Existence of Three Solutions for $p$-biharmonic Equation

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doi:10.5899/2011/jnaa-00073 Research Article

Existence of Three Solutions for

p-biharmonic

Equation

Lin Li ∗

Department of Science, Sichuan University of Science and Engineering, Zigong 643000, PR China

Copyright 2011 c⃝ Lin Li. This is an open access article distributed under the Creative Commons Attri-bution License, which permits unrestricted use, distriAttri-bution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the existence of at least three solutions to a Navier boundary problem in-volving thep-biharmonic equation, will be established. The technical approach is mainly based on the three critical points theorem of B. Ricceri.

Keywords: p-biharmonic, Navier condition; Multiple solutions, Three critical points theorem

1 Introduction and main results

Consider the Navier boundary value problem involving the p-biharmonic equation

{ −∆(

|∆u|p−2_∆_u)

=λf(x, u) +µg(x, u), in Ω,

u= ∆u= 0, on∂Ω, (P)

whereλ,µ∈[0,+∞), Ω⊂RN(N ≥1) is a non-empty bounded open set with a sufficient smooth boundary∂Ω,p > max{1, N/2},f,g: Ω×_R→_Rare the Carath´eodory functions.

We recall that a functionf: Ω×R→Ris said to be Carath´eodory if

• t→f(t, x) is measurable for everyx∈_R;

• x→f(t, x) is continuous for a.e. t∈Ω.

Here in the sequel, X will be denoted as the Sobolev space W2,p_(Ω)_∩_W1,p

0 (Ω). The

space X will be endowed with the norm

∥u∥=

(∫

Ω

|∆u|pdx

)1_p

.

∗_{Email address: lilin420@gmail.com}

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Let

K = sup

u∈W2,p_(Ω)_∩_W1,p

0 (Ω)\{0}

sup_x_∈_Ω|u(x)|

∥u∥ . (1.1)

Since p > max{

1,N₂}

,W2,p(Ω)∩W₀1,p(Ω)֒→ C0(Ω) is compact, and one has K <+∞. As always, a weak solution of problem (P) is anyu∈X such that

− ∫

Ω

|∆u|p−2∆u∆ξdx=λ

∫

Ω

f(x, u, v)ξdx+µ

∫

Ω

g(x, u, v)ξdx (1.2)

for everyξ ∈X.

In recent years, the three critical points theorem of B. Ricceri has been widely used to solve differential equations, see [1, 2, 3, 5, 6, 7, 9, 12, 13] and reference therein.

The fourth-order equation of nonlinearity furnishes a model to study traveling waves in suspension bridges; therefore this becomes very significant in Physics. Many authors consider this type of equation, we refer to [4, 8, 10, 11] and there reference therein.

To the best of our knowledge, there are only very few results regarding multiple solu-tions to thep-biharmonic equation. In this paper, the existence of at least three solutions of problem (P) will be proved. The technical approach is based on the three critical points theorem of B. Ricceri [14]. Our theorem, under the new assumptions, ensures the existence of an open interval Λ⊆[0,+∞) and a positive real number ρ such that, for eachλ∈Λ, problem (P) admits at least three weak solutions whose norms inX are less thanρ.

Now, for every x0 ∈ Ω and selected r1, r2 with r2 > r1 > 0, such that B(x0, r1) ⊂

B(x0, r2) ⊆ Ω, where B(x0, r1) denotes the ball with the center at x0 and radius of r1.

Put

θ=

    

   

3KN

(r2−r1)(r2+r1)

(

πN2₍₍_r₂₊_r₁₎N₋₍₂_r₁₎N₎

2N_Γ₍₁₊N

2)

)_p1

, N < 4r1

r2−r1,

12Kr1

(r2−r1)2(r2+r1)

(

πN2₍₍_r₂₊_r₁₎N₋₍₂_r₁₎N₎

2N_Γ₍₁₊N

2)

)1_p

, N ≥ 4r1

r2−r1,

(1.3)

where Γ(·) is the Gamma function. Also, letF(x, t) =∫t

0f(x, ξ)dξ. Our main results are

the following theorems.

Theorem 1.1. Suppose that r2 > r1 > 0, such that B(x0, r2) ⊂ Ω and assuming there exist three positive constants c, d and γ with γ < p, c < θd, and a negative function α∈L1(Ω)such that

(j1) F(x, s)≤0 for a.e. x∈Ω\B(x0, r1) and all s∈[0, d];

(j2) m(Ω) inf(x,s)∈Ω×[−c,c]F(x, s)>

( _c

θd

)p∫

B(x0_,r

1)F(x, d)dx; (j3) F(x, s)≥α(x)(1 +|s|γ) for a.e. x∈Ωand all s∈R.

Then, there exist an open interval Λ ⊆ [0,+∞) and a positive real number ρ with the following property: for each λ ∈ Λ and for each Carath´eodory function g: Ω×R 7→ R, thus satisfying

(j4) sup{|s|≤ζ}|g(·, s)| ∈L1(Ω), for all ζ >0,

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Let h1 ∈C(Ω) be a positive function andh2 ∈C(R) be a negative function. Let

f(x, u) =h1(x)h2(u)

for each (x, u)∈Ω×R,

H(t) =

∫ t

0

h2(ξ)dξ

for allt∈Randα1(x) = _hα₁(₍x_x)₎. Then, using Theorem 1.1, the following result is obtained:

Corollary 1.1. Assume that there exist three positive constants, c, d, and γ withγ < p, c < θd, and a negative functionα∈L1(Ω)such that

(k1) F(x, s)≤0 for a.e. x∈Ω\B(x0, r1) and all s∈[0, d];

(k2) m(Ω) inf(x,s)∈Ω×[−c,c]F(x, s)>

( _c

θd

)p H(d)

H(c)

∫

B(x0

,r1)h1(x)dx;

(k3) F(x, s)≥α1(x)(1 +|s|γ) for a.e. x∈Ωand all s∈R.

Then there exist an open interval Λ ⊆ [0,+∞) and a positive real number ρ with the following property: for each λ ∈ Λ and for two Carath´eodory functions g: Ω×R 7→ R, satisfying

(k4) sup{|s|≤ζ}|g(·, s)| ∈L1(Ω), for all ζ >0,

there exists δ >0 such that, for each µ∈[0, δ], problem

{ −∆(

|∆u|p−2∆u)

=λh1(x)h2(u) +µg(x, u), inΩ,

u= ∆u= 0, on∂Ω,

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

A simple consequence of Theorem 1.1 is as follows:

Theorem 1.2. Let f:R → R be a continuous function. Put F(s) = ∫

Ωf(t)dt for each s∈Rand assume that there exist three positive constants c, d, γ with c < θd, γ < p and a negative constant α such that

(l1) F(s)≤0 for all s∈[0, d];

(l2) m(Ω) inf[−c,c]F(s)>

( _c

θd

)p rN1 π

N

2

Γ(1+N2)

F(d);

(l3) F(s)≥α(1 +|s|γ) for all s∈R.

Then there exist an open interval Λ ⊆ [0,+∞) and a positive real number ρ with the following property: for each λ ∈ Λ and for each Carath´eodory function g: Ω×R 7→ R, satisfying

(l4) sup{|s|≤ζ}|g(·, s)| ∈L1(Ω), for all ζ >0,

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2 Proof of theorems

For the reader’s convenience, the revised form of Ricceri’s three critical points theorem Theorem 1 in [14] and Proposition 3.1 of [12] are recalled.

Theorem 2.1 ([14],Theorem 1). Let X be a reflexive real Banach space. Φ : X 7→ R

is a continuously Gâteaux differentiable and a sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse onX∗ andΦis bounded on each bounded subset ofX;Ψ :X7→Ris a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact; I ⊆Ran interval. Assuming that

lim

∥x∥→+∞(Φ(x) +λΨ(x)) = +∞

for all λ∈I, and that there existsh∈_Rsuch that

sup

λ∈I

inf

x∈X(Φ(x) +λ(Ψ(x) +h))<xinf∈Xsup_λ_∈_I(Φ(x) +λ(Ψ(x) +h)). (2.4)

Then, there exists an open interval Λ⊆I and a positive real numberρ with the following property: for every λ ∈ Λ and every C1 _functional _J_:_X _7→ _R _{with compact derivative,} there exists δ >0 such that, for each µ∈[0, δ], the equation

Φ′(x) +λΨ′(x) +µJ′(x) = 0

has a least three solutions in X whose norms are less than ρ.

In Proposition 3.1 of [12], if a minus is added to Ψ, then hen the result is as follows:

Proposition 2.1 ([12],Proposition 3.1). Let X be a non-empty set and Φ,Ψ two real functions on X. Assuming that there are r >0 andx0, x1 ∈X such that

Φ(x0) = Ψ(x0) = 0, Φ(x1)> r, inf

x∈Φ−1

(]−∞,r])Ψ(x)> r

Ψ(x1)

Φ(x1).

Then, for each h satisfying

sup

x∈Φ−1

(]−∞,r])

−Ψ(x)< h < r−Ψ(x1)

Φ(x1) ,

one has

sup

λ≥0

inf

x∈X(Φ(x) +λ(h+ Ψ(x)))<xinf∈Xsup_λ_≥₀(Φ(x) +λ(h+ Ψ(x))).

Before giving the proof for Theorem 1.1, consider the following two lemmas.

Lemma 2.1. Assume that there exist two positive constants c, d with c < θd, such that

(j1) F(x, s)≤0 for a.e. x∈Ω\B(x0, r1) and all s∈[0, d];

(j2) m(Ω) inf(x,s)∈Ω×[−c,c]F(x, s)>

( _c

θd

) ∫

B(x0_,r

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Then there exists u∗_∈_W2,p_(Ω)_∩_W1,p

0 (Ω)such that

∥u∗∥p > pr,

and

m(Ω) inf

(x,s)∈Ω×[−c,c]F(x, s)>

(

c K∥u∗_∥

) ∫

Ω

F(x, u∗(x))dx.

Proof. Let

w(x) =

      

0, x∈Ω\B(x0, r2),

d(3(l4

−r4

2)−4(r1+r2)(l3−r23)+6r1r2(l2−r22))

(r2−r1)3(r1+r2) , x∈B(x

0_{, r2}₎_\_B₍_x0_{, r1}₎_,

d, x∈B(x0, r1),

(2.5)

whereu∗(x) =w(x) andl= dist(x, x0) =

√ ∑N

i=1(xi−x0i)2 .

We then have

∂u∗(x)

∂xi

=

 



0, x∈Ω\B(x0_{, r}

2)∪B(x0, r1), 12d(l2

(xi−x0i)−(r1+r2)l(xi−x0i)+r1r2(xi−x0i))

(r2−r1)3(r1+r2) , x∈B(x

0_{, r}

2)\B(x0, r1),

∂2u∗(x)

∂2_x

i

=

 



0, x∈Ω\B(x0_{, r}

2)∪B(x0, r1), 12d(r1r2+(2l−r1−r2)(xi−x0i)

2

/l−(r2+r1−l)l)

(r2−r1)3(r1+r2) , x∈B(x

0_{, r}

2)\B(x0, r1),

N

∑

i=1

∂2u∗(x)

∂2_x

i

=

 



0, x∈Ω\B(x0, r2)∪B(x0, r1), 12d((N+2)l2

−(N+1)(r1+r2)l+N r1r2)

(r2−r1)3(r1+r2) , x∈B(x

0_{, r}

2)\B(x0, r1),

It is easy to verify thatu∗∈W2,p(Ω)∩W₀1,p(Ω), and particularly, one has

∥u∗∥p= (12d)

p₂_πN

2 (r2−r1)3p₍_r1₊_r2₎p_Γ(N

2)

∫ r2

r1

|(N+ 2)r2−(N + 1)(r1+r2)r+N r1r2|prN−1dr.

(2.6) Here, we obtain from (1.3) and (2.6) that

θd K <∥u

∗_∥_. _(2.7)

By the assumption

c < θd,

it follows from (2.7) that

∥u∗∥p

p > 1 p ( dθ K )p > 1 p (c K )P .

Since, 0≤u∗≤d, for each x∈Ω, the condition (j1) ensures that

∫

Ω\B(x0_,r 2)

F(x, u∗(x))dx+

∫

B(x0_,r

2)\B(x0,r1)

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Hence, by condition (j2) and (2.7), we have

m(Ω) inf

(x,s)∈Ω×[−c,c]F(x, s)>

( c

θd

)p∫

B(x0_,r 1)

F(x, d)dx

>

(

c K∥u∗_∥

)p∫

B(x0_,r 1)

F(x, d)dx

≥ (

c K∥u∗_∥

)p∫

Ω

F(x, u∗)dx.

Lemma 2.2. Let T:X 7→X∗ _{be the operator defined by}

T(u)(ξ) =

∫

Ω

|∆u(x)|p−2∆u(x)∆ξ(x)dx

for all u, ξ ∈ X, where X∗ _{denotes the dual of} _X_{. Then} _T _{admits a continuous inverse}

onX∗.

Proof. If x, y ∈ RN and ⟨·,·⟩ denotes the usual inner product in RN, owing to (2.2) of [15], there exists a positive constant Cp such that

⟨|x|p−2x− |y|p−2y, x−y⟩ ≥Cp|x−y|p.

Thus, it becomes easy to verify that

(T(u)−T(v))(u−v)≥ ∥x−y∥p.

for every u and v belonging to X. This actually implies that T is a uniformly monotone in X. In addition, standard arguments ensure that T also appears to be coercive and hemicontinuos inX. Therefore, the conclusion follows immediately by applying Theorem 26. A. of [16].

The proof of our main results follows:

Proof of Theorem 1.1. For eachu∈X, let

Φ(u) =∥u∥

p

p , Ψ(u) =−

∫

Ω

F(x, u)dx, J(u) =− ∫

Ω

∫ u(x)

0

g(x, ξ)dξdx.

Under the condition of Theorem 1.1, Φ is a continuously Gâteaux differentiable and a se-quentially weakly lower semicontinuous functional. Moreover, from Lemma 2 the Gâteaux derivative of Φ admits a continuous inverse on X∗_{. Ψ and} _J _{are continuously Gâteaux}

differential functionals whose Gˆateaux derivative is compact. Obviously, Φ is bounded on each bounded subset ofX. Particularly, for each u, ξ∈X,

⟨Φ′(u),(ξ)⟩=

∫

Ω

|∆u(x)|p−2∆u(x)∆ξ(x)dx,

⟨Ψ′(u),(ξ)⟩=

∫

Ω

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⟨J′(u),(ξ)⟩=

∫

Ω

g(x, u)ξ(x)dx.

Hence, it follows from (1.2) that the weak solutions of equation (P) are exactly the solu-tions of the equation

Φ′(u) +λΨ′(u) +µJ′(u) = 0.

Due to (j3), for eachλ >0, it appears that

lim

∥u∥→+∞(Φ(u) +λΨ(u)) = +∞, (2.8)

and therefore the first assumption of Theorem 2.1 holds true. Due to Lemma 2.1, there exists u∗ _∈_X _{such that}

Φ(u∗) = ∥u

∗_∥p p p > 1 p ( c K )p

>0 = Φ(0) (2.9)

and

m(Ω) inf

(x,s)∈Ω×[−c,c]F(x, s)>

(

c K∥u∗_∥

) ∫

Ω

F(x, u∗(x))dx. (2.10)

Now, we can obtain from (1.1) that

sup

x∈Ω

|u(x)| ≤K∥u∥ (2.11)

for each u∈X. Letr = 1_p(_c

K

)p

, for each u∈X such that

Φ(u) = ∥u∥

p

p ≤r.

So, it follows from (2.9) and (2.10) that

inf

{(u)|Φ(u)≤r}(Ψ(u)) = inf  

(u)

∥u∥p

p ≤r

  

∫

Ω

F(x, u)dx

≥m(Ω) inf

(x,s)∈Ω×[−c,c]F(x, s) >

(

c K∥u∗_∥

) ∫

Ω

F(x, u∗(x))dx.

Therefore, using Proposition 1, withu0 = 0 and u1 =u∗, we obtain

sup

λ≥0

inf

x∈X(Φ(x) +λ(h+ Ψ(x)))<xinf∈Xsup_λ_≥₀(Φ(x) +λ(h+ Ψ(x))), (2.12)

and therefore the assumption (2.4) of Theorem 2.1 holds ture.

Now, setI = [0,+∞), by (2.8), (2.12), all the assumptions of Theorem 2.1 are satisfied. Hence, from Theorem 2.1 we conclude.

Proof of Theorem 1.2. From (l2) and since

∫

B(x0_,r

1)F(d)dx=r

N 1 π N 2 Γ(1+N 2)

F(d), we have

m(Ω) inf

s∈Ω×[−c,c]F(s)>

( c

θd

)p∫

B(x0

,r1)

F(d)dx.

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3 Acknowledgment

This work was Supported by Scientific Research Fund of School of Science SUSE (No.10LXYB06).

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