2 The main result and proof of the theorem

On an eigenvalue problem with variable exponents and sign-changing potential

Bin Ge

Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, P. R. China

Received 23 September 2015, appeared 27 December 2015 Communicated by Dimitri Mugnai

Abstract. In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. We prove that any λ > 0 sufficiently small is an eigenvalue of the nonhomogeneous eigenvalue problem

(−div(a(|∇u|)∇u) =λV(x)|u|^q(x)−2u, inΩ,

u=0, on∂Ω.

The proofs of the main results are based on Ekeland’s variational principle.

Keywords:nonhomogeneous differential operator, eigenvalue problem, Orlicz–Sobolev space, indefinite potential.

2010 Mathematics Subject Classification: 46N20, 47J10, 47J30, 35J60, 35J70.

1 Introduction

Let Ω ⊂ _R^N(N ≥ 3)be a bounded domain with smooth boundary∂Ω. We assume that the functiona:(0,∞)→_Ris such that the mappingφ:R→_Rdefined by

φ(t) =

(a(|t|)t, fort 6=0, 0, fort =0,

is an odd, increasing homeomorphism from R onto R. We also suppose throughout this paper that λ>0, Vis an indefinite sign-changing weight andq:Ω→ (1,∞)is a continuous function. In this note we study the following nonlinear eigenvalue problem:

(−div(a(|∇u|)∇u) =λV(x)|u|^q(x)−2u, inΩ,

u=0, on∂Ω. (P)

The interest in analyzing this kind of problems is motivated by some recent advances in the study of eigenvalue problems involving non-homogeneous operators in the divergence form. We refer especially to the results in [5,6,11,13–16,18].

BEmail: gebin791025@hrbeu.edu.cn

Mih˘ailescu and R˘adulescu, in [13], studied the same nonhomogeneous eigenvalue prob- lem in the particular case when V(x) = 1. The authors proved, under the assumption 1 < inf_x∈_Ωq(x) < p₀, that there exists λ₀ > 0 such that any λ ∈ (0,λ₀) is an eigenvalue for problem (P).

In order to go further we introduce the functional space setting where problem (P) will be discussed. In this context we notice that the operator in the divergence form is not ho- mogeneous and thus, we introduce an Orlicz–Sobolev space setting for problems of this type.

Orlicz–Sobolev spaces have been used in the last decades to model various phenomena. Chen, Levine and Rao [3] proposed a framework for image restoration based on a variable exponent Laplacian. A second application which uses variable exponent type Laplace operators is modelling electrorheological fluids [9]. On the other hand, the presence of the continuous functionssandqas exponents appeals to a suitable variable exponent Lebesgue space setting.

In the following, we give a brief description of the Orlicz–Sobolev spaces and of the variable exponent Lebesgue spaces.

We first recall some basic facts about Orlicz spaces. Define Φ(t) =

Z _t

0

φ(s)ds, Φ^∗(t) =

Z _t

0

φ⁻¹(s)ds, ∀t ∈_R.

We observe thatΦis a Young function, that is,Φ(0) =0,Φis convex, and limt→_∞Φ(t) = +_∞.

Furthermore, sinceΦ(0) =0 if and only ift =0, limt→0Φ(t)

t =0, and limt→_∞^Φ(t)

t = +_{∞, then} Φis called anN-function. The function Φ^∗ is called the complementary function of Φand it satisfies

Φ^∗(t) =sup{st−_Φ(s):s≥0}, ∀t ≥0.

We also observe thatΦ^∗is also anN-function and the following Young’s inequality holds true:

st≤_Φ(s) +_Φ^∗(t), ∀s,t≥0.

The Orlicz spacesL_Φ(_Ω)defined by theN-functionΦ(see [1,2,4]) is the space of measurable functionsu:Ω→_Rsuch that

kuk_LΦ =sup _Z

Ωuvdx: Z

ΩΦ^∗(|v|)dx≤1

<+_∞.

Then(L_Φ(_Ω),k · k_LΦ)is a reflexive Banach space whose norm is equivalent to the Luxemburg norm

kuk_Φ =_inf

µ>_{0 :}

Z

ΩΦ u

µ

dx≤1

. For Orlicz spaces, Hölder’s inequality reads as follows (see [17]):

Z

Ωuvdx ≤2kuk_LΦkvk_LΦ∗, ∀u∈ L_Φ(_Ω), ∀v∈ L_Φ^∗(_Ω).

We denote byW₀¹L_Φ(_Ω)the corresponding Orlicz–Sobolev space for problem (P), equipped with the norm

kuk=k∇uk_Φ (see [8]). The spaceW₀¹L_Φ(_Ω)is also a Banach space.

Throughout this paper we assume that 1<lim inf

t→_∞

tφ(t)

Φ(t) ≤lim sup

t>0

tφ(t)

Φ(t) <_∞ (1.1)

and the function [0,+_∞) 3 t → _Φ(√

t ) is convex. Due to assumption (1.1), we may define the numbers

p₀ =inf

t>₀

tφ(t)

Φ(t) ^{and p}

0=sup

t>0

tφ(t) Φ(t)^. Note that for a(|t|) =|t|^p−2,p>1, one has p₀ = p⁰ = p.

On the other hand, the following relations hold true:

kuk^p0 ≤

Z

ΩΦ(|∇u|)dx≤ kuk^p0, ∀u∈W₀¹L_Φ(_Ω)withkuk<1, (1.2) kuk^p0 ≤

Z

ΩΦ(|∇u|)dx≤ kuk^p0, ∀u∈W₀¹L_Φ(_Ω)withkuk>1 (1.3) (see [12, Lemma 1]).

Let us now introduce the Orlicz–Sobolev conjugateΦ∗ of Φ, which is given by Φ⁻_∗¹(t) =

Z _t

0

Φ⁻¹(s)

s^NN+1 ds, (1.4)

(see [1]), where we suppose that limt→0

Z ₁

t

Φ⁻¹(s)

s^NN+1 ds<+_∞ and lim

t→_∞ Z _t

1

Φ⁻¹(s)

s^NN+1 ds=_∞. (1.5) In the case Φ(_t) = ¹_p|t|^p, (1.5) holds if and only ifp< _N.

2 The main result and proof of the theorem

We say thatλ∈_Ris an eigenvalue of problem (P) if there existsu∈W₀¹L_Φ(_Ω)\{₀}_{such that}

Z

Ωa(|∇u|)∇u∇vdx−λ Z

ΩV(x)|u|^q(x)−2uvdx =0,

for all v ∈ W₀¹L_Φ(_Ω). We point out that ifλis an eigenvalue of problem (P), then the corre- sponding eigenfunctionv∈W₀¹L_Φ(_Ω)\{0}is a weak solution of problem (P).

Our main result is given by the following theorem.

Theorem 2.1. Suppose that(1.5)and the following conditions hold:

H(q,s): 1<q(x)< p₀≤ p⁰<s(x), ∀x∈_Ω. H(_Φ): lim

t→_∞

|t|

s−_q+ S− −q+

Φ∗(kt) =0, ∀k >0.

H(V): V ∈ L^s(x)(_Ω) and there exists a measurable set Ω0 ⊂ _Ω of positive measure such that V(x)>0,∀x ∈_Ω0.

Then there existsλ₀>₀such that anyλ∈(_0,λ₀)is an eigenvalue of the problem(P).

Proof. In order to formulate the variational problem (P), let us introduce the functionals F, G, ϕ_λ :W₀¹L_Φ(_Ω)→_Rdefined by

F(u) =

Z

ΩΦ(|∇u|)dx, G(u) =

Z

Ω

V(x)

q(x)|u|^q(x)dx

and

ϕ_λ(u) =F(u)−λG(u).

Denote bys⁰(x)the conjugate exponent of the functions(x)and putα(x):= _s^s₍⁽_x^x₎₋^)q_q^(x_(x⁾₎. From H(q,s), we haves⁰(x)q(x) < α(x), ∀x ∈ _Ω, α(x) < _s^s−⁻−^qq⁺⁺, ∀x ∈ Ω. Thus, by relation (1.5), condition H(_Φ) and Theorem 2.2 in [7], we deduce that W₀¹L_Φ(_Ω) is compactly embedded in L

s−_q+

s− −q+(_Ω). That fact combined with the continuous embedding of L

s−_q+

s− −q+(_Ω) in L^α(x)(_Ω) ensures thatW₀¹L_Φ(_Ω)is compactly embedded inL^α(x)(_Ω). In an analogous way, we can show that the embeddingX,→ L^s0(x)q(x)(_Ω)is compact.

The proof is divided into the following four steps.

Step 1.We will show that ϕ_λ ∈C¹(_W0¹_LΦ(_Ω)_,R)_.

Firstly, by Lemma 3.4 in [7] we deduce that F is a C¹ convex functional, with Fréchet derivative given by

hF⁰(u),vi=

Z

Ωa(|∇u|)∇u∇vdx.

Therefore, we only need to prove that G ∈ C¹(W₀¹L_Φ(_Ω),R), that is, we show that for all h∈W₀¹L_Φ(_Ω),

limt↓0

G(u+th)−G(u)

t =hdG(u),hi,

anddG:W₀¹L_Φ(_Ω)→ (W₀¹L_Φ(_Ω))^∗ is continuous, where we denote by(W₀¹L_Φ(_Ω))^∗ the dual space ofW₀¹L_Φ(_Ω),h·,·iis the pairing between(W₀¹L_Φ(_Ω))^∗ andW₀¹L_Φ(_Ω).

For allh∈W₀¹L_Φ(_Ω), we have limt↓0

G(u+th)−G(u)

t = ^d

dtG(u+th) t=0

= d

dt Z

Ω

V(x)

q(x)|u+th|^q(x)dx t=0

=

Z

Ω

d dt

V(x)

q(x)|u+th|^q(x) t=₀

dx

=

Z

ΩV(x)|u+th|^q(x)−2(u+th)h t=0dx

=

Z

ΩV(x)|u|^q(x)−2uhdx

=hdG(u),hi.

The differentiation under the integral is allowed for t close to zero. Indeed, for |t| < _1, using Hölder’s inequality and conditionH(q,s), we have

Z

Ω|V(x)|u+th|^q(x)−2(u+th)h|dx≤

Z

Ω|V(x)||u+th|^q(x)−1|h|dx

≤

Z

Ω|V(x)|(|u|+|h|)^q(x)−1|h|dx

≤3|V|_s(x)|u|+|h|

qⁱ−1 q(x)|h|_α(x)

< +_∞,

wherei= +if

|u|+|h|

q(x) >1 andi= −if

|u|+|h|

q(x) ≤1. SinceW₀¹L_Φ(_Ω),→L^α(x)(_Ω), W₀¹L_Φ(_Ω),→L^q(x)(_Ω)_andV ∈ L^s(x)(_Ω)_.

On the other hand, sinceW₀¹L_Φ(_Ω)is continuously embedded in L^α(x)(_Ω)it follows that there exists positive constantsc₁such that|h|_α(x)≤c₁khk. Therefore, by conditionH(q,s), we have

|hdG(u),hi|= Z

Ω|V(x)|u|^q(x)−2uhdx

≤

Z

Ω|V(x)||u|^q(x)−1|h|dx

≤ 1

s⁻ + ^q

+

q⁺−1 + ¹ α⁻

|V|_s(x)|u|^q(x)−1 _q(x)

q(x)−1

|h|_α(x)

≤ 1

s⁻ + ^q

+

q⁺−1 + ¹ α⁻

|V|_s(x)|u|^q_qⁱ₍⁻_x)¹|h|_α(x)

≤c₁ 1

s⁻ + ^q

+

q⁺−1 + ¹ α⁻

|V|_s(x)|u|^q_qⁱ₍⁻_x)¹khk, for any h∈W₀¹L_Φ(_Ω).

Thus there existsc₂=c_{1 s}¹− + _q+^q+−1 + ¹

α⁻

|V|_s(x)|u|^q_qⁱ₍⁻_x)¹ such that

|hdG(u)_,hi| ≤c₂khk_.

Using the linearity ofdG(u)and the above inequality we deduce thatdG(u)∈ (W₀¹L_Φ(_Ω))^∗ Note that map L^q(x)(_Ω) 3 u 7→ |u|^q(x)−2u ∈ L

q(x)

q(x)−1(_Ω) is continuous. For the Fréchet differentiability, we conclude that Gis Fréchet differentiable. Furthermore,

hG⁰(u)_,vi=

Z

ΩV(x)|u|^q(x)−2uvdx, for all u,v∈W₀¹L_Φ(_Ω). The Step 1 is completed.

It is clear that(u,λ)is a solution of (P) if and only if F⁰(u) =λG⁰(u)in(W₀¹L_Φ(_Ω))^∗. Step 2. There exists λ₀ > 0 such that for any λ ∈ (0,λ₀) there exist τ,a > 0 such that ϕ_λ(u)≥a>0 for any u∈W₀¹L_Φ(_Ω)withkuk= τ.

Since the embedding W₀¹L_Φ(_Ω) ,→ L^s0(x)q(x)(_Ω) is continuous, we can find a constant c₃ >0 such that

|u|_s0(x)q(x) ≤c₃kuk, ∀u∈W₀¹L_Φ(_Ω). (2.1) Let us fix τ ∈ (0, 1) such that τ < _c¹

3. Then relation (2.1) implies |u|_s0(x)q(x) < 1, for all u∈W₀¹L_Φ(_Ω)withkuk=τ. Thus,

Z

ΩV(x)|u|^q(x)dx≤ |V|_s(x)|u|^q(x)

s⁰(x)≤ |V|_s(x)|u|^q−

q(x)s⁰(x), (2.2) for all u∈W₀¹L_Φ(_Ω)withkuk=τ.

Combining (2.1) and (2.2), we obtain Z

ΩV(x)|u|^q(x)dx≤c^q₃⁻|V|_s(x)kuk^q−, (2.3) for all u∈W₀¹L_Φ(_Ω)_withkuk=_ρ.

Taking into account relations (1.2) and (2.3) we deduce that for any u ∈ W₀¹L_Φ(_Ω) with kuk=τ<1, we have

ϕ_λ(u) =

Z

ΩΦ(|∇u|)dx−λ Z

Ω

V(x)

q(x)|u|^q(x)dx

≥ kuk^p0 −^λc

q⁻ 3

q⁻ |V|_s(x)kuk^q−

=τ^q

−

τ^p

0−q⁻− ^λc

q⁻ 3

q⁻ |V|_s(x)

! . Putting

λ₀= ^τ

p⁰−q⁻

2

q⁻ c^q₃⁻|V|_s(x)^,

then for anyλ∈(_0,λ₀)_andu∈ Xwithkuk= τ, there existsa= ^τ₂^p0, such that ϕ_λ(u)≥a >0.

Step 3. There exists ξ ∈ W₀¹L_Φ(_Ω) such that ξ ≥ 0, ξ 6= 0 and ϕ_λ(tξ) < 0, for t > 0 small enough.

In fact, assumptionH(q,s)implies q(x)< p0, ∀x ∈_Ω0. In the sequel, we use the notation q⁻₀ =inf_Ω0q(x)andq⁺₀ = sup_Ω

0q(x). Thus, there existsε₀ >0 such thatq⁻₀ +ε₀ < p₀. Sinceq∈ C(_Ω0), there exists an open setΩ1⊂_Ω0 such that

|q(x)−q⁻₀|< ε0, ∀x ∈_Ω1. Thus, we deduce

q(_x)≤q⁻₀ +ε₀, ∀x∈_Ω1. (2.4) Takeξ ∈C^∞₀ (_Ω0)such thatΩ1 ⊂supp(ξ),ξ(x) =1 forx∈ _Ω1and 0< ξ <1 inΩ0. We also point out that there existst₀∈ (0, 1)such that for anyt ∈(0,t₀)we have

t|∇ξ|=tkξk<1. (2.5)

Using (1.3), (2.4) and (2.5), for allt ∈(_{0, 1}), we get the estimate ϕ_λ(tξ) =

Z

ΩΦ(t|∇ξ|)dx−λ Z

Ω

t^q(x)

q(x)^V(x)|ξ|^q(x)dx

≤t^p0kξk^p0−λ Z

Ω0

t^q(x)

q(x)^V(x)|ξ|^q(x)dx

≤t^p0kξk^p0− ^λ q⁺₀

Z

Ω1

t^q(x)V(x)|ξ|^q(x)dx

≤t^p0kξk^p0− ^λt

q⁻₀+ε0

q⁺₀ Z

Ω1

V(x)|ξ|^q(x)dx.

Then, for anyt <τ

1 p0−q−

0−ε0 with 0<τ<_minⁿ_1,^λ

R

Ω1V(x)|φ|^q(x)dx q⁺₀kξk^p0

o

, we conclude that ϕ_λ(tξ)<_0.

By Step 2, we have

v∈inf∂Bρ(0)ϕ_λ(v)>0. (2.6) On the other hand, by Step 3, there existsξ ∈ W₀¹L_Φ(_Ω)such that ϕ_λ(tξ) < 0 for t > 0 small enough. Using (2.3), it follows that

ϕ_λ(u)≥ kuk^p0−λc^q₃⁻|V|_s(x)kuk^q−, ∀u∈ B_ρ(0). Thus,

−_∞<c_λ := _inf

v∈Bρ(0)

ϕ_λ(v)<_0.

Now let ε be such that 0 < ε < _infv∈∂B

ρ(0)ϕ_λ(v)−_infv∈B

ρ(0)ϕ_λ(v). Then, by applying Ekeland’s variational principle to the functional

ϕ_λ : Bρ(0)→_R, there existsu_ε ∈B_ρ(0)such that

ϕ_λ(u_ε)≤ inf

v∈Bρ(0)

ϕ_λ(v) +ε,

ϕ_λ(u_ε)< ϕ_λ(u) +εku−u_εk_, u6= u_ε. Since

ϕ_λ(u_ε)≤ inf

v∈Bρ(0)

ϕ_λ(v) +ε ≤ inf

v∈Bρ(0)ϕ_λ(v) +ε< inf

v∈∂Bρ(0)ϕ_λ(v), we deduce thatu_ε ∈ B_ρ(0).

Now, we defineT_λ :Bρ(0)→_Rby

T_λ(u) = ϕ_λ(u) +εku−u_εk.

It is clear thatu_ε is a minimum ofT_λ. Therefore, for smallt>0 andv∈B₁(0), we have T_λ(u_ε+tv)−T_λ(u_ε)

t ≥ _0,

which implies that

ϕ_λ(u_ε+tv)−ϕ_λ(u_ε)

t +εkvk ≥0.

Ast→0, we have

hdϕ_λ(uε),vi+εkvk ≥0, ∀v∈ B₁(0).

Hence,kϕ⁰_λ(u_ε)k_X^∗ ≤ε. We deduce that there exists a sequence{u_n}^∞_n=1 ⊂B_ρ(₀)_{such that} ϕ_λ(u_n)→c_λ and ϕ⁰_λ(u_n)→0. (2.7) It is clear that {u_n}^∞_n=1 is bounded in W₀¹L_Φ(_Ω). Since W₀¹L_Φ(_Ω) is reflexive, there exists a subsequence, still denoted by{u_n}^∞_n=1, andu∈W₀¹L_Φ(_Ω)such that{u_n}^∞_n=1converges weakly to uinW₀¹L_Φ(_Ω).

Step 4. We will show thatu_n→u inW₀¹L_Φ(_Ω). Claim:

nlim→_∞ Z

ΩV(x)|un|^q(x)−2un(un−u)dx=0.

In fact, from the Hölder type inequality, we have Z

ΩV(x)|u_n|^q(x)−2u_n(u_n−u)dx

≤ |V|_s(x)|u_n|^q(x)−2u_n(u_n−u) s⁰(_x)

≤ |V|_s(x)|u_n|^q(x)−2u_n q(x)

q(x)−1

|u_n−u|_α(x)

≤ |V|_s(x)1+|u_n|^q_q⁺_(x⁻₎¹|u_n−u|_α(x).

SinceW₀¹L_Φ(_Ω)is continuously embedded inL^q(x)(_Ω)and{un}^∞_n is bounded inW₀¹L_Φ(_Ω), so {u_n}^∞_n is bounded in L^q(x)(_Ω). On the other hand, since the embedding W₀¹L_Φ(_Ω) ,→ L^α(x)(_Ω) is compact, we deduce that |u_n−u|_α(x) → 0 as n → +∞. Hence, the proof of the claim is complete.

Moreover, sincedϕ_λ(u_n)→0 and{u_n}^∞_n is bounded inW₀¹L_Φ(_Ω), we have

|hdϕ_λ(u_n),u_n−ui|

≤ |hdϕ_λ(u_n),u_ni|+|hdϕ_λ(u_n),ui|

≤ kdϕ_λ(un)k_(W1

0L_Φ(_Ω))^∗kunk+kdϕ_λ(un)k_(W1

0L_Φ(_Ω))^∗kuk, that is,

n→+lim_∞hdϕ_λ(u_n),u_n−ui=0.

Using the previous claim and the last relation we deduce that

n→+lim_∞ Z

Ωa(|∇u_n|)∇u_n∇(u_n−u)dx=0. (2.8) From (2.8) and the fact thatun *uinW₀¹L_Φ(_Ω)it follows that

n→→+lim_∞hF⁰(u_n)_,u_n−ui=_{0. (2.9)} Next, we show that un → u in W₀¹L_Φ(_Ω). Since {un} converges weakly to u in W₀¹L_Φ(_Ω) it follows that {ku_nk} is a bounded sequence of real numbers. That fact and relations (1.2) and (1.3) yield that the sequence{F(u_n)}is bounded. Then, up to a subsequence, we deduce that F(un) → c. The function F being convex, from Mazur’s lemma, it is also weakly lower semi-continuous. Hence

F(u)≤_{lim inf}

n→_∞ F(u_n) =c. (2.10)

On the other hand, sinceFis convex, we have

F(u)≥F(u_n) +hF⁰(u_n),u−u_ni. (2.11) Furthermore, relations (2.9), (2.10) and (2.11) imply

F(u) =c.

Taking into account that {^un₂^+u} converges weakly to u in W₀¹L_Φ(_Ω) and using the above method we find

c= F(u)≤ Fu_n+u 2

. (2.12)

We assume by contradiction that{u_n}does not converge to uin W₀¹L_Φ(_Ω). Furthermore, we deduce that {^un₂^−u}does not converge to u inW₀¹L_Φ(_Ω). It follows that there existε > 0 and a subsequence{u_nk}of {u_n}such that

u_n−u 2

≥_ε, ∀k ≥_{1. (2.13)}

Thus, relations (1.2), (1.3) and (2.13) imply that there existsε₁>0 Fu_n−u

2

≥ε₁, ∀k≥_{1. (2.14)}

Moreover, from hypotheses (1.1) we deduce that we can apply Lemma 2.1 in [10] in order to obtain

1 2

hΦ(|t|) +_Φ(|s|)ⁱ≥_Φ

|t+s| 2

+_Φ

|t−s| 2

, ∀t,s∈_R. The above inequality yields

1 2 h

F(u) +F(v)ⁱ≥ Fu+v 2

+Fu−v 2

, ∀u,v∈W₀¹L_Φ(_Ω)_{. (2.15)} Hence, from (2.14) and (2.16), we have

1 2 h

F(u) +F(u_nk)ⁱ−Fun_k+u 2

≥ Fun_k−u 2

≥ε₁, ∀k≥1. (2.16) Lettingk→_∞in the above inequality we have

c−ε₁ ≥lim sup

k→_∞

Fu_nk +u 2

,

and that is a contradiction with (2.12). We conclude thatu_n→ uin W₀¹L_Φ(_Ω). Thus, in view of (2.7), we obtain

ϕ_λ(u) =c_λ <0 and ϕ⁰_λ(u) =0. (2.17) The proof is complete.

Acknowledgement

B. Ge was supported by the NNSF of China (No. 11201095), the Youth Scholar Backbone Supporting Plan Project of Harbin Engineering University, the Fundamental Research Funds for the Central Universities, the Postdoctoral Research Startup Foundation of Heilongjiang (no. LBH-Q14044), the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502).

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