Introduction The aim of this article is to establish the existence of at least three weak solutions for the Navier boundary-value problem ∆2p(x)u=λf(x, u(x)) +µg(x, u(x

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

A VARIATIONAL APPROACH FOR SOLVING

p(x)-BIHARMONIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS

SHAPOUR HEIDARKHANI, GHASEM A. AFROUZI, SHAHIN MORADI, GIUSEPPE CARISTI

Communicated by Vicentiu Radulescu

Abstract. In this article, we show the existence of at least three weak so- lutions forp(x)-biharmonic equations with Navier boundary conditions. The proof of the main result is based on variational methods. We also provide an example to illustrate our results.

1. Introduction

The aim of this article is to establish the existence of at least three weak solutions for the Navier boundary-value problem

∆²_p(x)u=λf(x, u(x)) +µg(x, u(x)), x∈Ω,

u= ∆u= 0, x∈∂Ω (1.1)

where Ω ⊂R^N(N ≥2) is a bounded domain with boundary of class C¹, λ > 0, µ≥0,f, g∈C⁰(Ω×R),p(·)∈C⁰(Ω) with

max{2,N

2 }< p⁻ := inf

x∈Ω

p(x)≤p⁺:= sup

x∈Ω

p(x)

and ∆²_p(x)u:= ∆(|∆|^p(x)−2∆u) which is the operator of fourth order called thep(x)- biharmonic operator. This operator is a natural generalization of thep-biharmonic operator (wherep >1 is a constant).

The operator ∆_p(x)u:= div(|∇u|^p(x)−2∇u) is called thep(x)-Laplacian which is a generalization of thep-Laplacian and possesses more complicated nonlinearities than thep-Laplacian, for example, it is inhomogeneous.

Recently, the investigation of differential equations and variational problems with variable exponent has become a new and interesting topic. The study of various mathematical problems with variable exponent has been received considerable at- tention in recent years. These problems are interesting in applications, for example in nonlinear elasticity theory and in modelling electrorheological fluids (Acerbi and Mingione [1], Diening [11], Halsey [13], Ru˘zi˘cka [37], Rajagopal and Ru˘zi˘cka [33])

2010Mathematics Subject Classification. 35J20, 35J60.

Key words and phrases. p(x)-Laplace operator; variable exponent Sobolev spaces;

variational method; critical point theory.

c

2017 Texas State University.

Submitted May 22, 2016. Published January 23, 2017.

1

and from the study of elastic mechanics (Zhikov [42]), and raise many difficult mathematical problems. After this pioneering models, many other applications of differential operators with variable exponents have appeared in a large range of fields, such as image restoration (Chen et al. [9]) and mathematical biology (Fragnelli [12]).

Fourth-order equations can describe the static form change of beam or the sport of rigid body. In [22], Lazer and Mckenna have pointed out that this type of nonlinearity furnishes a model to study travelling waves in suspension bridges.

Numerous authors investigated the existence and multiplicity of solutions for the problems involving biharmonic, p-biharmonic andp(x)-biharmonic operators. We refer to [2, 4, 8, 10, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 38, 39, 40] for advances and references of this area. For example, Li and Tang in [24] by using a three critical points theorem obtained due to Ricceri, established the existence of at least three weak solutions for a class of Navier boundary value problem involving the p-biharmonic

∆(|∆u|^p−2∆u) =λf(x, u) +µg(x, u), x∈Ω, u= ∆u= 0, x∈∂Ω

whereλ, µ∈[0,+∞) andf : ¯Ω×R→Ris a continuous function, andg: Ω×R→R is a Carath´eodory function. Yin and Xu in [39] based on a three critical points theorem due to Ricceri, obtained the existence of at least three weak solutions for a class of quasilinear elliptic equations involving thep(x)-biharmonic operator with Navier boundary value conditions. Also in [2] by using critical point theory, the existence of infinitely many weak solutions for a class of Navier boundary-value problem depending on two parameters and involving thep(x)-biharmonic operator

∆²_p(x)u=λf(x, u(x)) +µg(x, u(x)), x∈Ω, u= ∆u= 0, x∈∂Ω

whereλis a positive parameter,µ is a non-negative parameter,f, g∈C⁰(Ω×R), was studied. Kong in [19] using variational arguments based on Ekeland’s varia- tional principle and some recent theory on the generalized Lebesgue-Sobolev spaces L^p(x)(Ω) and W^k,p(x)(Ω) studied a p(x)-biharmonic nonlinear eigenvalue prob- lem, while in [19] using variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces L^p(x)(Ω) and W^k,p(x)(Ω) he studied the multiplicity of weak solutions to a fourth order nonlin- ear elliptic problem with ap(x)-biharmonic operator. In [17], based on variational methods and critical point theory, the existence of solutions for the problem (1.1), in the caseµ= 0, was investigated. In fact, the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosetti-Rabinowitz condition on the nonlinear term was established. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin the ex- istence of the third solution for the problem was ensured, while in [16] based on variational methods the existence of at least one weak solution for the same problem was discussed.

We refer the reader to the recent monograph by Molica Bisci, R˘adulescu and Servadei [25] for related problems concerning the variational analysis of solutions of

some classes of boundary value problems. Also for further studies on this subject, we refer the reader to [3, 7, 31, 32, 34].

Inspired by the above works, in this article, we discuss the existence of at least three weak solutions for (1.1), in which two parameters are involved. Precise esti- mates of these two parametersλand µwill be given. No asymptotic condition at infinity is required on the nonlinear term. In Theorem 3.1 we establish the exis- tence of at least three weak solutions for the problem (1.1). We present example 3.2 which illustrates Theorem 3.1. Theorem 3.3 is a consequence of Theorem 3.1.

As a consequence of Theorem 3.3, we obtain Theorem 3.5 for the autonomous case andµ= 0. Finally, we present Example 3.6 in which the hypotheses of Theorems 3.5 are fulfilled.

2. Preliminaries

LetX be a nonempty set and Φ,Ψ :X →Rbe two functions. For allr, r1, r2>

infXΦ,r2> r1,r3>0, we define ϕ(r) := inf

u∈Φ⁻¹(−∞,r)

(sup_u∈Φ−1(−∞,r)Ψ(u))−Ψ(u)

r−Φ(u) ,

β(r₁, r₂) := inf

u∈Φ⁻¹(−∞,r1)

sup

v∈Φ⁻¹[r1,r2)

Ψ(v)−Ψ(u) Φ(v)−Φ(u), γ(r2, r3) :=sup_u∈Φ−1(−∞,r2+r3)Ψ(u)

r₃ ,

α(r1, r2, r3) := max{ϕ(r1), ϕ(r2), γ(r2, r3)}.

We shall discuss the existence of at least three solutions to (1.1). Our main tool to prove the results is [5, Theorem 3.3] that we now recall as follows.

Theorem 2.1. Let X be a reflexive real Banach space, Φ : X → R be a convex, coercive and continuously Gˆateaux differentiable functional whose Gˆateaux deriva- tive admits a continuous inverse on X^∗, Ψ : X → R be a continuously Gˆateaux differentiable functional whose Gˆateaux derivative is compact, such that

(A1) infXΦ = Φ(0) = Ψ(0) = 0;

(A2) for everyu1, u2∈X such thatΨ(u1)≥0and Ψ(u2)≥0, one has inf

s∈[0,1]Ψ(su1+ (1−s)u2)≥0.

Assume that there are three positive constantsr1, r2, r3 withr1< r2, such that (A3) ϕ(r1)< β(r1, r2);

(A4) ϕ(r2)< β(r1, r2);

(A5) γ(r2, r3)< β(r1, r2).

Then, for eachλ∈]_β(r¹

1,r2),_α(r ¹

1,r2,r3)[ the functional Φ−λΨadmits three critical pointsu1, u2, u3such thatu1∈Φ⁻¹(]−∞, r1[), u2∈Φ⁻¹([r1, r2[)andu3∈Φ⁻¹(]−

∞, r2+r3[).

Theorem 2.1 is a counter-part of a three critical point theorem by Ricceri [35, 36], which extends previous results by Pucci and Serrin [29, 30].

We refer the interested reader to the papers [6, 15, 20] in which Theorem 2.1 has been successfully used to ensure the existence of at least three solutions for boundary value problems.

For the reader’s convenience, we recall some background facts concerning the Lebesgue-Sobolev spaces with variable exponent and introduce some notation. For more details, we refer the reader to [31, 32]. Set

C₊(Ω) :={h:h∈C(Ω) andh(x)>1,∀x∈Ω}.

Forp(·)∈C+(Ω), define the variable exponent Lebesgue spaceL^p(·)(Ω) by L^p(·)(Ω) :={u: Ω→Rmeasurable and

Z

Ω

|u(x)|^p(x)dx <∞}.

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

|u|_p(·)= inf{β >0 : Z

Ω

|u(x)

β |^p(x)dx≤1}

and (L^p(·)(Ω),|u|_p(·)) becomes a Banach space, and we call it variable exponent Lebesgue space. Define the variable exponent Sobolev spaceW^m,p(·)(Ω) by

W^m,p(·)(Ω) ={u∈L^p(·)(Ω)|D^αu∈L^p(·)(Ω),|α| ≤m}

where

D^αu= ∂^|α|

∂x^α₁¹· · ·∂x^α_N^Nu withα= (α₁, . . . , α_N) is a multi-index and|α|=PN

i=1α_i. The spaceW^m,p(·)(Ω), equipped with the norm

kuk_m,p(·):= X

|α|≤m

|D^αu|_p(·),

becomes a separable, reflexive and uniformly convex Banach space. We denote by X^∗its dual.

We denote

X :=W^2,p(·)(Ω)∩W₀^1,p(·)(Ω) whereW₀^m,p(·)(Ω) denote the closure of C₀^∞(Ω) inW^m,p(·)(Ω).

Foru∈X, we define

kuk= inf{β >0 : Z

Ω

|∆u(x)

β |^p(x)dx≤1}.

Clearly, we observe thatXendowed with the above norm is a separable and reflexive Banach space.

Remark 2.2. From [41], the normkuk_2,p(·) is equivalent to the norm|∆u|p(·) in the space X. Consequently, the normskuk_2,p(·), kuk and |∆u|p(·) are equivalent.

For the rest of this article, we usekukinstead ofkuk_2,p(·)onX.

Proposition 2.3 ([34]). The conjugate space of L^p(·)(Ω) isL^q(·)(Ω) where q(·)is the conjugate function of p(·); i.e.,

1 p(·)+ 1

q(·) = 1.

Foru∈L^p(·)(Ω)andv∈L^q(·)(Ω), we have

Z

Ω

u(x)v(x)dx ≤( 1

p⁻ + 1

q⁻)|u|_p(·)|v|_q(·)≤2|u|_p(·)|v|_q(·). Proposition 2.4 ([34]). Let ρ(u) =R

Ω|u|^p(x)dx. For u, u_n ∈L^p(·)(Ω), we have

(1) |u|_p(·)< (=;>); 1⇔ρ(u)< (=;>) 1;

(2) |u|_p(·)>1⇒ |u|^p_p(·)⁻ ≤ρ(u)≤ |u|^p_p(·)⁺; (3) |u|_p(·)<1⇒ |u|^p_p(·)⁺ ≤ρ(u)≤ |u|^p_p(·)⁻; (4) |un|_p(·)→0⇔ρ(u_n)→0;

(5) |un|_p(·)→ ∞ ⇔ρ(u_n)→ ∞.

From Proposition 2.4, foru∈L^p(·)(Ω) the following inequalities hold:

kuk^p− ≤ Z

Ω

|∆u|^p(x)dx≤ kuk^p+, ifkuk ≥1, (2.1) kuk^p+≤

Z

Ω

|∆u|^p(x)dx≤ kuk^p−, ifkuk ≤1. (2.2) Proposition 2.5 ([38]). If Ω ⊂ R^N is a bounded domain, then the embedding X ,→C⁰(Ω) is compact whenever ^N₂ < p⁻.

From Proposition 2.5, there exists a positive constant c depending onp(·), N and Ω such that

kuk_∞= sup

x∈Ω

|u(x)| ≤ckuk, ∀u∈X. (2.3) Corresponding to f and g we introduce the functions F : Ω×R → R and G : Ω×R→R, as follows

F(x, t) :=

Z t 0

f(x, ξ)dξ for (x, t)∈Ω×R, G(x, t) :=

Z t 0

f(x, ξ)dξ for (x, t)∈Ω×R. We say that a functionu∈X is a weak solution of (1.1) if Z

Ω

|∆u(x)|^p(x)−2∆u(x)∆v(x)dx−λ Z

Ω

f(x, u(x))v(x)dx−µ Z

Ω

g(x, u(x))v(x)dx= 0 holds for allv∈X.

In the sequel meas(Ω) denotes the Lebesgue measure of the set Ω.

3. Main results

In this section, we formulate our main results on the existence of at least three weak solutions for problem (1.1). For our convenience, set

G^θ:=

Z

Ω

max

|ξ|≤θG(x, ξ)dx forθ >0, Gη := meas(Ω) inf

Ω×[0,η]

G(x, t) forη >0.

Ifg is sign-changing, then clearlyG^θ≥0 andGη≤0.

Fixx⁰∈Ω and choose s1, s2 with 0< s1 < s2, such that B(x⁰, s2)⊆Ω where B(x, s) stands for the open ball inR^N of radiussand centerx. Let

σ:= 2c^p−π^N2(s^N₂ −s^N₁ ) NΓ(^N₂)

×maxn12(N+ 2)²(s₁+s₂) (s2−s1)³

p⁻

,12(N+ 2)²(s₁+s₂) (s2−s1)³

p⁺o

and

ρ:= 2c^p−π^N2(s^N₂ −s^N₁) NΓ(^N₂)

×minn

12(N+ 2)²(s1+s2) (s₂−s₁)³

^p−

,12(N+ 2)²(s1+s2) (s₂−s₁)³

^p+o . Fixing four positive constantsθ1,θ2,θ3 andη≥1, we put

δ_λ,g := minn 1

p⁺c^p− minnθ₁^p−−λp⁺c^p−R

ΩF(x, θ1)dx G^θ1

,θ^p₂⁻−λp⁺c^p−R

ΩF(x, θ₂)dx

G^θ2 ,(θ^p₃⁻−θ₂^p−)−λp⁺c^p−R

ΩF(x, θ₃)dx G^θ3

o

,

ση^p+ p⁻c^p− −λ R

B(x⁰,s₁)F(x, η)dx−R

ΩF(x, θ1)dx G_η−G^θ1

o .

(3.1)

Theorem 3.1. Assume that there exist positive constantsθ1, θ2,θ3andη≥1with θ1< ρ^p1−η,η <min{(_σp^p+−)^p1+θ^p

−/p⁺

2 , θ2} andθ2< θ3 such that (A6) f(x, t)≥0 for each(x, t)∈Ω×[−θ₃, θ₃];

(A7)

maxnR

ΩF(x, θ₁)dx θ^p₁⁻ ,

R

ΩF(x, θ₂)dx θ^p₂⁻ ,

R

ΩF(x, θ₃)dx θ^p₃⁻−θ₂^p−

o

< p⁻ p⁺σ

R

B(x⁰,s1)F(x, η)dx−R

ΩF(x, θ1)dx

η^p+ .

Then, for every

λ∈Λ :=

ση^p+ p⁻c^p−

R

B(x⁰,s₁)F(x, η)dx−R

ΩF(x, θ1)dx, 1

p⁺c^p− minn θ₁^p− R

ΩF(x, θ1)dx, θ^p₂⁻ R

ΩF(x, θ2)dx, θ^p₃⁻−θ₂^p− R

ΩF(x, θ3)dx o

and for every non-negative continuous functiong: Ω×R→R, there existsδλ,g >0 given by (3.1)such that, for eachµ∈[0, δλ,g), problem(1.1)has at least three weak solutions u1, u2 and u3 such that max_x∈Ω|u1(x)| < θ1, max_x∈Ω|u2(x)|< θ2 and maxx∈Ω|u3(x)|< θ3.

Proof. Our goal is to apply Theorem 2.1 to problem (1.1). We consider the auxiliary problem

∆²_p(x)u=λfˆ(x, u(x)) +µg(x, u(x)), x∈Ω,

u= ∆u= 0, x∈∂Ω (3.2)

where ˆf ∈C⁰(Ω×R) defined setting fˆ(x, ξ) =







f(x,0), ifξ <−θ3, f(x, ξ), if −θ₃≤ξ≤θ₃, f(x, θ3), ifξ > θ3.

If a weak solution of (3.2) satisfies the condition−θ3≤u(x)≤θ3 for everyx∈Ω, then, clearly it turns to be also a weak solution of (1.1). Therefore, for our goal, it is sufficient to show that our conclusion holds for (1.1). Consider the functionals Φ,Ψ for everyu∈X, defined by

Φ(u) = Z

Ω

1

p(x)|∆u(x)|^p(x)dx, (3.3) Ψ(u) =

Z

Ω

F(x, u(x))dx+µ λ Z

Ω

G(x, u(x))dx. (3.4)

Let us prove that the functionals Φ and Ψ satisfy the required conditions in Theorem 2.1. It is well known that Ψ is a differentiable functional whose differential at the pointu∈X is

Ψ⁰(u)(v) = Z

Ω

f(x, u(x))v(x)dx+µ λ Z

Ω

g(x, u(x))v(x)dx

for everyv∈X, as well as it is sequentially weakly upper semicontinuous. Recalling (2.1), we have

Φ(u)≥ 1

p⁺kuk^p−,

for allu∈Xwithkuk>1, which implies Φ is coercive. Moreover, Φ is continuously differentiable whose differential at the pointu∈X is

Φ⁰(u)(v) = Z

Ω

|∆u(x)|^p(x)−2∆u(x)∆v(x)dx

for every v ∈ X. Also, Φ⁰ : X → X^∗ is a compact operator (see [38, Lemma 3.1]). Furthermore, Φ is sequentially weakly lower semicontinuous. Therefore, we observe that the regularity assumptions on Φ and Ψ, as requested of Theorem 2.1, are satisfied. Definewby setting

w(x) :=











0, x∈Ω\B(x⁰, s2),

η[3(l⁴−s⁴₂)−4(s1+s₂)(l³−s³₂)+6s₁s₂(l²−s²₂)]

(s₂−s1)³(s₁+s₂) , x∈B(x⁰, s2)\B(x⁰, s1),

η, x∈B(x⁰, s₁)

wherel= dist(x, x⁰) = q

PN

i=1(xi−x⁰_i)². Then

∂w(x)

∂xi

=











0, ifx∈Ω\B(x⁰, s2)∪B(x⁰, s1),

12η[l²(xi−x⁰_i)−l(s1+s2)(xi−x⁰_i)+s1s2(xi−x⁰_i)]

(s₂−s1)³(s₁+s₂) , ifx∈B(x⁰, s2)\B(x⁰, s1),

∂²w(x)

∂x²_i =











0, ifx∈Ω\B(x⁰, s2)∪B(x⁰, s1),

12η[s1s2+(2l−s1−s2)(xi−x⁰_i)²/l−(s1+s2−l)l]

(s2−s1)³(s1+s2) , ifx∈B(x⁰, s₂)\B(x⁰, s₁),

N

X

i=1

∂²w(x)

∂x²_i =











0, ifx∈Ω\B(x⁰, s2)∪B(x⁰, s1),

12η[(N+2)l²−(N+1)(s₁+s₂)l+N s₁s₂] (s₂−s1)³(s₁+s₂) , ifx∈B(x⁰, s2)\B(x⁰, s1).

So, one has ρη^p−

p⁺c^p− ≤Φ(w) = Z

B(x⁰,s₂)\B(x⁰,s₁)

1

p(x)|∆w(x)|^p(x)dx≤ ση^p+ p⁻c^p−.

On the other hand, bearing (A6) in mind and since g is non-negative, from the definition of Ψ, we infer

Ψ(w) = Z

Ω

h

F(x, w(x)) +µ

λG(x, w(x))i dx≥

Z

B(x⁰,s₁)

F(x, η)dx.

Choose r1 = _p¹+

θ1

c

p⁻

, r2 = _p¹+

θ2

c

p⁻

and r3 = _p¹+

θ^p₃⁻−θ₂^p− c^p−

. From the condi- tionsθ1< ρ^p1−η,η <(_σp^p+−)^p1+θ^p

−/p⁺

2 and θ2< θ3, we achieve r1<Φ(w)< r2 and r3>0. For allu∈X with Φ(u)< r1, taking (2.1) and (2.2) into account, one has

kuk ≤max

(p⁺r₁)^p1+,(p⁺r₁)^p1− .

So, thanks to the embeddingX ,→C⁰(Ω) (see (2.3)), one has kuk_∞ < θ1. From the definition ofr1, it follows that

Φ⁻¹(−∞, r1) ={u∈X; Φ(u)< r1} ⊆ {u∈X;|u| ≤θ1}.

Hence, by using assumption (A6), one has sup

u∈Φ⁻¹(−∞,r1)

Z

Ω

F(x, u(x))dx≤ Z

Ω

sup

|t|≤θ1

F(x, t)dx≤ Z

Ω

F(x, θ₁)dx.

As above, we can obtain that sup

u∈Φ⁻¹(−∞,r2)

Z

Ω

F(x, u(x))dx≤ Z

Ω

F(x, θ2)dx, sup

u∈Φ⁻¹(−∞,r2+r₃)

Z

Ω

F(x, u(x))dx≤ Z

Ω

F(x, θ3)dx.

Therefore, since 0∈Φ⁻¹(−∞, r₁) and Φ(0) = Ψ(0) = 0, one has ϕ(r1) = inf

u∈Φ⁻¹(−∞,r1)

(sup_u∈Φ−1(−∞,r1)Ψ(u))−Ψ(u) r₁−Φ(u)

≤ sup_u∈Φ−1(−∞,r1)Ψ(u) r₁

= sup_u∈Φ−1(−∞,r1)

R

Ω[F(x, u(x)) +^µ_λG(x, u(x))]dx r1

≤ R

ΩF(x, θ1)dx+^µ_λG^θ1

1 p⁺

θ₁ c

^p− ,

ϕ(r2)≤ sup_u∈Φ−1(−∞,r2)Ψ(u) r2

= sup_u∈Φ−1(−∞,r2)

R

Ω[F(x, u(x)) +^µ_λG(x, u(x))]dx r2

≤ R

ΩF(x, θ2)dx+^µ_λG^θ2

1 p⁺

θ₂ c

^p− and

γ(r2, r3)≤ sup_u∈Φ−1(−∞,r2+r3)Ψ(u) r₃

= sup_u∈Φ−1(−∞,r2+r3)

R

Ω[F(x, u(x)) +^µ_λG(x, u(x))]dx r3

≤ R

ΩF(x, θ3)dx+^µ_λG^θ3

1 p⁺

_θp− 3 −θ^p₂⁻

c^p−

.

On the other hand, for eachu∈Φ⁻¹(−∞, r1) one has β(r1, r2)≥

R

B(x⁰,s₁)F(x, η)dx−R

ΩF(x, θ1)dx+^µ_λ(Gη−G^θ1) Φ(w)−Φ(u)

≥ R

B(x⁰,s1)F(x, η)dx−R

ΩF(x, θ₁)dx+^µ_λ(G_η−G^θ1)

ση^p+ p⁻c^p−

.

From (A7) we obtain α(r₁, r₂, r₃) < β(r₁, r₂). Finally, we verify that Φ−λΨ satisfies assumption (A2) of Theorem 2.1. Let u₁ andu₂ be two local minima for Φ−λΨ. Then u₁ and u₂ are critical points for Φ−λΨ, and so, they are weak solutions of (1.1). Since we assumedf is nonnegative and since g is non-negative, for fixedλ >0 andµ≥0 we have (λf+µg)(x, su1+ (1−s)u2)≥0 for alls∈[0,1], and consequently, Ψ(su1+ (1−s)u2)≥0, for everys∈[0,1]. Hence, Theorem 2.1 implies that for every

λ∈

ση^p+ p⁻c^p−

R

B(x⁰,s1)F(x, η)dx−R

ΩF(x, θ1)dx, 1

p⁺c^p− minn θ^p₁⁻ R

ΩF(x, θ₁)dx, θ₂^p− R

ΩF(x, θ₂)dx, θ₃^p−−θ^p₂⁻ R

ΩF(x, θ₃)dx o

andµ∈[0, δ_λ,g), the functional Φ−λΨ has three critical pointsu_i,i= 1,2,3,inX such that Φ(u1)< r1, Φ(u2)< r2and Φ(u3)< r2+r3, that is, max_x∈Ω|u1(x)|< θ1, max_x∈Ω|u2(x)|< θ2 and max_x∈Ω|u3(x)|< θ3. Then, taking into account the fact that the solutions of problem (1.1) are exactly critical points of the functional

Φ−λΨ we have the desired conclusion.

The following example illustrates the result of Theorem 3.1.

Example 3.2. Let Ω ={(x, y)∈R²:x²+y²≤2}. Consider the problem (∆²_p(x,y)u=λf(u) +µg(u), (x, y)∈Ω,

u= ∆u= 0, (x, y)∈∂Ω (3.5)

wherep(x, y) =x²+y²+ 2 for all (x, y)∈Ω and f(t) =

(5t⁴, ift≤1,

√5

t, ift >1.

By the expression off we have F(t) =

(t⁵, ift≤1, 10√

t−9, ift >1.

Direct calculations give p⁻ = 2 and p⁺ = 4. By choosing x₀ = 0, s₁ = 1 and s₂= 2, we obtainσ= 3⁹×2²⁴πc² andρ= 3⁵×2¹²πc². We consider two cases for

c. First, suppose that c≤1. Choosingη= 1, θ1= 10⁻⁸c,θ2= ^10√12

2 and θ3= 10¹² we see that

maxnmeas(Ω)F(θ₁)

θ₁² , meas(Ω)F(θ₂)

θ₂² , meas(Ω)F(θ₃) θ₃²−θ²₂

o

=8×10⁷π−72π 10²⁴

< 1

3⁹×2²⁵πc²(π−4×10⁻²⁴c³π)

= p⁻ p⁺σ

R

B(x⁰,s1)F(x, η)dx−R

ΩF(x, θ₁)dx

η^p+ ,

which means the assumption (A7) is satisfied. It is easy to see that other assump- tions of Theorem 3.1 are also fulfilled. Therefore, in this case, it follows that for every

λ∈ 3⁹×2²³π

π−4×10⁻²⁴c³π, 10²⁴

32×10⁷πc²−288πc²

and for every non-negative continuous function g : R → R, there exists ˆδ > 0 such that for each µ ∈ [0,δ), then problem (3.5) has at least three weak solu-ˆ tionsu1,u2 andu3 such that max_x∈Ω|u1(x)|<10⁻⁸c, max_x∈Ω|u2(x)|< ^10√12

2 and maxx∈Ω|u3(x)|<10¹².

Now, suppose that c > 1. Choosing η = 1, θ₁ = ¹⁰_c⁻⁸, θ₂ = ^10√12

2 c^3/2 and θ₃= 10¹²c^3/2, we have

maxnmeas(Ω)F(θ₁)

θ₁² , meas(Ω)F(θ₂)

θ₂² , meas(Ω)F(θ₃) θ₃²−θ²₂

o

=8×10⁷πc³⁴ −72π 10²⁴c³

< 1

3⁹×2²⁵πc²(π−4×10⁻²⁴π c³ )

= p⁻ p⁺σ

R

B(x⁰,s1)F(x, η)dx−R

ΩF(x, θ₁)dx

η^p+ ,

which means the assumption (A7) is fulfilled. Clearly, other assumptions of Theo- rem 3.1 in this case are satisfied too. Then, in this case, it follows for every

λ∈ 3⁹×2²³π

π−^4×10_c3^−24π, 10²⁴c³

32×10⁷πc¹¹⁴ −288πc²

and for every non-negative continuous function g : R → R, there exists δ > 0 such that for each µ ∈ [0, δ), the problem (3.5) has at least three weak solutions u1, u2 and u3 such that maxx∈Ω|u1(x)| < ¹⁰_c⁻⁸, maxx∈Ω|u2(x)| < ^10√12₂c^3/2 and max_x∈Ω|u3(x)|<10¹²c^3/2.

For given positive constantsθ1, θ4andη≥1, we set δ⁰_λ,g:= minn 1

p⁺c^p−minnθ^p₁⁻−p⁺c^p−λR

ΩF(x, θ1)dx

G^θ1 ,

θ^p₄⁻−2p⁺c^p−λR

ΩF(x, ¹

p−√ 2

θ₄)dx 2G

1 p−√

2

θ₄ ,θ^p₄⁻−2p⁺c^p−λR

ΩF(x, θ4)dx 2G^θ4

o ,

ση^p+ p⁻c^p− −λ R

B(x⁰,s1)F(x, η)dx−R

ΩF(x, θ₁)dx Gη−G^θ1

o .

(3.6)

Now, we deduce the following straightforward consequence of Theorem 3.1.

Theorem 3.3. Assume that there exist positive constants θ₁,θ₄ and η ≥1 with θ1<min{η^p+/p−, ρ^p1−η} andη <min{(_2σp^p+−)^p1+θ^p

−/p⁺

4 , θ4}such that (A8) f(x, t)≥0 for each(x, t)∈Ω×[−θ4, θ4];

(A9) maxnR

ΩF(x, θ1)dx θ^p₁⁻

, 2R

ΩF(x, θ4)dx θ₄^p−

o

< p⁻ p⁺σ+p⁻

R

B(x⁰,s1)F(x, η)dx

η^p+ .

Then, for every

λ∈Λ⁰:= (p⁺σ+p⁻)η^p+ p⁻p⁺c^p−R

B(x⁰,s1)F(x, η)dx, 1

p⁺c^p−minn θ^p₁⁻ R

ΩF(x, θ₁)dx, θ₄^p− 2R

ΩF(x, θ₄)dx o

and for every non-negative continuous functiong: Ω×R→R, there existsδ_λ,g⁰ >0 given by (3.6)such that, for eachµ∈[0, δ_λ,g⁰ ), problem(1.1)has at least three weak solutions u1,u2 and u3 such that max_x∈Ω|u1(x)| < θ1, max_x∈Ω|u2(x)| < _p−¹√

2θ4

andmax_x∈Ω|u₃(x)|< θ₄. Proof. Chooseθ₂= ¹

p−√

2θ₄and θ₃=θ₄. So, from (A9) one has R

ΩF(x, θ₂)dx θ^p₂⁻

= 2R

ΩF(x, _p−¹√ 2θ4)dx θ^p₄⁻ ≤ 2R

ΩF(x, θ₄)dx θ^p₄⁻

< p⁻ p⁺σ+p⁻

R

B(x⁰,s₁)F(x, η)dx η^p+

(3.7)

and R

ΩF(x, θ3)dx θ₃^p−−θ^p₂⁻

= 2R

ΩF(x, θ4)dx θ^p₄⁻

< p⁻ p⁺σ+p⁻

R

B(x⁰,s1)F(x, η)dx

η^p+ . (3.8)

Moreover, sinceθ₁< η^p+/p−, from (A9) we have p⁻

p⁺σ R

B(x⁰,s₁)F(x, η)dx−R

ΩF(x, θ1)dx η^p+

> p⁻ p⁺σ

R

B(x⁰,s1)F(x, η)dx

η^p+ − p⁻

p⁺σ R

ΩF(x, θ1)dx θ^p₁⁻

> p⁻ p⁺σ

R

B(x⁰,s₁)F(x, η)dx

η^p+ − (p⁻)² p⁺σ(p⁺σ+p⁻)

R

B(x⁰,s₁)F(x, η)dx η^p+

= p⁻

p⁺σ+p⁻ R

B(x⁰,s1)F(x, η)dx

η^p+ .

Hence, from (A9), (3.7) and (3.8), it is easy to observe that the assumption (A7) of Theorem 3.1 is satisfied, and it follows the conclusion.

Remark 3.4. We observe that, in our results, no asymptotic conditions onf and g are needed and only algebraic conditions on f are imposed to guarantee the existence of solutions. Moreover, in the conclusions of the above results, one of the three solutions may be trivial since the values off(x,0) andg(x,0) forx∈Ω are not determined.

Here, we want to point out a simple consequence of Theorem 3.3 when f does not depend uponxandµ= 0. To be precise, consider the problem

∆²_p(x)u=λf(u(x)), x∈Ω,

u= ∆u= 0, x∈∂Ω (3.9)

wheref :R→Ris a continues function. Put F(t) =

Z t 0

f(ξ)dξ fort∈R.

Theorem 3.5. Let f be a non-negative and nonzero function such that lim

t→0⁺

f(t)

|t|^p−−1 = lim

t→+∞

f(t)

|t|^p−−1 = 0. (3.10)

Then, for everyλ > λ^∗ where

λ^∗= inf (p⁺σ+p⁻)η^p+

p⁻p⁺c^p−meas(B(x⁰, s₁))F(η) :η≥1, F(η)>0 problem (3.9)has at least two non-trivial weak solutions.

Proof. Fixλ > λ^∗ and letη ≥1 such thatF(η)>0 and λ > (p⁺σ+p⁻)η^p+

p⁻p⁺c^p−meas(B(x⁰, s₁))F(η). From (3.10) there isθ₁>0 such that

θ1<min{η^p+/p−, ρ^p1−η}and F(θ1) θ₁^p−

< 1

λmeas(Ω)p⁺c^p−, andθ4>0 such that

η <min ( p⁺

2σp⁻)^p1+θ^p

−/p⁺

4 , θ4 , F(θ4)

θ₄^p− < 1

2λmeas(Ω)p⁺c^p−.

Therefore, all assumptions of Theorem 3.3 are fulfilled and it ensures the conclusion.

Finally, we present an example in which the hypotheses of Theorem 3.5 are satisfied.

Example 3.6. We consider the problem

∆²_p(x,y)u=λf(u), (x, y)∈Ω,

u= ∆u= 0, (x, y)∈∂Ω (3.11)

where Ω ={(x, y)∈R²:x²+y²≤2},p(x, y) =x²+y²+ 2 for (x, y)∈Ω and f(t) =

(4t³, ift≤1,

√4

t, ift >1.

A direct calculation shows that F(t) =

(t⁴, ift≤1, 8√

t−7, ift >1.

By simple calculations, we obtain p⁻ = 2 and p⁺ = 4. Choosing x0 = 0, s1 = 1, s2 = 2 and η = 1, we observe that all assumptions of Theorem 3.5 are fulfilled.

Therefore, it follows that for every

λ > 2²⁶×3⁹πc²+ 2 8πc² ,

problem (3.11) has at least two distinct non-trivial weak solutions.

Acknowledgements. This article was written while the first author was visiting Department of Economics at University of Messina in March 2016. He expresses his gratitude to the department for warm hospitality.

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Shapour Heidarkhani

Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran

E-mail address:[email protected]

Ghasem A. Afrouzi

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazan- daran, Babolsar, Iran

E-mail address:[email protected]

Shahin Moradi

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazan- daran, Babolsar, Iran

E-mail address:[email protected]

Giuseppe Caristi

Department of Economics, University of Messina, via dei Verdi, 75, Messina, Italy E-mail address:[email protected]