Nonlinear Analysis 67 (2007) 1419–1425www.elsevier.com/locate/na

Existence and multiplicity of solutions for a Neumann probleminvolving the p(x)-Laplace operator

Mihai Mihailescu∗

Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Received 15 May 2006; accepted 18 July 2006

Abstract

In this paper we study an elliptic equation with nonstandard growth conditions and the Neumann boundary condition. Weestablish the existence of at least three solutions by using as the main tool a variational principle due to Ricceri.c© 2006 Elsevier Ltd. All rights reserved.

MSC: 35D05; 35J60; 35J70; 58E05

Keywords: p(x)-Laplace operator; Sobolev space with variable exponent; Ricceri’s variational principle

1. Introduction and preliminary results

In this paper we are concerned with the study of a Neumann problem of the type−div(|∇u|p(x)−2

∇u) + |u|p(x)−2u = λ f (x, u), for x ∈ Ω

∂u∂ν

= 0, for x ∈ ∂Ω ,(1)

where Ω ⊂ RN (N ≥ 3) is a bounded domain with a smooth boundary, λ > 0 is a real number, p is a continuousfunction on Ω with infy∈Ω p(y) > N and f : Ω × R → R is a continuous function which will be specified later.We denote by ν the outward unit normal to ∂Ω . The main interest in studying such problems arises from the presenceof the p(x)-Laplace operator, namely div(|∇u|

p(x)−2∇u). This is a generalization of the classical p-Laplace operator

div(|∇u|p−2

∇u) obtained in the case when p is a positive constant. We point out that elliptic equations involving thep(x)-Laplace equations are not trivial generalizations of similar problems studied in the constant case since the p(x)-Laplace operator is not homogeneous and, thus, some techniques which can be applied in the case of the p-Laplaceoperators will fail in that new situation, such as the Lagrange Multiplier Theorem. On the other hand, problemsinvolving p(x)-growth conditions are extremely attractive because they can model phenomena which arise from thestudy of electrorheological fluids or elastic mechanics. In that context we refer the reader to Diening [6], Halsey [13],Ruzicka [19], Zhikov [22], and the references therein.

∗ Tel.: +40 744771405; fax: +40 251412673.E-mail address: mmihailes@yahoo.com.

0362-546X/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2006.07.027

1420 M. Mihailescu / Nonlinear Analysis 67 (2007) 1419–1425

In the case when 1 < p(x) < N for any x ∈ Ω there were extensive studies in the last few decades dealingwith problems of type (1) but with Dirichlet boundary conditions. We just recall the papers by Alves and Souto [1],Chabrowski and Fu [5], Fan and Zhang [11], Mihailescu and Radulescu [15], where different techniques for findingsolutions are illustrated.

In this paper we will study problem (1) in the case when p(x) > N for any x ∈ Ω . We will prove the existence ofat least three weak solutions in a variable Sobolev space by using as the main tool a three critical point theorem dueto Ricceri (see Theorem 1 in [17]). We recall Ricceri’s result in a convenient form (see Theorem 1 in [3]):

Theorem 1. Let X be a separable and reflexive real Banach space; Φ : X → R a continuously Gateaux differentiableand sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse onX?; Ψ : X → R a continuously Gateaux differentiable functional whose Gateaux derivative is compact. Assume that

(i) lim‖u‖→∞(Φ(u) + λΨ(u)) = ∞ for all λ > 0; and that there are r ∈ R and u0, u1 ∈ X such that(ii) Φ(u0) < r < Φ(u1);

(iii) infu∈Φ−1((−∞,r ]) Ψ(u) >(Φ(u1)−r)Ψ (u0)+(r−Φ(u0))Ψ (u1)

Φ(u1)−Φ(u0).

Then there exist an open interval Λ ⊂ (0, ∞) and a positive real number q such that for each λ ∈ Λ the equation

Φ′(u) + λΨ ′(u) = 0

has at least three solutions in X whose norms are less than q.

We start with some preliminary basic results on the theory of Lebesgue–Sobolev spaces with variable exponent.For more details we refer the reader to the book by Musielak [16] and the papers by Edmunds et al. [7–9], Kovacikand Rakosnık [14], Fan et al. [10,12] and Mihailescu and Radulescu [15].

Assume that p ∈ C(Ω) and p(x) > 1, for all x ∈ Ω .Set

C+(Ω) = h; h ∈ C(Ω), h(x) > 1 for all x ∈ Ω.

For any h ∈ C+(Ω) we define

h+= sup

x∈Ωh(x) and h−

= infx∈Ω

h(x).

For any p(x) ∈ C+(Ω), we define the variable exponent Lebesgue space

L p(x)(Ω) =

u; u is a measurable real-valued function such that

∫Ω

|u(x)|p(x) dx < ∞

.

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

|u|p(x) = inf

µ > 0;

∫Ω

∣∣∣∣u(x)

µ

∣∣∣∣p(x)

dx ≤ 1

.

We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces. If 0 < |Ω | < ∞

and p1, p2 are variable exponents so that p1(x) ≤ p2(x) almost everywhere in Ω then there exists the continuousembedding L p2(x)(Ω) → L p1(x)(Ω).

We denote by L p′(x)(Ω) the conjugate space of L p(x)(Ω), where 1/p(x) + 1/p′(x) = 1. For any u ∈ L p(x)(Ω)

and v ∈ L p′(x)(Ω) the Holder type inequality∣∣∣∣∫Ω

uv dx∣∣∣∣ ≤

(1

p−+

1p′−

)|u|p(x)|v|p′(x) (2)

holds true.An important role in manipulating the generalized Lebesgue–Sobolev spaces is played by the modular of the

L p(x)(Ω) space, which is the mapping ρp(x) : L p(x)(Ω) → R defined by

ρp(x)(u) =

∫Ω

|u|p(x) dx .

M. Mihailescu / Nonlinear Analysis 67 (2007) 1419–1425 1421

If u ∈ L p(x)(Ω) then the following relations hold: true

|u|p(x) > 1 ⇒ |u|p−

p(x) ≤ ρp(x)(u) ≤ |u|p+

p(x) (3)

|u|p(x) < 1 ⇒ |u|p+

p(x) ≤ ρp(x)(u) ≤ |u|p−

p(x). (4)

Next, we define the variable exponent Sobolev space

W 1,p(x)(Ω) = u ∈ L p(x); |∇u| ∈ L p(x)(Ω).

That space endowed with the norm

‖u‖W 1,p(x) = |u|p(x) + |∇u|p(x)

is a separable and reflexive Banach space. We note that we can use the following equivalent norm on W 1,p(x)(Ω):

‖u‖ = inf

µ > 0;

∫Ω

(∣∣∣∣u(x)

µ

∣∣∣∣p(x)

+

∣∣∣∣∇u(x)

µ

∣∣∣∣p(x))

dx ≤ 1

.

We set

I (u) =

∫Ω

(|u|p(x)

+ |∇u|p(x)) dx .

If u ∈ W 1,p(x)(Ω) then the following relations hold true:

‖u‖ > 1 ⇒ ‖u‖p−

≤ I (u) ≤ ‖u‖p+

(5)

‖u‖ < 1 ⇒ ‖u‖p+

≤ I (u) ≤ ‖u‖p−

. (6)

Remark 1. If N < p−≤ p(x) for any x ∈ Ω , by Theorem 2.2 in [12] we deduce that W 1,p(x)(Ω) is continuously

embedded in W 1,p−

(Ω). Since N < p− it follows that W 1,p−

(Ω) is compactly embedded in C(Ω). Thus, we deducethat W 1,p(x)(Ω) is compactly embedded in C(Ω). Defining ‖u‖∞ = supx∈Ω |u(x)| we find that there exists a positiveconstant c > 0 such that

‖u‖∞ ≤ c ‖u‖, ∀ u ∈ W 1,p(x)(Ω).

2. The main result

In this paper we study problem (1) in the particular case

f (x, t) = |t |q(x)−2t − t,

where q(x) ∈ C+(Ω) satisfies 2 < q(x) < infy∈Ω p(y) for any x ∈ Ω .Thus, problem (1) becomes−div(|∇u|

p(x)−2∇u) + |u|

p(x)−2u = λ(|u|q(x)−2u − u), for x ∈ Ω

∂u∂ν

= 0, for x ∈ ∂Ω .(7)

We say that u ∈ W 1,p(x)(Ω) is a weak solution of problem (7) if∫Ω

(|∇u|p(x)−2

∇u∇v + |u|p(x)−2uv) dx − λ

∫Ω

|u|q(x)−2uv dx + λ

∫Ω

uv dx = 0,

for any v ∈ W 1,p(x)(Ω).

1422 M. Mihailescu / Nonlinear Analysis 67 (2007) 1419–1425

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

Theorem 2. Assume that infy∈Ω p(y) > N and 2 < q(x) < infy∈Ω p(y) for any x ∈ Ω . Then there exist an openinterval Λ ⊂ (0, ∞) and a positive constant ρ > 0 such that for any λ ∈ Λ, problem (7) has at least three weaksolutions whose norms are less than ρ.

Remark 2. We remark that results similar to those given by Theorem 2 were obtained in the case of the classicalp-Laplace operator in a large number of papers. In that context we refer the reader to [2,18,3,4] and the referencestherein.

On the other hand, we point out the fact that problem (7) can be regarded as an eigenvalue problem. In that context,Theorem 2 establishes the existence of a continuous family of eigenvalues for problem (7) which are not simple.

3. Proof of Theorem 2

Let E denote the generalized Sobolev space W 1,p(x)(Ω). In order to apply Ricceri’s result we define the functionalsΦ, Ψ : E → R by

Φ(u) =

∫Ω

1p(x)

(|∇u(x)|p(x)+ |u(x)|p(x)) dx,

Ψ(u) = −

∫Ω

1q(x)

|u(x)|q(x) dx +12

∫Ω

u2(x) dx .

Arguments similar to those used in the proof of Proposition 3.1 in [15] imply that Φ, Ψ ∈ C1(E, R) with thederivatives given by

〈Φ′(u), v〉 =

∫Ω

(|∇u(x)|p(x)−2∇u(x)∇v(x) + |u(x)|p(x)−2u(x)v(x)) dx,

〈Ψ ′(u), v〉 = −

∫Ω

|u(x)|q(x)−2u(x)v(x) dx +

∫Ω

u(x)v(x) dx,

for any u, v ∈ E . Thus, we deduce that u ∈ E is a weak solution of Eq. (7) if there exists λ > 0 such that u is a criticalpoint of the operator Φ + λΨ . It follows that we can seek for weak solutions of problem (7) by applying Theorem 1.We show that it can indeed be used by verifying the fact that its hypotheses are fulfilled.

With that end in view, we prove first that (Φ′)−1: E?

→ E exists and it is continuous. The main tool for showingthat fact will be Theorem 26.A(d) in [21] (see also [20] pp. 56–57). According to that theorem it is enough to verifythat Φ′ is coercive, hemicontinuous and uniformly monotone.

Indeed, by relation (5) it is clear that for any u ∈ E with ‖u‖ > 1 we have

〈Φ′(u), u〉

‖u‖=

I (u)

‖u‖≥ ‖u‖

p−−1

and thus,

lim‖u‖→∞

〈Φ′(u), u〉

‖u‖= ∞,

i.e. Φ′ is coercive. The fact that Φ′ is hemicontinuous can be verified using standard arguments. Finally, we show thatΦ′ is uniformly monotone. In that context, we recall the inequality

(|ξ |s−2ξ − |η|

s−2η) · (ξ − η) ≥12s |ξ − η|

s, ∀ ξ, η ∈ RN , ∀ s ≥ 2.

Thus, we deduce that

〈Φ′(u) − Φ′(v), u − v〉 ≥1

2p+

∫Ω

(|∇(u(x) − v(x))|p(x)+ |u(x) − v(x)|p(x)) dx, ∀ u, v ∈ E . (8)

M. Mihailescu / Nonlinear Analysis 67 (2007) 1419–1425 1423

Define the function α : [0, ∞) → [0, ∞) by

α(t) =1

2p+·

t p+

−1, for t ≤ 1t p−

−1, for t ≥ 1.

It is easy to check that α is an increasing function with α(0) = 0 and limt→∞ α(t) = ∞. Relations (5), (6) and (8)imply

〈Φ′(u) − Φ′(v), u − v〉 ≥ α(‖u − v‖) · ‖u − v‖, ∀ u, v ∈ E,

i.e. Φ′ is uniformly monotone. We conclude that (Φ′)−1 exists and it is continuous.Next, we verify that condition (i) in Theorem 1 is fulfilled. Indeed, by relation (5) we deduce that for any u ∈ E

with ‖u‖ > 1 we have

Φ(u) ≥1

p+· ‖u‖

p−

.

On the other hand, since q−≤ q(x) ≤ q+ for any x ∈ Ω it follows that∫

Ω

1q(x)

|u(x)|q(x) dx ≤1

q−

∫Ω

(|u(x)|q−

+ |u(x)|q+

) dx =1

q−· (|u|

q−

q− + |u|q+

q+), ∀ u ∈ E .

Using Remark 1 we have that E is continuously embedded in Lq∓

(Ω) and, thus, we find two positive constants d1,d2 > 0 such that

|u|q− ≤ d1 ‖u‖, |u|q+ ≤ d2 ‖u‖, ∀ u ∈ E .

All the above pieces of information yield that for any λ > 0 the following inequality holds true:

Φ(u) + λΨ(u) ≥1

p+· ‖u‖

p−

−λ

q−· (d1 ‖u‖

q−

+ d2 ‖u‖q+

), ∀ u ∈ E .

Since q−≤ q+ < p− it follows that lim‖u‖→∞(Φ(u) + λΨ(u)) = ∞ and (i) is verified.

In the sequel we verify that conditions (ii) and (iii) in Theorem 1 are satisfied. In order to do that we define thefunction H : Ω × [0, ∞) → R by

H(x, t) =1

q(x)· tq(x)

−12

· t2, ∀ x ∈ E and t ∈ [0, ∞).

It is clear that H is of class C1 with respect to t , uniformly when x ∈ Ω and

Ht (x, t) = t · (tq(x)−2− 1), ∀ x ∈ E and t ∈ (0, ∞).

Thus, Ht (x, t) ≥ 0 for all t ≥ 1 and all x ∈ Ω and Ht (x, t) ≤ 0 for all t ≤ 1 and all x ∈ Ω . It follows that H(x, t) isincreasing when t ∈ (1, ∞) and decreasing when t ∈ (0, 1), uniformly with respect to x . Furthermore,

limt→∞

H(x, t) = ∞, uniformly with respect to x ∈ E .

Using that fact we get that there exists δ > 1 such that

H(x, t) ≥ 0 = H(x, 0) ≥ H(x, τ ), ∀ x ∈ E, t > δ, and τ ∈ (0, 1). (9)

Let a, b be two real numbers such that 0 < a < min1, c, with c given in Remark 1, and b > δ satisfies bp−

· |Ω | > 1.Relation (9) implies∫

Ωsup

0≤t≤aH(x, t) dx ≤ 0 <

1cp+

·a p+

bp−·

∫Ω

H(x, b) dx .

We consider u0, u1 ∈ E , u0(x) = 0, u1(x) = b for any x ∈ Ω . We also define r =1

p+ ·( a

c

)p+

. Clearly, r ∈ (0, 1). Asimple computation implies

Φ(u0) = Ψ(u0) = 0,

1424 M. Mihailescu / Nonlinear Analysis 67 (2007) 1419–1425

Φ(u1) =

∫Ω

1p(x)

· bp(x) dx ≥1

p+· bp−

· |Ω | >1

p+·

(ac

)p+

= r,

Ψ(u1) = −

∫Ω

H(x, b) dx .

Thus, we deduce that

Φ(u0) < r < Φ(u1),

and (ii) in Theorem 1 is verified.On the other hand, we have

−(Φ(u1) − r)Ψ(u0) + (r − Φ(u0))Ψ(u1)

Φ(u1) − Φ(u0)= −r ·

Ψ(u1)

Φ(u1)= r ·

∫Ω H(x, b) dx∫

Ω1

p(x)· bp(x) dx

> 0.

Next, we focus our attention on the case when u ∈ E with Φ(u) ≤ r < 1. Then, by (6) it is clear that

1p+

· ‖u‖p+

≤1

p+· I (u) ≤ Φ(u) ≤ r =

1p+

·

(ac

)p+

< 1.

Thus, using Remark 1 we deduce that for any u ∈ E with Φ(u) ≤ r we have

|u(x)| ≤ c · ‖u‖ ≤ c · (p+· r)1/p+

= a, ∀ x ∈ Ω .

The above inequality shows that

− infu∈Φ−1((−∞,r ])

Ψ(u) = supu∈Φ−1((−∞,r ])

−Ψ(u) ≤

∫Ω

sup0≤t≤a

H(x, t) dx ≤ 0.

We find that

− infu∈Φ−1((−∞,r ])

Ψ(u) < r ·

∫Ω H(x, b) dx∫

Ω1

p(x)· bp(x) dx

,

or

infu∈Φ−1((−∞,r ])

Ψ(u) >(Φ(u1) − r)Ψ(u0) + (r − Φ(u0))Ψ(u1)

Φ(u1) − Φ(u0),

and, thus, condition (iii) in Theorem 1 is verified.We showed that all the assumptions of Theorem 1 are satisfied. We conclude that there exists an open interval

Λ ⊂ (0, ∞) and a positive constant ρ > 0 such that for any λ ∈ Λ, the equation

Φ′(u) + λΨ ′(u) = 0

has at least three solutions in E whose norms are less than ρ. The proof of Theorem 2 is complete.

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