lim

m→∞

Z

Ω

|x|^−ap|∇u_m|^p−2∇u_m·(∇u_m−∇u)dx=0.

On the other hand, since{u_m}converges weakly touinX, we have

(2.20) lim

m→∞

Z

Ω

|x|^−ap|∇u|^p−2∇u·(∇u_m−∇u)dx=0.

By (2.19) and (2.20),

m→∞lim Z

Ω

|x|^−ap |∇u_m|^p−2∇u_m− |∇u|^p−2∇u

·(∇u_m−∇u)dx=0.

or

(2.21) lim

m→∞

Z

Ω

|∇v_m|^p−2∇v_m− |∇v|^p−2∇v

·(∇v_m−∇v)dx=0, where∇v_m=|x|^−a∇u_m,∇v=|x|^−a∇u.

We recall that the following inequalities hold |ξ|^p−2ξ− |η|^p−2η,ξ−η

≥C₇(|ξ|+|η|)^p−2|ξ−η|²if 1<p<2, (2.22)

|ξ|^p−2ξ− |η|^p−2η,ξ−η

≥C₈|ξ−η|^pifp≥2, for allξ,η∈R^N, whereh., .idenotes the usual product inR^N.

If 1<p<2, using the H¨older inequality, by (2.21), (2.22) we have 0≤ ku_m−uk_a,p^p =k|∇v_m−∇v|k_L^pp(Ω)

≤ Z

Ω

|∇v_m−∇v|^p(|∇v_m|+|∇v|)^p(p−2)2 (|∇v_m|+|∇v|)^p(2−p)2 dx

≤ _Z

Ω

|∇v_m−∇v|²(|∇v_m|+|∇v|)^p−2dx ^p_2Z

Ω

(|∇v_m|+|∇v|)^pdx ^2−p₂

≤C_{9 Z}

Ω

|∇v_m|^p−2∇v_m− |∇v|^p−2∇v,∇v_m−∇v dx

^p₂

× _Z

Ω

(|∇v_m|+|∇v|)^pdx ^2−p₂

≤C10

Z

Ω

|∇v_m|^p−2∇v_m− |∇v|^p−2∇v,∇v_m−∇v dx

^p₂ , which converges to 0 asm→∞. Ifp≥2, one has

0≤ ku_m−uk_a,p^p =k|∇v_m−∇v|k_L^p_p(Ω)

≤C₁₁ Z

Ω

|∇v_m|^p−2∇v_m− |∇v|^p−2∇v,∇v_m−∇v dx,

which converges to 0 asm→∞. So we conclude that{u_m}converges strongly touinXand the functionalT_λsatisfies the Palais-Smale condition.

Proof of Theorem 1.2. By Lemmas 2.1-2.4, problem (1.1) admits a non-negative, nontrivial weak solutionu1as the global minimizer ofT_λ. Setting

(2.23) c:=inf

χ∈Γ max

u∈χ([0,1])T_λ(u), whereΓ:={χ∈C([0,1],X):χ(0) =0,χ(1) =u₁}.

Lemmas 2.5, 2.6 show that all assumptions of the mountain pass theorem in [1] are satisfied,T_λ(u₁)<0 andku₁ka,p>ρ. Then,cis a critical value ofT_λ, i.e. there exists u2∈X such thatT_λ⁰(u₂)(ϕ) =0 for allϕ∈X oru2is a weak solution of (1.1). Moreover, u₂ is not trivial andu₂6≡u₁sinceT_λ(u₂) =c>0>T_λ(u₁). Theorem 1.2 is completely proved.

Acknowledgement. The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript. This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED).

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