Multiplicity of Solutions for a Nonlinear Degenerate Problem in Anisotropic Variable Exponent Spaces

BULLETINof the MALAYSIANMATHEMATICAL

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

Bull. Malays. Math. Sci. Soc. (2)36(1) (2013), 117–130

Multiplicity of Solutions for a Nonlinear Degenerate Problem in Anisotropic Variable Exponent Spaces

DENISASTANCU-DUMITRU

Department of Mathematics, University of Craiova, 200585 Craiova, Romania [email protected]

Abstract. We study a nonlinear elliptic problem with Dirichlet boundary condition involv- ing an anisotropic operator with variable exponents on a smooth bounded domainΩ⊂R^N. For that equation we prove the existence of at least two nonnegative and nontrivial weak solutions. Our main result is proved using as main tools the Mountain Pass Theorem and a direct method in Calculus of Variation.

2010 Mathematics Subject Classification: 35J60, 35J70, 46E30

Keywords and phrases: Anisotropic variable exponent Sobolev spaces, weak solution, criti- cal point, Mountain Pass Theorem, variational methods.

1. Introduction

Equations involving variable exponent growth conditions have been extensively studied in the last decade. The large number of papers studying problems involving variable exponent growth conditions is motivated by the fact that this type of equations can serve as models in the theory of electrorheological fluids (see, e.g. [15]), image processing (see, e.g. [3]), the theory of elasticity (see, e.g. [18]) or biology (see, e.g. [10]). In this context, we just refer to the survey paper [11] and the references therein.

Typical models of elliptic equations with variable exponent growth conditions appeal to the so called p(x)-Laplace operator, that is

∆_p(x)u:=div(|∇u|^p(x)−2∇u),

wherep(x)is a function satisfyingp(x)>1 for eachx. Recently, Mih˘ailescu-Pucci-R˘adulescu extended in [13] the study involving the p(x)-Laplace operator to the case of anisotropic equations with variable exponent growth conditions, where the differential operator consid- ered has the form

(1.1)

N

∑

i=1

∂_xi(|∂_xiu|^pi(x)−2∂_xiu),

Communicated bySriwulan Adji.

Received:January 18, 2011;Revised:May 8, 2011.

with p_i(x)functions satisfying inf_xp_i(x)>1 for eachi∈ {1, ...,N}. Undoubtedly, in the particular case whenp_i(x) =p(x)for eachi∈ {1, ...,N}the above differential operator be- comes

N

∑

i=1

∂_xi(|∂_xiu|^p(x)−2∂_xiu)and has similar properties with thep(x)-Laplace operator. On the other hand, the anisotropic equations with variable exponent growth conditions enable the study of equations with more complicated nonlinearities since the differential operator (1.1) allows a distinct behavior for partial derivatives in various directions.

In this paper we study the existence of nontrivial solutions of an nonhomogeneous anisotropic problem of type

(1.2)











−∑^N

i=1

∂xi(|∂_xiu|^pi(x)−2∂xiu) =λf(x,u), forx∈Ω,

u=0, forx∈∂Ω,

u≥0, forx∈Ω,

whereΩ ⊂R^N(N ≥3) is a bounded domain with smooth boundary, pi are continuous functions onΩsuch that 2≤p_i(x)for anyx∈Ωandi∈ {1, ...,N}.

2. Preliminary results Set

C+(Ω) ={h;h∈C(Ω),h(x)>1 for allx∈Ω}.

For anyh∈C₊(Ω)we define h⁺=sup

x∈Ω

h(x) and h⁻=inf

x∈Ωh(x).

For anyp∈C+(Ω), we definethe variable exponent Lebesgue space L^p(·)(Ω) ={u; uis a measurable real-valued function such that

Z

Ω

|u(x)|^p(x)dx<∞}.

We define a norm, the so-calledLuxemburg norm, on this space by the formula

|u|_p(·)=inf (

µ>0;

Z

Ω

u(x) µ

p(x)

dx≤1 )

.

Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects:

they are Banach spaces [12, Theorem 2.5], the H¨older inequality holds [12, Theorem 2.1], they are reflexive if and only if 1<p⁻≤p⁺<∞[12, Corollary 2.7] and continuous func- tions are dense ifp⁺<∞[12, Theorem 2.11]. The inclusion between Lebesgue spaces also generalizes naturally [12, Theorem 2.8]: if 0<|Ω|<∞andp₁,p₂are variable exponents, so thatp₁(x)≤p₂(x)almost everywhere inΩ, then there exists the continuous embedding L^p2(·)(Ω),→L^p1(·)(Ω), whose norm does not exceed|Ω|+1.

We denote byL^q(·)(Ω)the conjugate space ofL^p(·)(Ω), where 1/p(x) +1/q(x) =1 for anyx∈Ω. For anyu∈L^p(·)(Ω)andv∈L^q(·)(Ω)the H¨older type inequality

(2.1)

Z

Ω

uv dx

≤ 1

p⁻+ 1 q⁻

|u|_p(·)|v|_q(·) holds true.

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by themodularof theL^p(·)(Ω)space, which is the mappingρp(·):L^p(·)(Ω)→Rdefined by

ρp(·)(u) = Z

Ω

|u|^p(x)dx.

If(u_n),u∈L^p(·)(Ω)andp⁺<∞then the following relations hold true (2.2) |u|_p(·)>1 ⇒ |u|^p_p(·)⁻ ≤ρ_p(·)(u)≤ |u|^p_p(·)⁺

(2.3) |u|_p(·)<1 ⇒ |u|^p_p(·)⁺ ≤ρ_p(·)(u)≤ |u|^p_p(·)⁻ (2.4) |u_n−u|_p(·)→0 ⇔ ρp(·)(u_n−u)→0.

Spaces withp⁺=∞have been studied by Edmunds, Lang and Nekvinda [4].

Next, we defineW₀^1,p(·)(Ω)as the closure ofC₀^∞(Ω)under the norm kuk=|∇u|_p(·).

The space (W₀^1,p(·)(Ω),k · k)is a separable and reflexive Banach space. We note that if q∈C₊(Ω)andq(x)<p^?(x)for all x∈Ω then the embeddingW₀^1,p(·)(Ω),→L^q(·)(Ω)is compact and continuous, where p^?(x) =N p(x)/(N−p(x))if p(x)<N or p^?(x) = +∞if p(x)≥N. We refer to [5, 6, 7, 8, 12] for further properties of variable exponent Lebesgue- Sobolev spaces.

Finally, we introduce a natural generalization of the variable exponent Sobolev space W₀^1,p(·)(Ω) that will enable us to study with sufficient accuracy problem (1.2). For this purpose, let us denote by−→p :Ω→R^Nthe vectorial function−→p = (p1, ...,pN), wherep1,..., pN:Ω→(1,∞)are continuous functions. We defineW^1,

−

→p(·)

0 (Ω), theanisotropic variable exponent Sobolev space,as the closure ofC₀^∞(Ω)with respect to the norm

kuk^−→_p(·)=

N i=1

∑

|∂_xiu|_p

i(·).

W^1,

−

→p(·)

0 (Ω)endowed with the above norm is a reflexive Banach space (see [13]).

On the other hand, in order to facilitate the manipulation of the spaceW^1,

−

→p(·)

0 (Ω)we

introduced−→ P₊,−→

P−inR^Nas

−

→P₊= (p⁺₁, ...,p⁺_N), −→

P−= (p⁻₁, ...,p⁻_N), andP₊⁺,P₋⁺,P₋⁻∈R⁺as

P₊⁺=max{p⁺₁, ...,p⁺_N}, P₋⁺=max{p⁻₁, ...,p⁻_N}, P₋⁻=min{p⁻₁, ...,p⁻_N}.

Throughout this paper we assume that (2.5)

N

∑

i=1

1 p⁻_i >1

and defineP₋^?∈R⁺andP−,∞∈R⁺by P₋^?= N

N

∑

i=1 1 p⁻_i −1

, P−,∞=max{P₋⁺,P₋^?}.

Next, we recall Theorem 1 in [13], regarding a compactness embedding ofW^1,

−

→p(·)

0 (Ω)

intovariable exponent Lebesgue space:

Theorem 2.1. Assume thatΩ⊂R^N (N≥3) is a bounded domain with smooth boundary.

Assume relation (2.5) is fulfilled. For any q∈C(Ω)verifying (2.6) 1<q(x)<P−,∞ f or all x∈Ω, the embedding

W^1,

−

→p(·)

0 (Ω),→L^q(·)(Ω) is continuous and compact.

3. The main result

In this paper we study problem (1.2) in the particular case f(x,t) =t^α(x)−1−t^β(x)−1, t≥0,x∈Ω, whereα:Ω→R,β:Ω→Rare continuous functions such that

1<β⁻≤β(x)≤α(x)≤α⁺<P₋⁻≤P₊⁺<P−,∞ for x∈Ω and there existsx₀∈Ωsuch that

β(x₀)<α(x₀).

More precisely, we consider the following problem

(3.1)











−∑^N

i=1

∂xi(|∂_xiu|^pi(x)−2∂xiu) =λ(u^α(x)−1−u^β(x)−1), forx∈Ω,

u=0, forx∈∂Ω,

u≥0, forx∈Ω.

We seek solutions for problem (3.1) belonging to the spaceW^1,

−

→p(·)

0 (Ω)in the sense below.

Definition 3.1. We say that u∈W^1,

−

→p(·)

0 (Ω)is a weak solution for problem (3.1) if u≥0 almost everywhere inΩand

Z

Ω

(N

∑

i=1

|∂_xiu|^pi(x)−2∂_xiu∂_xiv

−λ

u^α(x)−1v−u^β(x)−1v )

dx=0, for all v∈W^1,

−

→p(·)

0 (Ω).

The main result of this paper is given by the following theorem.

Theorem 3.1. There existsλ>0such that problem (3.1) has at least two distinct nonneg- ative, nontrivial weak solutions for allλ≥λ.

Theorem 3.1 supplements [14] with the case of anisotropic spaces.

4. Proof of Theorem 3.1

We start by introducing the energy functional corresponding to problem (3.1) which is de- fined asJ_λ :W^1,

−

→p(·)

0 (Ω)→R,

(4.1) J_λ(u) = Z

Ω N

∑

i=1

|∂_xiu|^pi(x) p_i(x) dx−λ

Z

Ω

u^α(x)₊ α(x) dx+λ

Z

Ω

u^β₊^(x) β(x)dx,

whereu+(x) =max{u(x),0}for eachx∈Ω. Standard arguments assure thatJ_λ∈C¹(W₀^{1,−→p(·)}(Ω),R) and the Fr´echet derivative is given by

(4.2) hJ_λ⁰(u),vi= Z

Ω N

∑

i=1

|∂_xiu|^pi(x)−2∂_xiu∂_xiv dx−λ Z

Ω

u^α(x)−1₊ v dx+λ Z

Ω

u^β₊^(x)−1v dx,

for allu,v∈W^1,

−

→p(·)

0 (Ω).

We prove some auxiliary results.

Lemma 4.1. If u∈W^1,

−

→p(·)

0 (Ω)then u₊,u−∈W^1,

−

→p(·)

0 (Ω)and

∂u+

∂x_i =







0, if [u≤0],

∂u

∂xi

, if [u>0],

∂u−

∂x_i =







0, if [u≥0],

∂u

∂xi

, if [u<0], where u±(x) =max{±u(x),0}for all x∈Ω.

Proof. The proof of this lemma runs similarly with the proof of Lemma 3.3 in [14] without any particular complication. Because of that fact we will omit it.

Lemma 4.2. If u is a critical point of functional J_λ then u≥0.

Proof. Ifuis a critical point ofJ_λ then using Lemma 4.1 we obtain 0=hJ_λ⁰(u),u−i

= Z

Ω N i=1

∑

|∂_xi u|^pi(x)−2∂x_iu∂x_iu−dx−λ Z

Ω

u^α(x)−1₊ u−dx+

λ Z

Ω

u^β₊^(x)−1u−dx

= Z

Ω N i=1

∑

|∂_xi u|^pi(x)−2∂x_iu∂x_iu−dx

= Z

Ω N

∑

i=1

|∂_xi u−|^pi(x)−2∂_xiu−∂_xiu−dx

= Z

Ω N i=1

∑

|∂_xi u−|^pi(x)dx, that is

(4.3)

Z

Ω N

∑

i=1

|∂_xiu−|^pi(x)dx=0.

Applying Jensen’s inequality to the convex functiong:[0,∞)→[0,∞)defined byg(s) =s^P−− it follows that for allu∈W^1,

−

→p(·)

0 (Ω)we have

(4.4)

ku₋k^P

−

−−

→p(·)

N^P−−−1

=N









N

∑

i=1

|∂_xiu−|_p

i(·)

N









P₋⁻

≤

N i=1

∑

|∂_xiu−|^P

−

−

pi(·).

For eachi∈ {1, . . . ,N}we define ξi=

P₊⁺, if |∂_xiu−|_pi(·)<1, P₋⁻, if |∂_xiu−|_p

i(·)>1.

Thus, using relation (2.2), (2.3) and (4.4) we get Z

Ω N i=1

∑

|∂_xiu−|^pi(x)dx≥

N i=1

∑

|∂_xiu−|^ξi

pi(·).

By above pieces of information and relation (4.3) we deduce thatku−k^−→_p(·)=

N i=1

∑

|∂_xiu−|_pi(·)= 0 which means thatu≥0.

We consider the functionalI:W^1,

−

→p(·)

0 (Ω)→Rdefined by I(u) =

Z

Ω N

∑

i=1

|∂_xiu|^pi(x) p_i(x) dx for everyu∈W^1,

−

→p(·)

0 (Ω). Standard arguments assure thatIis well-defined onW^1,

−

→p(·)

0 (Ω),

I∈C¹(W₀^{1,−→p(·)}(Ω),R)and the Fr´echet derivative is given by hI⁰(u),vi=

Z

Ω N i=1

∑

|∂_xiu|^pi(x)−2∂x_iu∂x_iv dx

for allu,v∈W^1,

−

→p(·)

0 (Ω).

Lemma 4.3. The functional I is weakly lower semicontinuous.

Proof. The conclusion of this lemma is obvious since we deal with a functionalIwhich is continuous and convex on the Banach spaceW^1,

−

→p(·)

0 (Ω).

Lemma 4.4. There exists a positive constantA such that

N

∑

i=1 Z

Ω

|∂_xiu|^pi(x)

p_i(x) dx≥A ^Z

Ω

|u|^P−−dx

for all u∈S:={v∈W^1,

−

→p(·)

0 (Ω): kvk^−→_p(·)>N}.

Proof. We fix arbitraryu∈S. Sincekuk^−→_p(·)>Nit follows that there exists j∈ {1, ...,N}

such that|∂_xju|_pj(·)>1. Then we deduce that (4.5)

Z

Ω

|∂_xju|^pj(x)

p_j(x) dx≥ 1 P₊⁺|∂_xju|^P

−

−

pj(·).

On the other hand, sinceu∈W₀^{1,−→p(·)}(Ω)we infer that∂_xju∈L^pj(·)(Ω). SinceP₋⁻≤p_j(x)for everyx∈Ωwe get thatL^pj(·)(Ω)is continuously embedded inL^P−−(Ω)and consequently, there exists a positive constant, sayCj>0, such that

|∂_xju|_P−

− ≤C_j|∂_xju|_pj(·), or

(4.6) |∂_xju|^P

−

−

pj(·)≥ 1 C^P

−−

j Z

Ω

|∂_xju|^P−−dx.

Now, using similar arguments as those used in the proof of relation (11) in [9] we obtain the existence of a positive constant, sayD_j>0, such that

(4.7)

Z

Ω

|∂_xju|^P−−dx≥Dj Z

Ω

|u|^P−−dx.

Relations (4.5), (4.6) and (4.7) imply the existence of a positive constant A := min

j∈{1,...,N}

D_j P₊⁺C^P

−

−

j

for which the conclusion of Lemma 4.4 holds true.

Remark 4.1. By Lemma 4.4 we deduce that there existsλ^?>0 such that

(4.8) λ^?=inf

u∈S N

∑

i=1 Z

Ω

|∂_xiu|^pi(x) p_i(x) dx Z

Ω

|u|^P−−dx whereS:={v∈W^1,

−

→p(·)

0 (Ω): kvk^−→_p(·)>N}.

Lemma 4.5.

1^◦ The functional J_λ is coercive and bounded from below.

2^◦ The functional J_λ is weakly lower semicontinuous.

Proof. 1^◦. By relation 1<β⁻≤β(x)≤α(x)≤α⁺<P₋⁻for everyx∈Ω andβ(x₀)<

α(x₀), we get that

t→∞lim 1

α(x)t^α(x)− 1 β(x)t^β(x) t^P−−

=0

for everyx∈Ω. Then for anyλ >0 there exists a positive constantC_λ such that λ

1

α(x)t^α(x)− 1 β(x)t^β(x)

≤λ^?

2 t^P−− +C_λ for allt≥0 andx∈Ω, whereλ^?is given by relation (4.8).

The above inequality shows that for anyu∈W^1,

−

→p(·)

0 (Ω)withkuk^−→_p(·)>Nwe obtain J_λ(u) =

N

∑

i=1 Z

Ω

|∂_xiu|^pi(x) p_i(x) dx−λ

Z

Ω

1

α(x)u^α(x)₊ − 1 β(x)u^β(x)₊

dx

≥

N

∑

Ω

|∂_xiu|^pi(x) p_i(x) dx−λ

Z

Ω

1

α(x)|u|^α(x) − 1 β(x)u^β(x)₊

dx

≥

N

∑

Ω

|∂_xiu|^pi(x)

p_i(x) dx−λ^? 2

Z

Ω

|u|^P−− dx−C_λ· |Ω|

≥

N

∑

i=1

1 2 Z

Ω

|∂_xiu|^pi(x)

p_i(x) dx−C_λ· |Ω|

≥ 1 2P₊⁺

N i=1

∑

Z

Ω

|∂_xiu|^pi(x) dx−C_λ· |Ω|.

In order to go further, we define for each i∈ {1, ...,N} and each u∈W^1,

−

→p(·)

0 (Ω) with

kuk^−→_p(·)>N

κi,u=

P₊⁺, if |∂_xiu|_p

i(·)<1, P₋⁻, if |∂_xiu|_pi(·)>1.

Using relations (2.2) and (2.3) we infer that for eachu∈W^1,

−

→p(·)

0 (Ω)withkuk^−→_p(·)>Nwe have

N

∑

i=1 Z

Ω

|∂_xiu|^pi(x)dx≥

N

∑

i=1

|∂_xiu|^κ_p^i,u

i(·)

≥

N i=1

∑

|∂_xiu|^P

−

−

pi(·)−

∑

{i:κi,u=P₊⁺}

(|∂_xiu|^P

−

−

pi(·)− |∂_xiu|^P++

pi(·))

≥ 1 N^P−−

kuk^P

−−

−

→p(·)−N.

From the above pieces of information we find that for eachu∈W^1,

−

→p(·)

0 (Ω)withkuk^−→_p(·)>N the following estimate holds true

J_λ(u)≥ 1 2P₊⁺N^P−−

kuk^P

−

−−

→p(·)− N

2P₊⁺−C_λ|Ω|. This inequalities show thatJ_λ is coercive and bounded from below.

2^◦. By Lemma 4.3 we have that the functionalI:W^1,

−

→p(·)

0 (Ω)→Rdefined by I(u) =

Z

Ω N

∑

|∂_xiu|^pi(x) p_i(x) dx for everyu∈W^1,

−

→p(·)

0 (Ω)is weakly lower semicontinuous. Next, we prove thatJ_λ is weakly lower semicontinuous. Let{u_n} ⊂W^1,

−

→p(·)

0 (Ω)be a sequence that converges weakly touin W^1,

−

→p(·)

0 (Ω). SinceIis weakly lower semicontinuous we deduce that

(4.9) I(u)≤lim inf

n→∞ I(u_n).

On the other hand, asW^1,

−

→p(·)

0 (Ω)is continuously and compactly embedded inL^α(·)(Ω)and L^β(·)(Ω)(by Theorem 2.1) it follows that{(u_n)₊}converges strongly tou₊inL^α(·)(Ω)and

L^β(·)(Ω). This fact and relation (4.9) imply that J_λ(u)≤lim inf

n→∞ J_λ(u_n),

that is the functionalJ_λ is weakly lower semicontinuous. The proof of Lemma 4.5 is com- pleted.

By Lemma 4.5 and Theorem 1.2 in[16], we deduce that there exists v₁∈W^1,

−

→p(·)

0 (Ω)a

global minimizer ofJ_λ,thus v₁≥0inΩby Lemma 4.2.

Lemma 4.6. There existsλ >0such that inf

W^1,

−

→p(·)

0 (Ω)

J_λ <0, for allλ ≥λ.

Proof. Since β(x₀)<α(x₀)we can choose a small neighborhoodΩ₁⊂Ωof x₀and we deduce that there exists an elementv₀∈W^1,

−

→p(·)

0 (Ω₁)⊂W^1,

−

→p(·)

0 (Ω)such that Z

Ω

v^α(x)₀ α(x)−v^β(x)₀

β(x)

dx>0.

Consequently, there existsλ>0 such thatJ_λ(v₀)<0 for anyλ ≥λ.

Remark 4.2. By Lemma 4.6 and the fact thatv1is a global minimizer ofJ_λ it follows that J_λ(v₁)<0 for anyλ≥λ and thus, we find thatv₁is a nontrivial weak solution of problem (3.1) forλ>0 large enough.

We fixλ≥λ and consider functionh:Ω×R→Rdefined by

h(x,t) =







0, if t<0,

t^α(x)−1−t^β(x)−1, if 0≤t≤v1(x), v^α(x)−1₁ (x)−v^β₁^(x)−1(x), if t>v₁(x) and functionH:Ω×R→R,

H(x,t) = Z t

0

h(x,s) ds,

that is the primitive of functionhwith respect to the second variable.

Define the functionalK_λ :W^1,

−

→p(·)

0 (Ω)→Rby

K_λ(v) = Z

Ω N i=1

∑

|∂_xiv|^pi(x) p_i(x) dx−λ

Z

Ω

H(x,v) dx.

Standard arguments assures thatK_λ ∈C¹(W₀^{1,−→p(·)}(Ω),R)and its Frech´et derivative is given by

hK_λ⁰(v),wi= Z

Ω N i=1

∑

|∂_xi v|^pi(x)−2∂x_iv∂x_iw dx−λ Z

Ω

h(x,v)w dx, for allv,w∈W^1,

−

→p(·)

0 (Ω).

Remark 4.3. We point out that ifv∈W^1,

−

→p(·)

0 (Ω)is a critical point of functionalK_λ then v≥0. The proof is similar with the one considered in the case ofJ_λ.

Lemma 4.7. If v is a critical point of functional K_λ then v≤v₁.

Proof. We have

0=hK_λ⁰(v),(v−v₁)₊i − hJ_λ⁰(v₁),(v−v₁)₊i

= Z

Ω N

∑

i=1

h|∂_xiv|^pi(x)−2∂xiv− |∂_xiv₁|^pi(x)−2∂xiv₁i

∂xi(v−v₁)₊dx−

λ Z

Ω

hh(x,v)−

v^α(x)−1₁ −v^β₁^(x)−1i

(v−v₁)₊dx

= Z

[v>v₁] N i=1

∑

h|∂_xiv|^pi(x)−2∂xiv− |∂_xiv₁|^pi(x)−2∂xiv₁i

∂xi(v−v₁)dx, that is

(4.10)

Z

[v>v₁] N i=1

∑

h|∂_xiv|^pi(x)−2∂x_iv− |∂_xiv₁|^pi(x)−2∂x_iv₁i

(∂_xiv−∂x_iv₁)dx=0.

Next, we recall that the following elementary inequality

(|η|^t−2η− |ζ|^t−2ζ)(η−ζ)≥2^−t|η−ζ|^t for allη,ζ∈R^N

which is valid for allt≥2. By equality (4.10), applying the above inequality we get

N

∑

i=1 Z

[v>v₁]

|∂_xiv−∂xiv₁|^pi(x) dx=0,

so∂x_i v(x) =∂x_iv₁(x)for alli∈ {1, . . . ,N}andx∈Ω2:={y∈Ω: v(y)>v₁(y)}.

Hence

N

∑

i=1 Z

Ω₂

|∂_xiv−∂_xiv₁|^pi(x) dx=0, and thus,

N i=1

∑

Z

Ω

|∂_xi(v−v1)₊|^pi(x) dx=0.

By relations (2.2) and (2.3) we obtaink(v−v₁)₊k^−→_p(·)=0. Sincev−v₁∈W^1,

−

→p(·)

0 (Ω)

by Lemma 4.1 we have that(v−v₁)₊∈W^1,

−

→p(·)

0 (Ω). Thus, we obtain that(v−v₁)₊=0 in Ωwhich means thatv≤v₁inΩ.

Next, we will determinate a nontrivial critical point for functionalK_λ using as a main tool the Mountain Pass Theorem. In order to do this, we prove the following lemma.

Lemma 4.8. There exist two constantsθ∈(0,kv₁k^−→_p(·))and a>0such that K_λ(v)≥a for all v∈W^1,

−

→p(·)

0 (Ω)withkvk^−→_p(·)=θ.

Proof. We fixv∈W^1,

−

→p(·)

0 (Ω)arbitrary withkvk^−→_p(·)<1. Obviously we have that 1

α(x) s^α(x)− 1

β(x) s^β(x)≤0 for anys∈[0,1]and anyx∈Ω.

We defineΩ3:=

x∈Ω: v(x)>min{1,v₁(x)} .

Ifx∈Ω\Ω3we have thatv(x)≤v₁(x)andv(x)≤1 and we deduce that H(x,v) = 1

α(x) v^α(x)₊ − 1

β(x) v^β₊^(x)≤0.

Ifx∈Ω₃∩ {x∈Ω;v₁(x)<v(x)<1}we have that H(x,v) = 1

α(x) v^α(x)₁ − 1

β(x) v^β₁^(x)+ (v^α(x)−1₁ −v^β(x)−1₁ )·(v−v₁)≤0.

Therefore, we conclude thatH(x,v)≤0 on(Ω\Ω3)∪(Ω₃∩ {x∈Ω; v₁(x)<v(x)<1}).

We define the set

Ω⁰₃:=Ω3\ {x∈Ω;v1(x)<v(x)<1}.

By relation (2.3), for allw∈W^1,

−

→p(·)

0 (Ω)withkwk^−→_p(·)<1, we obtain using Jensen’s in- equality

(4.11) kwk^P

+

−+

→p(·)

N^P++−1

=N









N

∑

i=1

|∂_xiw|_pi(·) N









P₊⁺

≤

N

∑

i=1

|∂_xiw|^P

+ +

pi(·)≤

N

∑

i=1

|∂_xiw|^p

+ i

pi(·)≤

N

∑

i=1 Z

Ω

|∂_xiw|^pi(x)dx.

Provided thatkvk^−→_p(·)<1 by relation (4.11) we get

(4.12) K_λ(v)≥ 1

P₊⁺ kvk^P

+

−+

→p(·)

N^P++−1

−λ Z

Ω⁰₃

H(x,v) dx.

We choose a constantrsuch that 1<P₊⁺<r<P−,∞. By Theorem 2.1 it follows that W^1,

−

→p(·)

0 (Ω)is continuously embedded inL^r(Ω)that means there exists a positive constant C₁such that

(4.13) |v|_Lr(Ω)≤C₁kvk^−→_p(·) for all v∈W^1,

−

→p(·)

0 (Ω).

Using the definition of functionalHand (4.13) we have λ

Z

Ω⁰₃

H(x,v)dx=λ Z

Ω⁰₃∩[v>v₁]

v^α(x)₁ α(x)−v^β(x)₁

β(x)

! dx+

λ Z

Ω⁰₃∩[v>v₁]

(v^α(x)−1₁ −v^β(x)−1₁ )(v−v₁)dx+

λ Z

Ω⁰₃∩[v<v₁]

v^α(x)₊ α(x)−v^β(x)₊

β(x)

! dx

≤ λ α⁻

Z

Ω⁰₃∩[v>v1]

v^α(x)₁ dx+λ Z

Ω⁰₃∩[v>v₁]

v^α(x)−1₁ v dx+

λ α⁻

Z

Ω⁰₃∩[v<v₁]

v^α(x)₊ dx

≤λ C₂ Z

Ω⁰₃

v^α(x)₊ dx

≤λ C₂ Z

Ω⁰₃

v^r₊ dx

≤λ C₃kvk^r−→p(·), whereC₂andC₃are positive constants.

By the above inequalities we deduce that for aθ ∈(0,min{1,kv₁k^−→_p(·)})small enough we get

K_λ(v)≥ 1 P₊⁺N^P++−1

−λ C₃kvk^r−P

+

− +

→p(·)

! kvk^P

+

−+

→p(·).

Sincer>P₊⁺the proof of Lemma 4.8 is completed.

Lemma 4.9. Functional K_λis coercive.

Proof. This proof can be carried out in a similar manner as the proof of Lemma 4.5 and for that reason we shall omit it.

Proof of Theorem 3.1. By Lemma 4.8 and the Mountain Pass Theorem (see [1] with the variant given by [17, Theorem 1.15]), we deduce that there exists a sequence {v_n} ⊂ W^1,

−

→p(·)

0 (Ω)such that

(4.14) K_λ(v_n)→c>0 and K_λ⁰(v_n)→0, where

c=inf

γ∈Γmax

t∈[0,1]K_λ(γ(t))≥a>0, withagiven by Lemma 4.8 and

Γ={γ∈C([0,1],W^1,

−

→p(·)

0 (Ω)): γ(0) =0,γ(1) =v₁}.

By relation (4.14) and Lemma 4.9, we obtain that {v_n} is a bounded sequence and thus, passing eventually to a subsequence of {v_n}, still denoted by {v_n} we may assume that there existsv₂∈W^1,

−

→p(·)

0 (Ω)such that{v_n}converges weakly tov₂inW^1,

−

→p(·)

0 (Ω).

We will show that{v_n}converges strongly tov₂inW^1,

−

→p(·)

0 (Ω).

By relation (4.14) we have that

(4.15) lim

n→∞hK_λ⁰(v_n),v_n−v₂i=0.

We get (4.16)

hI⁰(v_n)−I⁰(v₂),v_n−v₂i=hK_λ⁰(v_n)−K_λ⁰(v₂)i+λ Z

Ω

[h(x,v_n)−h(x,v₂)] (v_n−v₂)dx.

Since by Theorem 2.1 the anisotropic variable exponent spaceW^1,

−

→p(·)

0 (Ω)is continu- ously and compactly embedded in the Lebesgue spacesL^α(·)(Ω)andL^β(·)(Ω), we conclude that{v_n}converges strongly tov₂inL^α(·)(Ω)andL^β(·)(Ω).

Then by (4.15) and (4.16) we deduce that

hI⁰(v_n)−I⁰(v₂),v_n−v₂i=o(1) which is equivalent with

(4.17) lim

n→∞

N i=1

∑

Z

Ω

h|∂_xiv_n|^pi(x)−2∂x_iv_n− |∂_xiv₂|^pi(x)−2∂x_iv₂i

(∂_xiv_n−∂x_iv₂) dx=0.

Next, we recall again the inequality

(|η|^t−2η− |ζ|^t−2ζ)(η−ζ)≥2^−t|η−ζ|^t for allη,ζ∈R^N

which is valid for allt≥2. Applying the above inequality in equality (4.17), we get

n→∞lim

N i=1

∑

Z

Ω

|∂_xiv_n−∂x_iv₂|^pi(x) dx=0, and, consequently, the sequence{v_n}converges strongly tov2inW^1,

−

→p(·)

0 (Ω).

SinceK_λ ∈C¹(W₀^{1,−→p(·)}(Ω),R)and relation (4.14) holds true, we findK_λ(v₂) =c>0 andK⁰

λ(v₂) =0 in W^1,

−

→p(·)

0 (Ω)?

, the dual space ofW^1,

−

→p(·)

0 (Ω).

By Lemma 4.7 and Remark 4.3 we deduce that 0≤v₂≤v₁inΩ. Therefore,h(x,v₂) = v^α(x)−1₂ −v^β₂^(x)−1and

H(x,v₂) =v^α(x)₂ α(x)−v^β₂^(x)

β(x) and thus

K_λ(v₂) =J_λ(v₂) and K_λ⁰(v₂) =J_λ⁰(v₂).

We conclude thatv₂is a critical point of functionalJ_λ and thus a weak solution of Prob- lem (3.1).

Moreover, sinceJ_λ(v₂) =c>0=J_λ(0)it follows thatv₂is nontrivial. On the other hand, by relationJ_λ(v₂) =c>0>J_λ(v₁), where the latter inequality is given by Remark 4.2, we have thatv₂6=v₁.

In conclusion, we proved that problem (3.1) has two distinct nonnegative and nontrivial weak solutions forλ large enough. The proof of our main result is complete.

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