Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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For all u ∈ W

1,p(x)

(Ω) the following relations hold

(2.1) kuk > 1 ⇒ kuk

−

≤ I

p(x)

(u) ≤ kuk

;

(2.2) kuk < 1 ⇒ kuk

≤ I

p(x)

(u) ≤ kuk

−

Finally, we remember some embedding results regarding variable exponent

Lebesgue-Sobolev spaces. For the continuous embedding between variable

exponent Lebesgue-Sobolev spaces we refer to (Theorem 1.1, Fan et al.

if p : Ω → IR is Lipschitz continuous and p

< N , then for any q ∈ L

∞

(Ω)

with p(x) ≤ q(x) ≤

Np(x)

N−p(x)

, there is a continuous embedding W

1,p(x)

(Ω) ,→

q(x)

(Ω). In what concerns if N < p

−

= ess inf

x∈

Ω

p(x) ≤ p(x) for any

x ∈ Ω, by Theorem 2.2 in Fan et al.

we deduce that W

1,p(x)

(Ω) is contin-

uously embedded in W

1,p

−

(Ω). Since N < p

−

it follows that W

1,p

−

(Ω) is

compactly embedded in C(

Ω). Thus, we obtain that W

1,p(x)

(Ω) is contin-

uously embedded in C(Ω).

From now on, E is the space W

1,p(x)

(Ω) with the norm

kuk =



Ω

|∇u|

p(x)

+ |u|

p(x)



1/p(x)

which is clearly equivalent to the usual one, on the space C(

Ω) we consider

the norm kuk

∞

= sup

x∈

Ω

|u(x)|. When p

−

> N, Sobolev’s Theorem implies

that E is compactly embedded in C(Ω), hence

(2.3) c = sup

u∈E\{0}

kuk

∞

kuk

< +∞.

Proposition 2.1. Let I : E → E

∗

be the operator deﬁned by

I(u)ϕ =

Ω



|∇u|

p(x)−2

∇u∇ϕ + |u|

p(x)−2

uϕ



for all u, ϕ ∈ E. Then I admits a continuous inverse on E

∗

Proof.

Denoting by h., .i the usual inner product in IR

. We have for some positive

constant C

hI(u), ui

kuk

≥ C

p(x)

(u)

kuk

≥ C

kuk

−

−1

Hence,

lim

kuk→+∞

hI(u), ui

kuk

= +∞.

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Then I is coercive. Moreover, by simple arguments we can easily verify

that I is hemicontinuous. It remains to show that I is uniformly monotone.

Indeed, recall ﬁrst the well known inequality

|x − y|

≤ 2

h|x|

m−2

x −|y|

m−2

y, x − yi, ∀x, y ∈ IR

, ∀m ≥ 2.

Thus, it is easy to see that

hI(u

) − I(u

), u

− u

i ≥

Ω

|∇u

− ∇u

p(x)

+ |u

− u

p(x)

≥ c(p

)ku

− u

for every u

and u

belonging to E. This means that I is uniformly mono-

tone operator in E. Therefore, the conclusion of Proposition 2.1 follows

directly from the result (Theorem 26. A) of Zeidler.

Our main result are the following

Theorem 2.1. Suppose p

−

> N and let α satisfying (1.2). Assume that

f : Ω × IR → IR is a function which is measurable in Ω and C

in IR

satisfying (1.1), (1.3) and (1.4). Then, there exist an open interval A of

]0, +∞[ and a positive real number t such that, for every λ ∈ A and every

function g : Ω × IR → IR which is measurable in Ω and C

in IR satisfying

(1.5), there exists σ > 0 such that for each µ ∈ [0, σ] the system (P

) has

at least three weak solutions whose norms in E are less than t.

3. Proof of the main result

We begin by setting

Φ(u) =

Ω

p(x)

(|∇u|

p(x)

+ |u|

p(x)

) dx

J(u) = −

Ω

F (x, u) dx

and

Ψ(u) =

Ω

γ(x)

|u|

γ(x)

for each u ∈ E. It is well known that Φ and J are well deﬁned and contin-

uously Gˆateaux diﬀerentiable with

(u)ϕ =

Ω

(|∇u|

p(x)−2

∇u∇ϕ + |u|

p(x)−2

uϕ) dx

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

212

and

(u)ϕ = −

Ω

f(x, u)ϕ dx

for all u, ϕ in E. Note that J

is compact and Φ is clearly weakly lower

semi-continuous and bounded on each bounded subset of E. Proposition

2.1 ensures that Φ

admits a continuous inverse on E

∗

. Moreover, it is easy

to see that

lim

kuk−→+∞

(Φ(u) + λJ(u)) = +∞

for all λ ∈]0, +∞[. Indeed, since E is embedded in C

(

Ω), we have

J(u) ≥ −|Ω|(

−

kuk

∞

+ c

kuk

∞

where |Ω| denotes the measure of Ω. Then, the coercivity of Φ + λJ can be

easily deduced from (2.1) since α

< p

−

and λ > 0 and from the fact that

kuk

∞

≤ Dkuk for some positive constant D. Then (i) is veriﬁed.

In the sequel, we will verify the conditions (ii) and (iii) of Theorem 1.1.

By (1.3) and (1.4), it follows that F (x, t) is increasing for t ∈ (1, ∞) and

decreasing for t ∈ (0, 1), uniformly with respect to x and F (x, 0) = 0 is

obvious. Also, by (1.4), we can choose δ > 1 such that

F (x, u) ≥ 0 = F (x, 0) ≥ F (x, τ), ∀u > δ, τ ∈]0, 1[.

Let b, d be two numbers such that 0 < b < min{1, c}, with c given by (2.3)

and d > δ such that d

−

|Ω| > 1. Then we obtain

Ω

sup

0≤u≤b

F (x, u) ≤ 0 <

−

Ω

F (x, d) dx.

Deﬁne the real number r by

r =





By choosing

(x) = 0 and u

(x) = d for every x ∈ Ω

we have

Φ(u

) = J(u

) = 0, Φ(u

) =

Ω

p(x)

|d|

p(x)

and

J(u

) = −

Ω

F (x, d) dx.

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213

Then we clearly have

Φ(u

) ≥

−

|Ω| > r.

Thus we deduce that

Φ(u

) < r < Φ(u

Then (ii) in Theorem 1.1 is veriﬁed.

On the other hand, we have

−

(Φ(u

) − r)J(u

) + (r − Φ(u

))J(u

)

Φ(u

) − Φ(u

)

= r

Ω

F (x, d) dx

Ω

p(x)

|d|

p(x)

> 0.

We also have

p(x)

(u) ≤ Φ(u),

which implies that

p(x)

(u) ≤ p

r < 1,

for all x ∈ Ω and for all u ∈ E such that Φ(u) ≤ r. Using (2.1), we get

kuk ≤ 1. This implies that

kuk ≤ Φ(u) ≤ r.

Taking into account that

|u(x)| ≤ c(p

< b

for all x ∈ Ω and for all u ∈ E such that Φ(u) ≤ r, with c = sup

u∈E

kuk

∞

kuk

It follows

− inf

u∈Φ

−1

(]−∞,r])

J(u) = sup

Φ(u)≤r

−J(u) ≤

Ω

sup

0≤u≤b

F (x, u) dx ≤ 0.

Consequently, we obtain

inf

u∈Φ

−1

(]−∞,r])

J(u) >

(Φ(u

) − r)J(u

) + (r − Φ(u

))J(u

)

Φ(u

) − Φ(u

)

This means that condition (iii) in Theorem 1.1 is veriﬁed.

Moreover, since the function G : Ω × IR −→ IR is a measurable in

Ω and C

in IR × IR satisfying the condition (1.5), then the functional

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214

Ψ(u) =

Ω

u(x)

g(x, t) dt

dx is well deﬁned and continuously Gˆateaux

diﬀerentiable on E, with compact derivative, and one has

(u)ϕ =

Ω

g(x, u(x))ϕ(x) dx

for all u, ϕ ∈ E. So, in view of Proposition 2.1 and Theorem 1.1, the proof

of Theorem 2.1 is achieved.

Corollary 3.1. Suppose p

−

> N. Let α satisfying (1.2). Then there exists

an open interval A of ]0, +∞[ and a positive real number t such that, for

every λ ∈ A and every γ, with γ > 1 there exists σ > 0 such that for each

µ ∈ [0, σ] the system (P

)

has at least two weak nonzero solutions whose

norms in E are less than t.

Proof.

Let

Φ(u) =

Ω

p(x)

(|∇u|

p(x)

+ |u|

p(x)

) dx

J(u) = −

Ω

(

α(x)

|u|

α(x)

− u) dx

and

Ψ(u) =

Ω

γ(x)

|u|

γ(x)

for each u ∈ E. It is well known that Φ and J are well deﬁned and contin-

uously Gˆateaux diﬀerentiable with

(u)ϕ =

Ω

(|∇u|

p(x)−2

∇u∇ϕ + |u|

p(x)−2

uϕ) dx

and

(u)ϕ = −

Ω

|u|

α(x)−2

uϕ dx

for all u, ϕ in E. Note that J

is compact and Φ is clearly weakly lower

semi-continuous and bounded on each bounded subset of E. Proposition

2.1 ensures that Φ

admits a continuous inverse on E

∗

. Moreover, it is easy

to see that

lim

kuk−→+∞

(Φ(u) + λJ(u)) = +∞

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

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for all λ ∈]0, +∞[. Consider now

H(x, u) =

α(x)

|u|

α(x)

− u.

Since α(x) > 0 for all x ∈ Ω we have

lim

|u|→∞

H(x, u) = +∞.

Choose δ > 1 such that

α(x)

|u|

α(x)

− u > 0

for all u > δ. Hence we get

H(x, u) ≥ 0 = H(x, 0) ≥ H(x, τ), ∀u > δ, τ ∈]0, 1[.

Let us choose b, d two real numbers and u

and u

the same as in the

proof of Theorem 2.1. Then we obtain (ii) and (iii) of Theorem 1.1 when

adapting the techniques given in the proof of Theorem 2.1. On the other

hand, the functional

Ψ(u) =

Ω

γ(x)

|u|

γ(x)

is continuously Gˆateaux diﬀerentiable on E, with compact derivative. So,

in view of Theorem 1.1, the proof is completed.

Acknowledgement

The research of S. El Manouni is supported by Al-Imam University project

No. 28/12.

References

1. G. Anello and G. Cordaro, Existence of solutions of the Neumann problem

involving the p-Laplacian via variational principle of Ricceri. Arch. Math.

(Basel)79 (2002)274-287.

2. G. Bonanno, A minimax inequality and its applications to ordinary diﬀeren-

tial equations. J. Math. Anal. Appli. 270(2002) 210-219.

3. G. Bonanno, Some remarks on a three critical points theorem, Nonlinear

Anal. 54(2003) 651-665.

4. G. Bonanno and P. Candito, Three solutions to a Neumann problem for

elliptic equations involving the p-Laplacian. Arch. Math. 80(2003) 424-429.

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involving the p(x)-Laplacian, Nonlinear Anal. to appear

6. S. El Manouni and M. Kbiri Alaoui, A result on elliptic systems with Neu-

mann conditions via Ricceri’s three critical points theorem, Nonlinear Anal.

to appear

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7. O. Kov´aˇcik and J. R´akosn´ık, On spaces L

p(x)

and W

1,p(x)

, Czechoslovak

Math. J. 41(1991) 592-618.

8. X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces

k,p(x)

(Ω), J. Math. Anal. Appl. 262(2001) 749-760.

9. X. L. Fan and D. Zhao, On the spaces L

p(x)

(Ω) and W

m,p(x)

(Ω), J. Math.

Anal. Appl. 263(2001) 424-446.

10. M. Mihˇailescu, Existence and multiplicity of solutions for a Neumann prob-

lem involving the p(x)-Laplace operator, Nonlinear Anal., 67(2007) 1419-

1425.

11. J. Musielak, Orlicz spaces and modular spaces. Lecture Notes in Mathemat-

ics, 1034. Springer-Verlag, Berlin (1983).

12. B. Ricceri, On three critical points theorem, Arch. Math. (Basel) 75 (2000),

220-226.

13. B. Ricceri, A three critical points theorem revisited. Nonlinear Anal. to ap-

pear (2008).

14. B. Ricceri, A general variational principle and some of its applications. J.

Comput. Appl. Math. 113(2000) 401-410.

15. M. Ruzicka, Electrorheological ﬂuids: Modelling and Mathematical Theory,

Lecture Notes in Math., Springer-Verlag, Berlin (2002).

16. M. Ruzicka, Flow of shear dependent electrorheological ﬂuids. C. R. A. S.

Sci. Paris. S´er. I. Math. 329(1999) 393-398.

17. X. Shi and X. Ding, Existence and multiplicity of solutions for a general

p(x)-Laplacian Neumann problem, Nonlinear Anal. to appear

18. J. Simon, r´egularit´e de la solution d’une equation non lin´eaire dans IR

LMN 665, P. Benilan ed., Berlin-Heidelberg-New York 1978.

19. E. Zeidler, Nonlinear functional analysis and its applications. Vol. II/B.

Berlin-Heidelberg-New York 1978.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

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Positive solutions with changing sign energy to nonhomogeneous

elliptic problem of fourth order

M. Talbi

∗

and N. Tsouli

†

D´epartement de Math´ematiques et Informatique Facult´e des Sciences

Universit´e Mohamed 1, Oujda, Morocco

E-mails:

∗

talbi

md@yahoo.fr,

†

tsouli@hotmail.com

In this paper, we study the existence for two positive solutions to a nonhomoge-

neous elliptic equation of fourth order with a parameter λ such that 0 < λ <

λ.

The ﬁrst solution has a negative energy while the energy of the second one is

positive for 0 < λ < λ

and negative for λ

< λ <

λ. The values λ

and

are given under variational form and we show that every corresponding critical

point is solution of the nonlinear elliptic problem (with a suitable multiplicative

term).

Keywords: Ekeland’s principle; p-Laplacian operator; Palais-Smale condition.

1. Introduction

We consider the problem with Navier boundary conditions

)



∆

u = λ|u|

q−2

u + |u|

r−2

u in Ω

u = ∆u = 0 on ∂Ω.

Here Ω is a smooth domain in R

(N ≥ 1), ∆

is the p-biharmonic operator

deﬁned by ∆

u = ∆(| ∆u |

p−2

∆u), λ is a positive parameter, p, q and r

are reals such that

1 < q < p < r < p

∗

, where







∗

N − 2p

if p < N/2,

∗

= +∞ if p ≥ N/2.

Such kind of problems with combined concave and convex nonlinearities

were studied recently by several authors

2–7,9–11,17

in the case of operator

∆

. Our main results here can be summarized as follows:

Let us put X = W

2,p

(Ω) ∩ W

2,p

(Ω). We ﬁnd two characteristic values λ

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and

λ (λ

λ) under variational form, i.e.

(V )

= C

(p, q, r) inf

u∈X\{0}

F (u) and

λ =

C(p, q, r) inf

u∈X\{0}

F (u),

such that two branches of positive solutions to (P

) exist for λ ∈]0,

λ[ (the

functional F will be given below). Moreover, the energy of the ﬁrst positive

solution is negative for λ ∈]0,

λ[ while the energy of the second positive

solution changes sign at λ

, i.e. it is positive forλ ∈]0, λ

[ and negative for

λ ∈]λ

λ[. Notice that these two positive solutions are found simultaneously

and that our approach does not use the mountain-pass lemma.

On the other hand, we show that every solution of (V ) is a solution of

the problem (P

) (with a suitable multiplicative term). This second point

lets expect that the ﬁrst nonlinear eigenvalue ζ of (V ), i.e.

ζ = sup{λ > 0 : (P

) has a nonnegative solution}

may satisfy a variational problem similar to (V ) (see

for p = 2). Let us

precise that

λ coincides with ζ when q → p and that

λ constitutes a good

minoration of ζ in the general case 1 < q < p.

We consider the transformation of Poisson problem used by P.Dr´abek

and M.ˆotani (cf.

We recall some properties of the Dirichlet problem for the Poisson equation:



−∆u = f in Ω,

u = 0 on ∂Ω.

(1)

It is well known that (1) is uniquely solvable in W

2,p

(Ω) ∩ W

1,p

(Ω) for all

f ∈ L

(Ω) and for any p ∈]1, +∞[.

We denote by :

X = W

2,p

(Ω) ∩ W

1,p

(Ω),

kuk

= (

Ω

|u|

dx)

1/p

the norm in L

(Ω),

kuk

2,p

= (k∆uk

+ kuk

)

1/p

the norm in X,

kuk

∞

the norm in L

∞

(Ω),

and < ., . > is the duality bracket between L

(Ω) and L

(Ω), where p

p/(p −1). Denote by Λ the inverse operator of −∆ : X → L

(Ω).

The following lemma gives us some properties of the operator Λ (cf.,

1216

)

Lemma 1.1.

(i) (Continuity): There exists a constant c

> 0 such that

kΛfk

2,p

≤ c

kfk

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holds for all p ∈]1, +∞[ and f ∈ L

(Ω).

(ii) (Continuity) Given k ∈ N

∗

, there exists a constant c

p,k

> 0 such that

kΛfk

k+2,p ≤ c

p,k

kfk

k,p

holds for all p ∈]1, +∞[ and f ∈ W

k,p

(Ω).

(iii) (Symmetry) The following identity:

Ω

Λu · vdx =

Ω

u ·Λvdx

holds for all u ∈ L

(Ω) and v ∈ L

(Ω) with p ∈]1, +∞[.

(iv) (Regularity) Given f ∈ L

∞

(Ω), we have Λf ∈ C

1,α

(

Ω) for all α ∈]0, 1[;

moreover, there exists c

> 0 such that

kΛfk

1,α

≤ c

kfk

∞

(v) (Regularity and Hopf-type maximum principle) Let f ∈ C(

Ω) and f ≥ 0

then w = Λf ∈ C

1,α

(

Ω), for all α ∈]0, 1[ and w satisﬁes: w > 0 in

Ω,

∂w

∂n

< 0 on ∂Ω.

(vi) (Order preserving property) Given f, g ∈ L

(Ω) if f ≤ g in Ω, then

Λf < Λg in Ω.

Remark 1.1. (∀u ∈ X)(∀v ∈ L

(Ω)) v = −∆u ⇐⇒ u = Λv.

Let us denote N

the Nemytskii operator deﬁned by



(v)(x) = |v(x)|

p−2

v(x) if v(x) 6= 0

(v)(x) = 0 if v(x) = 0,

and we have ∀v ∈ L

(Ω), ∀w ∈ L

(Ω) :

(v) = w ⇐⇒ v = N

(w).

We deﬁne the functionals P, Q, R : L

(Ω) → R as follows:

P (v) = kvk

, Q(v) = ||Λv||

and R(v) = ||Λv||

The operator Λ enables us to transform problem (P

) to an other problem

which we will study in the space L

(Ω).

Deﬁnition 1.1. We say that u ∈ X \ {o} is a solution of problem (P

), if

v = −∆u is a solution of the following problem

)



Find v ∈ L

(Ω) \ {o}, such that

(v) = λΛ(N

(Λv)) + Λ(N

(Λv)) in L

(Ω).