Open Access. © 2020 Martin Bohner et al., published by De Gruyter. This work is licensed under the Creative Commons Attribu-tion alone 4.0 License.

Nonauton. Dyn. Syst. 2020; 7:53–64

Research Article Open Access

Martin Bohner*, Giuseppe Caristi, Fariba Gharehgazlouei, and Shapour Heidarkhani

Existence and Multiplicity of Weak Solutionsfor a Neumann Elliptic Problem with~p(x)-Laplacianhttps://doi.org/10.1515/msds-2020-0108Received May 6, 2020; accepted June 28, 2020

Abstract:We are interested in the existence of multiple weak solutions for the Neumann elliptic problem in-volving the anisotropic ~p(x)-Laplacian operator, on a bounded domain with smooth boundary. We work onthe anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theo-rem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on thenonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraicconditions including the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, byemploying a three critical point theorem due to Bonanno and Marano, we guarantee the existence of at leastthree weak solutions for the problem in a special case.

Dedicated to the Memory of Professor Constantin Corduneanu(July 26, 1928 - December 26, 2018)

Keywords: ~p(x)-Laplacian operator, Neumann elliptic problem, weak solution, variational principle,anisotropic variable exponent Sobolev space.

MSC: 34C27, 34K14, 35B15, 35K57, 37A30

1 IntroductionOur study is conducted in the framework of the anisotropic variable exponent Lebesgue–Sobolev space. Inthis article, we are interested in the existence of multiple weak solutions for the Neumann ~p(x)-elliptic prob-lem of the type {

−∆~p(x)u + a(x)|u|p0(x)−2u = λf (x, u) + µg(x, u) in Ω,∂u∂γ = 0 on ∂Ω,

(1.1)

where Ω ⊂ RN is a bounded open set with boundary ∂Ω of class C1, and where ~ν is the outward unit normalto ∂Ω. Let a(·) ∈ L∞(Ω), a0 := ess infx∈Ω a(x) > 0, suppose f , g : Ω ×R→ R are Carathéodory functions, andλ, µ > 0 are parameters.

In the last few decades, one of the topics from the �eld of partial di�erential equations that has contin-uously attracted interest is that concerning the Sobolev space with variable exponents, W1,p(·)(Ω), where pis a function depending on x, see, for example, the monograph [16] and the references therein. Naturally,

*Corresponding Author: Martin Bohner:Missouri S&T, Department of Mathematics and Statistics, Rolla, MO 65409, USA,E-mail: bohner@mst.eduGiuseppe Caristi: University of Messina, Department of Economics, Messina, Italy, E-mail: gcaristi@unime.itFariba Gharehgazlouei, Shapour Heidarkhani: Razi University, Department of Mathematics, Faculty of Sciences, 67149 Ker-manshah, Iran, E-mail: f.gharehgazloo@yahoo.com and sh.heidarkhani@razi.ac.ir

https://doi.org/10.1515/msds-2020-0108

54 | M. Bohner, G. Caristi, F. Gharehgazlouei, S. Heidarkhani

problems involving the p(·)-Laplace operator

∆p(x)u = div(|∇u|p(x)−2∇u)

were intensively studied. At the same time, due to the development of the theory regarding the anisotropicSobolev space, the anisotropic variable exponent Sobolev spaceW1,~p(·)(Ω)has captured the attention ofmanyresearchers, and a new operator has taken its place in the mathematical literature, namely

∆~p(x)u =N∑i=1

∂∂xi

(∣∣∣∣ ∂u∂xi∣∣∣∣pi(x)−2 ∂u∂xi

).

It is clear that this ~p(·)-Laplace operator is a generalization of the p(·)-Laplace operator. The interest in trans-posing the problems into new problems with variable exponents is linked to a large scale of applications thatare involving some nonhomogeneous materials. It was established that for an appropriate treatment of thesematerials we cannot rely on the classical Sobolev space, and that we have to allow the exponent to vary in-stead.Workingwith variable exponents, henceworking in the framework of variable exponent spaces, opensthe door for multiple applications. We can refer here to electrorheological �uids or to thermorheological �u-ids that havemultiple applications to hydraulic valves and clutches, brakes, shock absorbers, robotics, spacetechnology, tactile displays etc. (see [5, 28] and their references). In addition, the variable exponent spacesare involved in studies hat provide other types of applications, e.g., in image restoration [15] and contact me-chanics [13]. Recently, this theory has been expanded by many researchers, see [3, 18]. For example, Fan andJi have treated in [19] the problem{

−∆p(x)u + a(x)|u|p(x)−2u = f (x, u) + g(x, u) in Ω,∂u∂γ = 0 on ∂Ω,

and they proved the existence of in�nitely many weak solutions of the problem under weaker hypothesesby applying a variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spacesW1,p(·)(Ω). Ahmed, Hjiaj, and Touzani have studied in [3] the Neumann ~p(x)-elliptic problem{

−∆~p(x)u + a(x)|u|p0(x)−2u = f (x, u) + g(x, u) in Ω,∂u∂γ = 0 on ∂Ω,

and they proved the existence of in�nitelymanyweak solutions in the anisotropic variable exponent Sobolevspace W1,~p(·)(Ω) under some hypotheses. For other related results, we refer to [2, 4, 11, 17, 20–22, 25–27, 29].

Here, we deal with the problem (1.1) when the nonlinearity f satis�es a subcritical growth condition.Via variational methods, we obtain the existence of at least one, two, and three weak solutions wheneverthe parameters λ and µ are in explicitly given positive intervals. The main tools are critical points theoremsestablished in [9, 10, 12]. Variants of such theorems have been successfully applied for other problems, see,e.g., [1, 7, 8].

This paper is organized as follows: In Section 2, we present some preliminary knowledge on theanisotropic Sobolev spaces with variable exponent. Section 3 contains the main results and the proofs ofthe main results. For our Neumann elliptic problem, we prove the existence of one weak solution in Theorem3.1, the existence of two weak solutions in Theorem 3.3, and the existence of three weak solutions in Theorem3.4.

2 PreliminariesIn this section, we introduce some de�nitions and results which will be used in the next section.

Let Ω be an open bounded subset of RN , N ≥ 1. We de�ne

C+ ={p(·) : Ω → R is measurable : 1 < p− ≤ p+ < ∞

},

Neumann Elliptic Problems | 55

wherep− = ess inf{p(x) : x ∈ Ω} and p+ = ess sup{p(x) : x ∈ Ω}.

We de�ne the variable exponent Lebesgue space Lp(·)(Ω) as the set of all measurable functions u : Ω → R forwhich the convex modular

ρp(x)(u) =∫Ω

|u|p(x)dx

is �nite. We de�ne a norm, the so-called Luxemburg norm, on this space by the formula

‖u‖p(·) = inf{γ > 0 : ρp(·)

(uγ

)≤ 1}.

The space(

Lp(·)(Ω), ‖ · ‖p(·))is a separable and re�exive Banach space. Moreover, the space Lp(·)(Ω) is uni-

formly convex, hence re�exive, and its dual space is isomorphic to Lp′(·)(Ω), where 1p(·) +

1p′(·) = 1. Finally, we

have the Hölder type inequality: ∣∣∣∣∣∣∫Ω

uvdx

∣∣∣∣∣∣ ≤(

1p− +

1(p′)−

)‖u‖p(·)‖v‖p′(·)

for all u ∈ Lp(·)(Ω) and v ∈ Lp′(·)(Ω). An important role in manipulating the generalized Lebesgue spaces is

played by the ρp(·)-modular of the space Lp(·)(Ω). We have the following result.

Proposition 2.1 (See [31]). If u ∈ Lp(·)(Ω), un ∈ Lp(·)(Ω), and p+ < ∞, then(i) ‖u‖p(·) > 1 implies ‖u‖

p−p(·) ≤ ρp(·)(u) ≤ ‖u‖

p+p(·),

(ii) ‖u‖p(·) < 1 implies ‖u‖p+p(·) ≤ ρp(·)(u) ≤ ‖u‖

p−p(·),

(iii) limn→∞ ‖un − u‖p(·) = 0 i� limn→∞ ρp(·)(un − u) = 0.

We de�ne the variable exponent Sobolev space W1,p(·)(Ω) by

W1,p(·)(Ω) ={u ∈ Lp(·)(Ω) : |∇u| ∈ Lp(·)(Ω)

},

equipped with the norm‖u‖1,p(·) = ‖u‖p(·) + ‖∇u‖p(·).

The space(

W1,p(·)(Ω), ‖ · ‖1,p(·))is a separable and re�exive Banach space. Now, we present the anisotropic

variable exponent Sobolev space used in the study of the main problem. Let p0(·), p1(·), . . . , pN(·) be N + 1variable exponents in C+(Ω). We denote

~p(·) = {p0(·), p1(·), . . . , pN(·)}, D0u = u, Diu =∂u∂xi

.

We de�nep = min{p−i : i = 0, 1, . . . , N}, so p > 1

andp = max{p+i : i = 0, 1, . . . , N}.

The anisotropic variable exponent Sobolev space W1,~p(·)(Ω) is de�ned by

W1,~p(·)(Ω) ={u ∈ Lp0(·)(Ω) : Diu ∈ Lpi(·)(Ω) for i = 1, 2, . . . , N

},

endowed with the norm

‖u‖ = ‖u‖1,~p(·) =N∑i=0

∥∥∥Diu∥∥∥pi(·)

.

The space(

W1,~p(·)(Ω), ‖ · ‖)is a separable and re�exive Banach space (cf. [6, 24]). Throughout the rest of this

paper, we assumep > N .

56 | M. Bohner, G. Caristi, F. Gharehgazlouei, S. Heidarkhani

Remark 2.2. Since W1,~p(·)(Ω) is continuously embedded in W1,p(Ω), and since W1,p(Ω) is compactly embed-ded in C0(Ω) (the space of continuous functions), the continuous embedding of W1,~p(x)(Ω) in C0(Ω) is com-pact.

We setc0 = sup

u∈W1,~p(·)(Ω)\{0}

‖u‖∞‖u‖1,~p(·)

.

Assume that f , g : Ω ×R→ R are Carathéodory functions satisfying

sup|t|≤s|f (·, t)| ∈ L1(Ω) and sup

|t|≤s|g(·, t)| ∈ L1(Ω) for each s > 0.

We set

F(x, t) =t∫

0

f (x, ξ )dξ and G(x, t) =t∫

0

g(x, ξ )dξ for all (x, t) ∈ Ω ×R.

We de�ne, for any u ∈ X := W1,~p(x)(Ω), the functionals Φ, Ψλ,µ : X → R by

Φ(u) :=N∑i=1

∫Ω

1pi(x)

∣∣∣∣ ∂u∂xi∣∣∣∣pi(x) dx + ∫

Ω

a(x)p0(x)

|u|p0(x)dx (2.1)

andΨλ,µ(u) :=

∫Ω

F(x, u)dx + µλ

∫Ω

G(x, u)dx. (2.2)

We say that a function u ∈ X is a weak solution of (1.1) if

N∑i=1

∫Ω

∣∣∣∣ ∂u∂xi∣∣∣∣pi(x)−2 ∂u∂xi ∂v∂xi dx +

∫Ω

a(x)|u|p0(x)−2uvdx

− λ∫Ω

f (x, u)vdx − µ∫Ω

g(x, u)vdx = 0 for all v ∈ X.

As in [3, Proposition 4.1], using our assumptions, we need the following proposition in the proof of Theorem3.1.

Proposition 2.3. The functional Φ(u) de�ned in (2.1) is coercive, that is,

Φ(u)→∞ as ‖u‖ →∞ and u ∈ X.

Proof. We have

Φ(u) =N∑i=1

∫Ω

1pi(x)

∣∣∣∣ ∂u∂xi∣∣∣∣pi(x) dx + ∫

Ω

a(x)p0(x)

|u|p0(x)dx

≥N∑i=1

1p+i

(∥∥∥∥ ∂u∂xi∥∥∥∥ppi(x)

− 1)

+ a0p+0

(‖u‖pp0(x) − 1

)≥ min{1, a0}

p(N + 1)p−1‖u‖p − N + a0p

≥ K1‖u‖p − K2,

where K1, K2 > 0 are constants. Thus, if ‖u‖ →∞, then Φ(u)→∞.

Neumann Elliptic Problems | 57

De�nition 2.4. LetΦ andΨ be two continuouslyGâteaux-di�erentiable functionals de�nedona real Banachspace X. Fix r ∈ R. The functional I = Φ −Ψ is said to verify the Palais–Smale condition cut o� upper at r, inshort (PS)[r], if any sequence {un}n∈N in X such that

{I(un)} is bounded, limn→∞

∥∥∥I′(un)∥∥∥X*

= 0, Φ(un) < r for each n ∈ N

has a convergent subsequence.

The following three theorems are the main tools in the next section to prove the results. The �rst one is usedto prove the existence of at least one weak solution, the second one for the existence of at least two weaksolutions, and the third one for the existence of at least three weak solutions. While the �rst two results aredue to Bonanno, the third one is due to Bonanno and Marano.

Theorem 2.5 (See [10, Theorem 2.3]). Let X be a real Banach space. Assume Φ, Ψ : X → R are two continu-ously Gâteaux-di�erentiable functionals such that infX Φ = Φ(0) = Ψ(0) = 0. Assume that there exist r > 0 andv ∈ X with 0 < Φ(v̄) < r such that(E1)

supΦ(u) 0 and v ∈ X with r < Φ(v) such that(E3)

supΦ(u)

58 | M. Bohner, G. Caristi, F. Gharehgazlouei, S. Heidarkhani

Theorem 3.1. Assume that there exist two positive constants τ and δ such that(H1) pc

p0(N + 1)

p−1‖a‖∞ meas(Ω) max{δp , δp} < pmin{1, a0}τp,(H2)

∫Ω sup|t|≤τ F(x,t)dx

τp <pmin{1,a0}

∫Ω F(x,δ)dx

pcp0(N+1)p−1‖a‖∞ meas(Ω) max{δp ,δp}

,(H3) F(x, t) ≥ 0 for each (x, t) ∈ Ω ×R+.

Then, for each λ ∈ Λw, given by(‖a‖∞ meas(Ω) max{δp , δp}

p∫Ω F(x, δ)dx

, min{1, a0}τp

pcp0(N + 1)p−1 ∫

Ω sup|t|≤τ F(x, t)dx

)(3.1)

and for each g : Ω ×R→ R, there exists δλ,g > 0, given by

min{

min{1, a0}τp − λpcp0(N + 1)

p−1 ∫Ω sup|t|≤τ F(x, t)dx

pcp0(N + 1)p−1 ∫

Ω sup|t|≤τ G(x, t)dx,

λp∫Ω F(x, δ)dx − ‖a‖∞ meas(Ω) max{δ

p , δp}p∫Ω G(x, δ)dx

},

such that for each µ ∈[0, δλ,g

), the problem (1.1) admits at least one nontrivial weak solution uλ ∈ X such that

‖uλ‖∞ ≤ τ.

Proof. Our goal is to apply Theorem 2.5 to (1.1). To this end, take the real Banach space X = W1,~p(x)(Ω) withthe norm as de�ned in Section 2, with �xed λ and µ as in the conclusion, Φ, Ψλ,µ de�ned in (2.1) and (2.2).We can see that Φ, Ψλ,µ ∈ C1(X,R) (see [24, Lemma 3.4]) with derivatives given by〈

Φ′(u), v〉

=N∑i=1

∫Ω

∣∣∣∣ ∂u∂xi∣∣∣∣pi(x)−2 ∂u∂xi ∂v∂xi dx +

∫Ω

a(x)|u|p0(x)−2uvdx

and 〈Ψ ′λ,µ(u), v

〉=∫Ω

f (x, u)vdx + µλ

∫Ω

g(x, u)vdx.

The functionals Φ, Ψλ,µ are sequentially weakly lower semicontinuous [14, Lemma 3]. Moreover, Ψ ′λ,µ : X →X* is a compact operator. Indeed, it is enough to show that Ψ ′λ,µ is strongly continuous on X. To show this,for �xed u ∈ X, if un → u weakly in X as n →∞, then un(x) converges uniformly to u(x) on Ω as n →∞, see[30]. Since f , g are continuous in R for every x ∈ Ω, we get

f (x, un) +µλ g(x, un)→ f (x, u) +

µλ g(x, u) as n →∞.

Thus, Ψ ′λ,µ(un)→ Ψ′λ,µ(u) as n →∞. Hence, by [30, Proposition 26.2], Ψ

′λ,µ is a compact operator. Moreover,

Φ′ admits a continuous inverse on X*. Indeed, according to [30, Theorem 26.A(d)], it is enough to verify thatΦ′ is coercive, hemicontinuous, and uniformly monotone. By Proposition 2.3, it is clear that for any u ∈ X,we have 〈

Φ′(u), u〉

‖u‖ =

∑Ni=1∫Ω

∣∣∣ ∂u∂xi ∣∣∣pi(x) dx + ∫Ω a(x)|u|p0(x)dx‖u‖ ≥

k3‖u‖p − k4‖u‖ ,

where k3, k4 are positive constants. Thus,

lim‖u‖→∞

〈Φ′(u), u

〉‖u‖ = ∞,

i.e., Φ′ is coercive. The fact that Φ′ is hemicontinuous can be veri�ed using standard arguments. Finally, weshow that Φ′ is uniformly monotone. In fact, for any ξi , ψi ∈ R, we have the inequality (see [23])(

|ξi|ri−2ξi − |ψi|ri−2ψi)

(ξi − ψi) ≥ 2−ri |ξi − ψi|ri for all ri > 2. (3.2)

Neumann Elliptic Problems | 59

Thus, for every u, v ∈ X, we deduce that〈Φ′(u) − Φ′(v), u − v

〉=

N∑i=1

∫Ω

(∣∣∣∣ ∂u∂xi∣∣∣∣pi(x)−2 ∂u∂xi −

∣∣∣∣ ∂v∂xi∣∣∣∣pi(x)−2 ∂v∂xi

)(∂u∂xi

− ∂v∂xi

)dx

+∫Ω

a(x)(|u|p0(x)−2u − |v|p0(x)−2v

)(u − v)dx

≥ 2−p N∑

i=1

∫Ω

∣∣∣∣ ∂u∂xi − ∂v∂xi∣∣∣∣pi(x) dx + ∫

Ω

a(x)|u − v|p0(x)dx

≥ min{2−p , a02−p}

N∑i=0

∫Ω

|Di(u − v)|pi(x)dx

≥

C1∑N

i=0

∥∥∥Di(u − v)∥∥∥pp(·)

if ‖u − v‖p(·) ,∥∥∥ ∂u∂xi − ∂v∂xi ∥∥∥p(·) > 1,

C2∑N

i=0

∥∥∥Di(u − v)∥∥∥pp(·)

if ‖u − v‖p(·) ,∥∥∥ ∂u∂xi − ∂v∂xi ∥∥∥p(·) < 1,

where the last inequality is obtained from Proposition 2.1. Thus, by [9, Proposition 2.1], the functional Iλ,µ =Φ − λΨλ,µ satis�es the (PS)

[r] condition for each r > 0, and so (E2) from Theorem 2.5 is satis�ed. Therefore, it

remains to verify (E1) from Theorem 2.5. To this end, we put r := min{1,a0}p(N+1)p−1(τc0

)pand pick w ∈ X, de�ned as

w(x) ={δ if x ∈ Ω,0 otherwise.

(3.3)

Then we haveΦ(w) =

∫Ω

a(x)p0(x)

|w(x)|p0(x)dx ≤ meas(Ω)‖a‖∞p max{δp , δp}. (3.4)

Hence, it follows from (H1) that0 < Φ(w) < r. Now, let u ∈ X be such that u ∈ Φ−1([0, r]). Then, by Proposition2.1(i), for any u ∈ X with ‖u‖ > 1, we obtain

min{1, a0}p(N + 1)p−1

(τc0

)p≥ Φ(u) ≥ min{1, a0}

p(N + 1)p−1‖u‖p − N + a0p .

Similarly, by Proposition 2.1(ii), for any u ∈ X with ‖u‖ < 1, we obtain

min{1, a0}p(N + 1)p−1

(τc0

)p≥ Φ(u) ≥ min{1, a0}

p(N + 1)p−1‖u‖p − N + a0p .

Then,‖u‖ ≤ max

{τc0

, (N + 1)pp −1( τc0

)pp

}= τc0

.

Hence, we obtain|u(x)| ≤ ‖u‖L∞(Ω) ≤ c0‖u‖ ≤ τ for all x ∈ Ω.

Therefore, one has

supΦ(u)

60 | M. Bohner, G. Caristi, F. Gharehgazlouei, S. Heidarkhani

+µλcp0p(N + 1)

p−1 ∫Ω sup|t|≤τ G(x, u)dx

min{1, a0}τp.

Moreover, thanks to (H3) and (3.4), one has

Ψ(w)Φ(w) ≥

p∫Ω F(x, δ)dx

‖a‖∞ meas(Ω) max{δp , δp}+ µλ

p∫Ω G(x, δ)dx

‖a‖∞ meas(Ω) max{δp , δp}.

Since µ < δλ,µ, we have

µ <min{1, a0}τp − λpc

p0(N + 1)

p−1 ∫Ω sup|t|≤τ F(x, t)dx

pcp0(N + 1)p−1 ∫

Ω sup|t|≤τ G(x, t)dx(3.5)

and

µ <λp∫Ω F(x, δ)dx − ‖a‖∞ meas(Ω) max{δ

p , δp}−p∫Ω G(x, δ)dx

. (3.6)

From (3.5) and (3.6), we get

pcp0(N + 1)p−1 ∫

Ω sup|t|≤τ F(x, t)dxmin{1, a0}τp

+ µλpcp0(N + 1)

p−1 ∫Ω sup|t|≤τ G(x, t)dx

min{1, a0}τp< 1λ

andp∫Ω F(x, δ)dx

‖a‖∞ meas(Ω) max{δp , δp}+ µλ

p∫Ω G(x, δ)dx

‖a‖∞ meas(Ω) max{δp , δp}> 1λ .

Then,supΦ(x)≤r Ψ(u)

r <1λ <

Φ(w)Ψ(w) .

Therefore, (E1) from Theorem 2.5 is satis�ed. Now, since λ ∈(Φ(w)Ψ(w) ,

rsupΦ(x)≤r Ψ(u)

), Theorem 2.5 with v̄ = w

guarantees the existence of a local minimum point uλ for the functional Iλ,µ such that 0 < Φ(uλ) < r, and souλ is a nontrivial weak solution of (1.1) such that ‖uλ‖∞ < τ.

The following example is an application of Theorem 3.1.

Example 3.2. Given the domain Ω = {(x1, x2) ∈ R2 : x21 + x22 ≤ 1}. Let pi(x) for i = 0, 1, 2 be the functionsde�ned by

p0(x1, x2) = 4 + x1x2 for (x1, x2) ∈ R2,p1(x1, x2) = 2(2 + x1 + x2) for (x1, x2) ∈ R2,p2(x1, x2) = 2(2 + x1x2) for (x1, x2) ∈ R2.

Then, p = 4, p = 6, and meas(Ω) = π. for all ((x1, x2), t) ∈ Ω × R, put f ((x1, x2), t) = pi(x1, x2)et2 . By the

expression of f , we have

F((x1, x2), t) = 2pi(x1, x2)et2 for all ((x1, x2), t) ∈ Ω ×R.

By choosing a(x1, x2) = 1, δ = 112 , τ = 1, and c0 = 4π− 14 , by simple calculations, obviously all assumptions

of Theorem 3.1 are satis�ed. Hence, by applying Theorem 3.1, for every

λ ∈(

14 × 125e 124

, 12 × 125e 12

)and for each g : Ω × R→ R, there exists δλ,g > 0 such that for each µ ∈

[0, δλ,g

), the problem (1.1) admits at

least one nontrivial weak solution uλ ∈ X such that ‖uλ‖∞ ≤ 1.

Next, our goal is to obtain the existence of twodistinctweak solutions for (1.1). The following result is obtainedby applying Theorem 2.6. Note that (H4) below is the well-known Ambrosetti–Rabinowitz condition.

Neumann Elliptic Problems | 61

Theorem 3.3. Assume that there exist two positive constants τ and δ such that (H1) from Theorem 3.1 holds.Moreover, assume(H4) there exist ρ > p and R > 0 such that

0 < ρF(x, t) < tf (x, t) for all x ∈ Ω and |t| ≥ R.

Then, for each

λ ∈ Λr :=(

0, min{1, a0}τp

pcp0(N + 1)p−1 ∫

Ω sup|t|≤τ F(x, t)dx

)and for each Carathéodory function g : Ω ×R→ R satisfying(H5) there exist c1, c2 > 0 and r ∈ C+(Ω) with 0 < r(x) ≤ r+ < p such that

|g(x, t)| ≤ c1 + c2|t|r(x)−1 for every (x, t) ∈ Ω ×R,

there exists δλ,g > 0, given as in Theorem 3.1, such that for each µ ∈[0, δλ,g

), the problem (1.1) admits at least

two nontrivial weak solutions.

Proof. LetΦ, Ψ be the functionals de�ned in Theorem 3.1, which satisfy all regularity assumptions requestedin Theorem 2.6. Arguing as in the proof of Theorem 3.1, choose r = min{1,a0}p(N+1)p−1 (

τc0 )

p and pick w ∈ X. Now, from(H4), by standard computations, there is a positive constant m such that

F(x, t) ≥ m|t|ρ for all x ∈ Ω.

Hence, for every λ ∈ Λr, u ∈ X \ {0}, and t > 1, we obtain

Iλ,µ(tu) = Φ(tu) − λ∫Ω

F(x, tu)dx − µ∫Ω

G(x, tu)dx

≤ tp

p

N∑i=1

∫Ω

∣∣∣∣ ∂u∂xi∣∣∣∣pi(x) dx + ∫

Ω

a(x)|u|p0(x)dx

−mλtρ

∫Ω

|u(k)|ρ + c1t∫Ω

|u|dx + c2tr+∫Ω

|u|r(x)dx.

Since ρ > p > r+, this condition guarantees that Iλ,µ is unbounded frombelow.We recall that Iλ,µ is a Gâteaux-di�erentiable functional whose Gâteaux derivative at the point u ∈ X is the functional I′λ,µ(u) ∈ X* given by

I′λ,µ(u)(v) =N∑i=1

∫Ω

∣∣∣∣ ∂u∂xi∣∣∣∣pi(x)−2 ∂u∂xi ∂v∂xi dx +

∫Ω

a(x)|u|p0(x)−2uvdx

−λ∫Ω

f (x, u)vdx − µ∫Ω

g(x, u)vdx

for every v ∈ X. Finally, we verify that Iλ,µ satis�es the (PS) condition. Indeed, if {un}n∈N ⊂ X such that{Iλ,µ(un)}n∈N is bounded and I′λ,µ(un) → 0 in X* as n → +∞, Then there exists a positive constant s0 suchthat ∣∣Iλ,µ(un)∣∣ ≤ s0 and ∥∥∥I′λ,µ(un)∥∥∥ ≤ s0 for all n ∈ N.Using also (H4), (H5), and the de�nition of I′λ,µ, we deduce that, for all n ∈ N,

s0 + s1 ‖un‖ ≥ ρIλ,µ(un) − I′λ,µ(un)un

≥(ρp − 1

)(min{1, a0}(N + 1)p−1

‖un‖p − N0 − a0)

62 | M. Bohner, G. Caristi, F. Gharehgazlouei, S. Heidarkhani

−µρ∫Ω

G(x, un)dx + µ∫Ω

g(x, un)undx

≥(ρp − 1

)(min{1, a0}(N + 1)p−1

‖un‖p − N0 − a0)

−µc3 ‖un‖∞ − µc4 ‖un‖r+∞

≥(ρp − 1

)(min{1, a0}(N + 1)p−1

‖un‖p − N0 − a0)− s2 ‖un‖ − s3 ‖un‖r

+

for some s1, s2, s3 > 0. Since ρ > p, it follows that {un}n∈N is bounded. Consequently, since X is a re�exiveBanach space, we have, up to a subsequence,

un ⇀ u in X.

By I′λ,µ(un)→ 0 and un ⇀ u in X, we get(I′λ,µ(un) − I

′λ,µ(u)

)(un − u)→ 0.

From the continuity of f and g, we have∫Ω

(f (x, un) − f (x, u)

)(un − u)dx → 0 as n →∞

and ∫Ω

(g(x, un) − g(x, u)

)(un − u)dx → 0 as n →∞.

Moreover, an easy computation shows(I′λ,µ(un) − I

′λ,µ(u)

)(un − u)

=N∑i=1

∫Ω

(∣∣∣∣∂un∂xi∣∣∣∣pi(x)−2 ∂un∂xi −

∣∣∣∣ ∂u∂xi∣∣∣∣pi(x)−2 ∂u∂xi

)(∂un∂xi

− ∂u∂xi

)dx

+∫Ω

a(x)(|un|p0(x)−2un − |u|p0(x)−2u

)(un − u)dx

−λ∫Ω

(f (x, un) − f (x, u)

)(un − u)dx

−µ∫Ω

(g(x, un) − g(x, u)

)(un − u)dx

≥ min{2−p , a02−p}N∑i=0

∫Ω

∣∣∣Di(un − u)∣∣∣pi(x) dx,where the last inequality is obtained by using (3.2). Combining the last relation with Proposition 2.1(iii), we�nd that the sequence {un}n∈N converges strongly to u in X. Therefore, Iλ,µ satis�es the (PS) condition, andso all hypotheses of Theorem 2.6 are satis�ed. Hence, applying Theorem 2.6, for each λ ∈ Λr, the functionIλ,µ admits at least two distinct critical points that are the weak solutions of (1.1).

Finally, we discuss the existence of at least three weak solutions for (1.1).

Theorem 3.4. Assume(H6) there exist c > 0 and r ∈ C+ with 0 < r(x) ≤ r+ < p such that

F(x, t),∣∣G(x, t)∣∣ ≤ c (1 + |t|r(x)) for all (x, t) ∈ Ω ×R.

Neumann Elliptic Problems | 63

Assume that there exist two positive constants τ and η such that(H7) a0pc

p0(N + 1)

p−1 meas(Ω) min{δp , δp} > min{1, a0}τp

and let (H2) and (H3) from Theorem 3.1 hold. Then, for every λ ∈ Λw as in (3.1) and for each g : Ω × R → R,there exists δλ,g > 0, given as in Theorem 3.1, such that for each µ ∈

[0, δλ,g

), the problem (1.1) admits at least

three distinct weak solutions.

Proof. Our aim is to apply Theorem 2.7.We consider the functionalsΦ andΨλ,µ, which, as seen before, satisfythe regularity assumptions requested in Theorem 2.7. Now, arguing as in the proof of Theorem 3.1, put w asin (3.3) and r = min{1,a0}p(N+1)p−1

(τc0

)p. Bearing in mind (H7), we obtain

Φ(w) > r > 0.

Therefore, according to the proof of Theorem 3.1, (E3) from Theorem 2.7 holds. Now, we prove that, for eachλ ∈ Λw, the functional Iλ,µ is coercive. By using (H6) and the Sobolev embedding theorem, we easily obtainfor all u ∈ X

Iλ,µ(u) ≥min{1, a0}p(N + 1)p−1

‖u‖p − N + a0p

− cλr+(‖u‖r

++ µλ ‖u‖

r+),

which implies Iλ,µ → ∞ as ‖u‖ → ∞. Hence, the functional Iλ,µ is coercive, and (E4) holds. So, for eachλ ∈ Λw, Theorem 2.7 implies that the functional Iλ,µ admits at least three critical points in X, and these areweak solutions of (1.1).

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Professor M. Bohner is an Editor of the Journal Nonautonomous Dynamical Systems, and therefore this sub-mission was handled by another Editor of the journal.

1 Introduction2 Preliminaries3 Main Results