Existence of infinitely many solutions for a Steklov problem involving the p ( x ) -Laplace operator

Existence of infinitely many solutions for a Steklov problem involving the p ( x ) -Laplace operator

Mostafa Allaoui

, Abdel Rachid El Amrouss and Anass Ourraoui

University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco Received 5 January 2013, appeared 16 May 2014

Communicated by Ondˇrej Došlý

Abstract. In this article, we study the nonlinear Steklov boundary-value problem

∆p(x)u=|u|^p(x)−2u inΩ,

|∇u|^p(x)−2∂u

∂ν = f(x,u) on∂Ω.

We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

Keywords: p(x)-Laplace operator, infinitely many solutions, Ricceri’s variational prin- ciple.

2010 Mathematics Subject Classification: 35J48, 35J60, 35J66.

1 Introduction

Motivated by the developments in elastic mechanics, electrorheological fluids and image restoration [4, 16, 17, 20, 21], the interest in variational problems and differential equations with variable exponent has grown in recent decades; see for example [6,12, 13,15]. We refer the reader to [3,7,8,18,19] for developments in p(x)-Laplacian equations.

The purpose of this article is to study the existence and multiplicity of solutions for the Steklov problem involving the p(x)-Laplacian,

∆_p(x)u= |u|^p(x)−2u inΩ,

|∇u|^p(x)−2∂u

∂ν = f(x,u) on∂Ω. (1.1)

where Ω⊂_R^N (N ≥2)is a bounded smooth domain, ^∂u_∂ν is the outer unit normal derivative on ∂Ω, pis a continuous function on Ωwith N < p⁻ :=inf_x∈Ωp(x)≤ p⁺ := sup_x∈Ωp(x)<

+_∞ and f: ∂Ω×_R → _Ris a continuous function. The main interest in studying such prob- lems arises from the presence of the p(x)-Laplace operator div(|∇u|^p(x)−2∇u), which is a

BCorresponding author. Email: allaoui19@hotmail.com

generalization of the classical p-Laplace operator div(|∇u|^p−2∇u)obtained in the case when p is a positive constant. Many authors have studied the inhomogeneous Steklov problems involving the p-Laplacian [14]. The authors have studied this class of inhomogeneous Steklov problems in the cases of p(x)≡ p = 2 and of p(x)≡ p > 1, respectively. From now, we put X=W^1,p(x)(_Ω)andw:= ^2πN/2

NΓ(^N₂) the measure of theN-dimensional unit ball.

The main results of this paper are as follows.

Theorem 1.1. We assume that f(x,t) = 0for all t≤ 0, a.e x ∈ _{∂Ω, and}infη≥0F(x,η)≥ 0for a.e.

x ∈ ∂Ω. Moreover, suppose that there exist two sequences {a_k}_k∈N and{b_k}_k∈N in (0,+_∞) with a_k <b_k,lim_k→+_∞b_k = +_∞such that

(1) lim_k→+_∞ ^b

p− k

a^p_k^{+ +β} = +_∞for some non-negative constantβ;

(2) max_Ω×[a

k,b_k] f ≤0for all k∈ _N;

(3) there exists a sequence{ξ_k}_k∈N ⊂ _Rsuch thatlim_k→+∞ξ_k = +_∞ and a constant h₀ > ^|Ω|

p⁻|∂Ω|, such that F(x,ξ_k)≥h₀ξ^p

+

k for a.e. x∈ _∂Ω;

(4) lim sup_k→+∞^{max∂Ω×[0,ak]F(x,η)}

b^p_k⁻ < ¹

C₀^p−p⁺|_∂Ω|, where C₀=sup_u∈X\{0} ^|_k^u_u^|∞_k.

Then problem(1.1)admits an unbounded sequence of non-negative weak solutions in X.

Theorem 1.2. We assume that f(x,t) = 0 for all t ≤ 0, a.e. x ∈ ∂Ω, and inf_η≥0F(x,η) ≥ 0 for a.e. x∈∂Ω. Moreover, suppose that there exist two sequences{a_k}_k∈Nand{b_k}_k∈Nin(0,+_∞)with a_k <b_k,lim_k→+∞b_k =0such that

(1) lim_k→+∞ ^b

p− k

a^p_k^{− −α} = +_∞for some non-negative constantα< p⁻; (2) max_Ω×[ak,bk] f ≤0for all k∈ _N;

(3) there exists a sequence{ξ_k}_k∈N ⊂ _R such thatlim_k→+_∞ξ_k = 0⁺ and a constant h0 > _p−^|Ω|_∂Ω^| |, such that F(x,ξ_k)≥h₀ξ^p

−

k for a.e. x∈ ∂Ω; (4) lim sup_k→+∞^{max∂Ω×[0,ak]F(x,η)}

b^p_k⁻ < ¹

C₀^p−p⁺|_∂Ω|, where C₀=_supu∈X\{0} ^|_k^u|∞

uk.

Then problem (1.1) admits a sequence of non-zero non-negative weak solutions, which strongly con- verges to0in X.

Example 1.3. An example of functions satisfying the assumptions of Theorem1.1

F(x,t) =







e^{2z(x)ln 2}h(x)_a^t−bk

k+1−bk

z(x)

ak+1−t ak+1−bk

z(x)

t^q(x), ift∈(b_k,a_k+1);

0, otherwise,

whereh ∈ C(_Ω)with min_x∈Ωh(x)≥ h0, z∈ C(_Ω)with min_x∈Ωz(x) >1 andq∈ C(_Ω)with p⁺ ≤q(x)≤ p⁺+βfor all x∈Ω. Note that in this occasion we can chooseξ_k = ^ak+1₂^+bk.

Example 1.4. An example of functions satisfying the assumptions of Theorem1.2

F(x,t) =







e^{2z(x)ln 2}h(x)_a^t−bk+1

k−bk+1

z(x)

a_k−t ak−bk+1

z(x)

t^q(x), ift ∈(b_k+1,a_k);

0, otherwise,

where h ∈C(_Ω)with min_x∈Ωh(x)≥ h₀,z ∈ C(_Ω)with min_x∈Ωz(x)> 1 andq∈ C(_Ω)with p⁻−α≤ q(x)≤ p⁻for all x∈_Ω. Note that in this occasion we can chooseξ_k = ^bk+1₂^+ak.

Existence of infinitely many solutions for boundary value problems have received a great deal of interest in recent years, see, for instance, the paper [2, 5] and references therein. In [1] we have considered the existence and multiplicity of solutions for the Steklov problem involving the p(x)-Laplacian of the type

(S)

(∆p(x)u= |u|^p(x)−2u inΩ,

|∇u|^p(x)−2∂u

∂ν = λf(x,u) on∂Ω.

Under the following assumptions of the function f, (f₁) | f(x,s)|≤a(x) +b|s |^α(x)−1, ∀(x,s)∈ _∂Ω×_R,

where a(x)∈ L

α(x)

α(x)−1(_∂Ω)andb≥0 is a constant, α(x)∈ C+(_∂Ω). (f2) f(x,t)<0 , when|t| ∈(0, 1),

f(x,t)≥m>0 , whent∈(t₀,∞),t₀>1,

we have established the existence of at least three solutions of this problem.

This article is organized as follows. First, we will introduce some basic preliminary results and lemmas in Section 2. In Section 3, we will give the proofs of our main results.

2 Preliminaries

For completeness, we first recall some facts on the variable exponent spaces L^p(x)(_Ω) and W^k,p(x)(_Ω). For more details, see [9, 10]. Suppose that Ωis a bounded open domain of R^N with smooth boundary ∂Ωand p∈C+(_Ω)where

C+(_Ω) =

p∈C(_Ω) and inf

x∈_Ωp(x)>1

.

Denote by p⁻ := inf_x∈Ωp(x)and p⁺ := sup_x∈Ωp(x). Define the variable exponent Lebesgue space L^p(x)(_Ω)by

L^p(x)(_Ω) =

u:Ω→_Ris a measurable and Z

Ω|u|^p(x)dx< +_∞

, with the norm

|u|_p(x)=inf

τ>0;

Z

Ω

u τ

p(x)

dx≤1

. Define the variable exponent Sobolev spaceW^1,p(x)(_Ω)by

W^1,p(x)(_Ω) =ⁿu∈ L^p(x)(_Ω):|∇u| ∈ L^p(x)(_Ω)^o,

with the norm

kuk=inf (

τ>0;

Z

Ω

∇u τ

p(x)

+^u τ

p(x)!

dx≤1 )

, kuk=|∇u|_p(x)+|u|_p(x).

We refer the reader to [9, 10] for the basic properties of the variable exponent Lebesgue and Sobolev spaces.

Lemma 2.1 ([10]). Both (L^p(x)(_Ω)_,| · |_p(x)) and (W^1,p(x)(_Ω)_,k · k) are separable and uniformly convex Banach spaces.

Lemma 2.2([10]). Hölder inequality holds, namely Z

Ω|uv|dx≤₂|u|_p(x)|v|_q(x) ∀u∈ L^p(x)(_Ω)_,v∈ L^q(x)(_Ω)_, where _p(¹_x)+ ¹

q(x) =1.

Lemma 2.3([10]). Let I(u) =R

Ω |∇u|^p(x)+|u|^p(x)dx, for u∈W^1,p(x)(_Ω)we have

• kuk<₁(=_1,>₁)⇔ I(u)<₁(=_1,>₁);

• kuk ≤1⇒ kuk^p+ ≤ I(u)≤ kuk^p−;

• kuk ≥₁⇒ kuk^p− ≤ I(u)≤ kuk^p+.

Lemma 2.4 ([9]). Assume that the boundary of Ω possesses the cone property and p ∈ C(_Ω)and 1≤q(x)< p^∗(x)for x∈ _Ω, then there is a compact embedding W^1,p(x)(_Ω),→ L^q(x)(_Ω), where

p^∗(x) =

( _{N p(x)}

N−p(x), if p(x)< N;

+_∞, if p(x)≥ N.

Lemma 2.5([9]). The embedding W^1,p(x)(_Ω),→C(_Ω)is compact whenever N< p⁻.

Lemma 2.6 ([11]). Let X be a separable and reflexive real Banach space, φ,ψ: X → _R be two se- quentially weakly lower semicontinuous and Gâteaux differentiable functionals. Assume also thatφis (strongly) continuous and satisfieslim_kuk→+∞φ(u) = +∞. For eachρ>inf_Xφ, put

ϕ(ρ) = inf

x∈φ⁻¹((−_∞,ρ))

ψ(x)−inf

φ⁻¹((−_∞,ρ))_wψ ρ−φ(x) ^, whereφ⁻¹((−_∞,ρ))_wis the closure ofφ⁻¹((−_∞,ρ))_win the weak topology.

1. If there exist a sequence{r_k} ⊂ (inf_Xφ,+_∞)with r_k → +_∞ and a sequence{u_k} ⊂ X such that for each k∈_N,

φ(u_k)<r_k (2.1)

and

ψ(u_k)− inf

φ⁻¹((−_∞,ρ))_w

ψ<r_k−φ(u_k), (2.2) and in addition,

lim inf

kuk→+_∞(φ(_u) +ψ(_u)) =−_∞, (2.3) then there exists a sequence{v_k}of local minima ofφ+ψsuch thatφ(v_k)→+_∞as k→+_∞.

2. If there exist a sequence{r_k} ⊂ (inf_Xφ,+_∞)with r_k → (inf_Xφ)⁺ and a sequence{u_k} ⊂ X such that for each k the conditions(2.1)and(2.2)are satisfied, and in addition, every global mini- mizer ofφis not a local minimizer ofφ+ψ, then there exists a sequence{v_k}of pairwise distinct local minimizers ofφ+ψsuch thatlim_k→+∞φ(v_k) =inf_Xφ, and{v_k}weakly converges to a global minimizer ofφ.

Definition 2.7. We say thatu ∈Xis a weak solution of (1.1) if Z

Ω|∇u|^p(x)−2∇u∇v dx+

Z

Ω|u|^p(x)−2uv dx=

Z

∂Ω f(x,u)v dσ for allv ∈X.

For eachu∈X, we define φ(u) =

Z

Ω

1 p(x)

|∇u|^p(x)+|u|^p(x)dx, ψ(u) =−

Z

∂ΩF(x,u)dσ, J(u) =φ(u) +ψ(u), where F(x,t) =Rt

0 f(x,s)ds. Then we have hφ⁰(u),vi=

Z

Ω

|∇u|^p(x)−2∇u∇v+|u|^p(x)−2uv dx, hψ⁰(u)_,vi=−

Z

∂Ω f(x,u)v dσ, for all v∈X.

Then it is easy to see thatφ,ψ∈C¹(X,R)andu∈ Xis a weak solution of (1.1) if and only ifu is a critical point of the functional J.

Notice thatφis convex and continuous functional so it is a weakly lower semi-continuous.

Since the embeddingX,→C(_Ω)is compact, we can see thatψ: X→_Ris sequentially weakly lower semi-continuous.

3 Proof of main results

For the proof of Theorems 1.1 and 1.2, we will use Lemma 2.6. We start with the following lemmas.

Lemma 3.1. φis coercive.

Proof. Whenkuk ≥1, we have φ(u) =

Z

Ω

1 p(x)

|∇u|^p(x)+|u|^p(x)dx ≥ ¹

p⁺kuk^p−_{, (3.1)} thenφis coercive. The proof is completed.

Sinceφ: X→_Ris coercive we can defineK(r)as K(r) =infn

τ>0 :φ⁻¹((−_∞,r))⊂ B_X(0,τ)^o, (3.2) forr>inf_Xφ, where

B_X(_0,τ) ={u∈ X:kuk<τ}_,

B_X(0,τ)denotes the closure of B_X(0,τ) in X. Since φis coercive, we know 0 < K(r)< +_∞ for each r > infXφ. From the definition of K(r), we have φ⁻¹((−_∞,r)) ⊂ BX(0,K(r)) and consequentlyφ⁻¹((−_∞,r))_w ⊂ B_X(0,K(r)). Since the embeddingX ,→ C(_Ω) is compact, so there is a constantC₀ >0 such that

C₀ = sup

u∈X\{0}

|u|_∞ kuk^. Therefore we have

B_X(0,K(r))⊂ {u∈C(_Ω):|u|_∞ ≤C₀K(r)}. So we have

inf

v∈φ⁻¹((−_∞,r))_w

ψ(v)≥ inf

kvk≤K(r)ψ(v)≥ inf

|v|_∞≤C0K(r)ψ(v). (3.3) Lemma 3.2. For r≥ _p¹+, we have

K(r)≤rp⁺_p¹₋

. (3.4)

Proof. Letr≥ _p¹+ andu∈Xbe such that φ(u)<r. Whenkuk ≥1, by (3.1), we obtain r >φ(u)≥ ¹

p⁺kuk^p−, thus we havekuk< rp^{+ 1}

p−

. Whenkuk<1, it is clear that kuk<₁≤rp⁺, which implies thatkuk< rp⁺_p¹₋

. By the definition ofK(r), (3.4) holds.

Proof of Theorem1.1. We use Lemma2.6(1) to prove Theorem1.1.

Now put r_k = _p¹+

b_k C₀

p⁻

, then lim_k→+_∞r_k = +∞. Using Lemma 3.2, we haveC₀K(r_k) ≤ b_k. Fixx₀∈_Ωand pickγ>0 such thatB(x₀,γ)⊆_Ω.

By condition (2), we have max_{∂Ω×[0,ak]}F = max_{∂Ω×[0,bk]}F. Now we consider the function u_k ∈ Xdefined by

u_k =











0, ifx∈ _Ω\B(x0,γ); η_k, ifx∈ B(x₀,^γ₂);

2ηk

γ γ− |x−x₀|, ifx∈ B(x₀,γ)\B(x₀,^γ₂),

(3.5)

withη_k ∈ (0,a_k]and x_k ∈ _∂Ωsuch that F(x_k,η_k) =max_{∂Ω×[0,ak}]F. Without loss of generality, we may assume thatη_k ≥max(γ, 1). In view of condition (1), we choosek₁∈ _Nsuch that

b_k^p− a^p_k⁺

> ^p

+C₀^p−wγ^N−p+ p⁻2^N

h

2^p+(2^N−1) +2^Nγ^p

+i

, (3.6)

for allk>k₁. For eachk>k₁, using (3.6), we have

φ(_uk) =

Z

Ω

1 p(x)

|∇u_k|^p(x)+|u_k|^p(x)dx

≤ ¹ p⁻

_Z

Ω|∇u_k|^p(x)dx+

Z

Ω|u_k|^p(x)dx

≤ ¹ p⁻

2η_k γ

p⁺h

B x₀,γ

−B x₀,γ 2

i

+

Z

B(x₀,γ)

|u_k|^p(x)dx

≤ ²

p⁺

η^p

+ k wγ^N p⁻γ^p+

1− ¹

2^N

+ ^η

p⁺ k wγ^N

p⁻

= ^η

p⁺ k wγ^N p⁻γ^p+2^N

h 2^p+

2^N−1

+2^Nγ^p

+i

≤ ^a

p⁺ k wγ^N p⁻γ^p+2^N

h 2^p+

2^N−1

+2^Nγ^p

+i

<r_k. Then for eachk >k₁, we have

φ(u_k)<r_k. (3.7)

For eachv∈ φ⁻¹((−_∞,r_k)), we can easily see that for each x∈ _∂Ω F(x,v(x))≤ max

∂Ω×[0,C0K(r_k)]F(x,v)

≤ max

∂Ω×[0,b_k]F(x,v)

= max

∂Ω×[0,a_k]F(x,v)

=F(x_k,η_k). By condition (4), there exists a k₂ ∈_Nsuch that

max_{∂Ω×[0,ak]}F(x,η) b_k^p−

= ^F(x_k,η_k) b_k^p−

< ¹

C₀^p−p⁺|_∂Ω|^{, (3.8)} for every k>k₂. Since lim_k→+∞ ^η

p+ k

b_k^p− =0 and (3.8), for everyk >k₂, we have F(x_k,η_k) + ^η

p+ k wγ^N

|_∂Ω|p⁻γ^p+2^N

h 2^p+

2^N−1

+2^Nγ^p

+i

b_k^p−

< ¹

C₀^p−p⁺|∂Ω|^{. (3.9)} Therefore, using (3.9), we obtain

sup

v∈C(_Ω),|v|_∞≤C0K(r_k) Z

∂ΩF(x,v(x))dσ−

Z

∂ΩF(x,u_k(x))dσ

≤F(x_k,η_k)|∂Ω|

< ^b

p⁻ k

C₀^p−p⁺

− ^η

p⁺ k wγ^N p⁻γ^p+2^N

h 2^p+

2^N−1

+2^Nγ^p

+i

≤r_k−φ(u_k),

for eachk> k₂. Then for everyk> k₂, we have sup

v∈C(_Ω),|v|_∞≤C0K(rk) Z

∂ΩF(x,v(x))dσ−

Z

∂ΩF(x,u_k(x))dσ<r_k−φ(u_k). (3.10) In view of (3.7) and (3.10), for each k > max{k₁,k₂}, we have proved (2.1) and (2.2). We need to verify that the functional φ+ψhas no global minimum, i.e. (2.3). By condition (3), we can find a sequence {ξ_k}_k∈N ⊂ _R such that lim_k→+_∞ξ_k = +_{∞ and F}(_x,ξ_k) ≥ h0ξ^p

+ k for a.e. x ∈ ∂Ω. Now we consider a function w_k ∈ X defined by w_k(x) = ξ_k. Without loss of generality, we may assume thatξ_k ≥1. So we have

φ(w_k) +ψ(w_k)≤ ^ξ

p⁺ k |_Ω|

p⁻ −h₀ξ^p

+ k |_∂Ω|

≤ξ^p

+ k |∂Ω|

|_Ω|

p⁻|_∂Ω|−h₀

. Sinceh₀> _p−^|Ω|_∂Ω^| |. It forces

k→+lim_∞ξ^p

+ k |∂Ω|

|_Ω| p⁻|∂Ω|−h₀

=−_∞.

Therefore, Lemma 2.6(1) assures that there is a sequence {v_k}_k∈N of local minima of φ+ψ such thatφ(v_k)→+_∞ask→+_∞.

It remains to show that the weak solutions obtained are non-negative.

Define

f+(x,t) =

(f(x,t), if t≥0;

0, otherwise, and consider the following problem

(S+)

(∆_p(x)u=|u|^p(x)−2u inΩ,

|∇u|^p(x)−2∂u

∂ν = f+(x,u) _on∂Ω, ifuis weak solution of the problem(S₊), then one has

Z

Ω|∇u|^p(x)−2∇u∇v dx+

Z

Ω|u|^p(x)−2uv dx−

Z

∂Ω f+(x,u)v dσ=0, (3.11) for all v ∈ X. Taking v = _u⁻ in (3.11) shows that ku⁻k = _{0, so u}⁻ = 0. Obviously, u is a non-negative solution of (1.1) in X. This completes the proof.

Proof of Theorem1.2. We use Lemma2.6(2) to prove Theorem1.2.

We putr_k = _p¹+ b_k C0

p⁻

, and consider the functionu_k ∈ Xdefined by

u_k =











0, ifx ∈_Ω\B(x₀,γ); η_k, ifx ∈B(x0,^γ₂);

2η_k

γ (γ− |x−x₀|), ifx ∈B(x₀,γ)\B(x₀,^γ₂).

(3.12)

We can easily get (2.1) and (2.2) using the same method as in the proof of Theorem 1.1. In view of condition (3), we can find a sequence{ξ_k}_k∈N ⊂ _R such that lim_k→+_∞ξ_k = ₀⁺ _and

F(x,ξ_k) ≥ h₀ξ^p

−

k for a.e. x ∈ ∂Ω. If we take w_k = ξ_k, of course the sequence {w_k} _strongly converges to 0 in Xandφ(w_k) +ψ(w_k)< 0 for allk ∈ _{N. Since} φ(0) +ψ(0) =0, this means that 0 is not a local minimum ofφ+ψ.

So, since 0 is the only global minimum of φ. Lemma 2.6(2) ensures that there exists a sequence {v_k} of pairwise distinct local minimizers of φ+ψ such that lim_k→+∞φ(v_k) = 0.

Using the same method as in the proof of Theorem1.1, we can get that each weak solution of problem (1.1) is non-negative. This completes the proof.

Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments.

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