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Fourth order elliptic boundary value problem with nonlinear term decaying at the origin

Abstract

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.

MSC:35J30, 35J40.

1 Introduction

Let Ω be a bounded domain in R n with smooth boundary Ω. Let c R 1 and g: Ω ¯ ×RR be a C 1 function. In this paper, we consider the number of the weak solutions for the following fourth order elliptic problem with the Dirichlet boundary condition

Δ 2 u + c Δ u = g ( x , u ( x ) ) in  Ω , u = 0 , Δ u = 0 on  Ω .
(1.1)

We assume that g C 1 ( Ω ¯ ×R,R) satisfies the following:

(g1) g C 1 ( Ω ¯ ×R,R),

(g2) g(x,0)=0, g(x,ξ)=o(|ξ|) uniformly with respect to x Ω ¯ ,

(g3) there exists C>0 such that |g(x,ξ)|<C(x,ξ) Ω ¯ ×R.

The eigenvalue problem

Δ u + λ u = 0 in  Ω , u = 0 on  Ω

has infinitely many eigenvalues λ j , j1, which is repeated as often as its multiplicity, and the corresponding eigenfunctions ϕ j , j1 suitably normalized with respect to L 2 (Ω) inner product. The eigenvalue problem

Δ 2 u + c Δ u = Λ u in  Ω , u = 0 , Δ u = 0 on  Ω ,

has also infinitely many eigenvalues Λ j = λ j ( λ j c), j1 and corresponding eigenfunctions ϕ j , j1. We note that

Λ 1 < Λ 2 Λ 3 , Λ j +.

Furthermore, we assume that c R 1 satisfies λ j <c< λ j + 1 .

Jung and Choi [1] proved that (1.1) has at least one nontrivial solution, when c< λ 1 and g satisfies the condition (g1), (g2) and additional conditions

(g3)′ there exists ξ0 such that p(x,ξ)0x Ω ¯ ,

(g4)′ there exist a constant r>0 and an element eH such that e=r, e<ξ and 1 2 r 2 Ω P(x,e)<0,

by reducing problem (1.1) to the problem with bounded nonlinear term and then applying the maximum principle for the elliptic operator −Δ and Δc two times and the mountain pass theorem in the critical point theory. Jung and Choi [2] showed the existence of at least two solutions, one of which is a bounded solution and a large norm solution of (1.1), when g(u) is polynomial growth or exponential growth nonlinear term. The authors proved these results by the variational method and the mountain pass theorem. For the constant coefficient semilinear case Choi and Jung [3] showed that the problem

Δ 2 u + c Δ u = b u + + s in  Ω , u = 0 , Δ u = 0 on  Ω ,
(1.2)

has at least two nontrivial solutions, when c< λ 1 , Λ 1 <b< Λ 2 and s<0 or when λ 1 <c< λ 2 , b< Λ 1 and s>0. The authors obtained these results by using the variational reduction method. The authors [4] also proved that when c< λ 1 , Λ 1 <b< Λ 2 and s<0, (1.2) has at least three nontrivial solutions by using the degree theory. Tarantello [5] also studied the problem

Δ 2 u + c Δ u = b ( ( u + 1 ) + 1 ) in  Ω , u = 0 , Δ u = 0 on  Ω .
(1.3)

She showed that if c< λ 1 and b Λ 1 , then (1.3) has a negative solution. She obtained this result by the degree theory. Micheletti and Pistoia [6] also proved that if c< λ 1 and b Λ 2 , then (1.3) has at least three solutions by the variational linking theorem and Leray-Schauder degree theory.

In this paper, we are trying to find weak solutions of (1.1), that is,

Ω [ Δ 2 u v + c Δ u v g ( x , u ) v ] dx=0,vH,

where H is introduced in Section 2.

We consider the associated functional of (1.1)

I(u)= Ω [ 1 2 | Δ u | 2 c 2 | u | 2 G ( x , u ) ] dx,
(1.4)

where G(x,s)= 0 s g(x,τ)dτ. By (g1), I is well defined.

Our main result is the following.

Theorem 1.1Assume that λ j <c< λ j + 1 , j1, andgsatisfies the conditions (g1)-(g3). Then (1.1) has at least one bounded weak solution.

We prove Theorem 1.1 by approaching the variational method and using the mountain pass theorem for the reduced fourth order elliptic problem with bounded nonlinear term. The outline of the proof of Theorem 1.1 is as follows: In Section 2, we prove that functional I(u) C 1 and the functional I satisfies the Palais-Smale condition. In Section 3, we show that the functional I satisfies the generalized mountain pass theorem, and so, prove that I has at least one nontrival critical point, from which we prove Theorem 1.1.

2 Variational approach

Let L 2 (Ω) be a square integrable function space defined on Ω. Any element u in L 2 (Ω) can be written as

u= h k ϕ k with  h k 2 <.

We define a subspace H of L 2 (Ω) as follows

H= { u L 2 ( Ω ) | | Λ k | h k 2 < } .
(2.1)

Then this is a complete normed space with a norm

u= [ | Λ k | h k 2 ] 1 2 .

Since λ k + and c is fixed, we have Λ k and

  1. (i)

    Δ 2 u+cΔuH implies uH,

  2. (ii)

    uC u L 2 ( Ω ) for some C>0,

  3. (iii)

    u L 2 ( Ω ) =0 if and only if u=0,

which is proved in [7].

Let

H + = { u H h k = 0  if  Λ k < 0 } , H = { u H h k = 0  if  Λ k > 0 } .

Then H= H H + , for uH, u= u + u + H H + . Let P + be the orthogonal projection from H onto H + and P be the orthogonal projection from H onto H . We can write P + u= u + , P u= u , for uH.

By the following Lemma 2.1, the weak solutions of (1.1) coincide with the critical points of the associated functional I(u).

Lemma 2.1Assume that λ j <c< λ j + 1 , j1, andgsatisfies the conditions (g1)-(g3). ThenI(u)is continuous, and Fréchet differentiable inHwith Fréchet derivative

I (u)h= Ω [ Δ u Δ h c u h g ( x , u ) h ] dx.

If we set

F(u)= 1 2 Ω G(x,u)dx,

then F (u)is continuous with respect to weak convergence, F (u)is compact and

F (u)h= Ω g(x,u)hdxfor allhH,

this implies thatI C 1 (H,R)andF(u)is weakly continuous.

The proof of Lemma 2.1 has the similar process to that of the proof in Appendix B in [8].

Now, we shall show that I(u) satisfies the Palais-Smale condition.

Lemma 2.2Assume that λ j <c< λ j + 1 , j1, andgsatisfies the conditions (g1)-(g3). Then the functionalIsatisfies the Palais-Smale condition: Any sequence( u m )inH, for which|I( u m )|Mand I ( u m )0asm, possesses a convergent subsequence.

Proof Let us choose uH. By g C 1 and (g1), G(x,u) is bounded. Then we have

I ( u ) = Ω [ 1 2 | Δ u | 2 c 2 | u | 2 G ( x , u ) ] d x 1 2 { λ 1 ( λ 1 c ) } u L 2 ( Ω ) 2 Ω G ( x , u ) d x .

Since u is bounded and Ω G(x,u)dx is bounded, I(u) is bounded from below. Thus, I satisfies the (PS) condition. □

3 Proof of Theorem 1.1

Now, we recall the generalized mountain pass theorem (cf. Theorem 5.3 in [8]).

Let

B r = { u H u r } , B r = { u H u = r } .

Theorem 3.1 (Generalized mountain pass theorem)

LetHbe a real Banach space withH=VX, whereV{0}and is finite-dimensional. Suppose thatI C 1 (H,R)satisfies(PS)condition, and

  1. (i)

    there are constantsρ,α>0and a bounded neighborhood B ρ of 0 such thatI | B ρ X α, and

  2. (ii)

    there is ane B 1 XandR>ρsuch that ifQ=( B ¯ R V){re0<r<R}, thenI | Q 0.

ThenIpossesses a critical valuebα. Moreover, bcan be characterized as

b= inf γ Γ max u Q I ( γ ( u ) ) ,

where

Γ= { γ C ( Q ¯ , H ) γ = id on Q } .

We shall show that the functionalIsatisfies the generalized mountain pass geometrical assumptions.

Let H j =span{ ϕ 1 ,, ϕ j }. Then H j is a subspace ofHsuch that

H= j N H j andH= H j H j .

Let

Q=( B ¯ R H j ) { r e e B 1 H j , 0 < r < R } .

Lemma 3.1Assume that λ j <c< λ j + 1 andgsatisfies (g1)-(g3). Then

  1. (i)

    there are constantsρ>0,α>0and a bounded neighborhood B ρ of 0 such thatI | B ρ H j α, and

  2. (ii)

    there is ane B 1 H j andR>ρsuch that ifQ=( B ¯ R H j ){re0<r<R}, thenI | Q 0, and

  3. (iii)

    there exists u 0 HQsuch that u 0 >RandI( u 0 )0.

Proof (i) Let u H j . We note that

if u H j , Ω ( Δ 2 u + c Δ u ) udx λ j + 1 ( λ j + 1 c) u L 2 ( Ω ) 2 >0.

Since G(x,u(x)) is bounded, there exists a constant C>0 such that CG(x,u(x))C. Thus, we have

I ( u ) = 1 2 P + u 2 1 2 P u 2 Ω G ( x , u ) 1 2 P + u 2 C

for C>0. There exist ρ>o and α>o such that if u B ρ H j , then I(u)α.

  1. (ii)

    Let us choose an element e B 1 H j . Let u( B r ¯ H j ){re0<r}. Then u=v+w, v B r H j , w=re. We note that

    if v B r H j , Ω ( Δ 2 v + c Δ v ) vdx λ j ( λ j c) v L 2 ( Ω ) 2 <0.

Thus, we have

I ( u ) = 1 2 r 2 1 2 v 2 Ω G ( x , v + r e ) 1 2 r 2 + 1 2 ( λ j ( λ j c ) ) v L 2 ( Ω ) 2 + C

for C>0. Then there exists R>0 such that if uQ=( B ¯ R H j ){re0<r<R}, then I(u) | Q 0, from which we can choose an element u 0 H B R such that I( u 0 )0.

  1. (iii)

    If we choose u 0 HQ, then by (ii), I( u 0 )0. □

Proof of Theorem 1.1 We will show that I(u) has a nontrivial critical point by the generalized mountain pass theorem. By Lemma 2.1, I(u) is continuous and Fréchet differentiable in H. By Lemma 2.2, the functional I satisfies (PS) condition. We note that I(0)=0. By Lemma 3.1, there are constants ρ>0,α>0 and a bounded neighborhood B ρ of 0 such that I | B ρ H j α, and there is an e B 1 H j and R>ρ such that if Q=( B ¯ R H j ){re0<r<R}. Let us set

Γ= { γ C ( Q ¯ , H ) γ = id  on  Q } .

By the generalized mountain pass theorem, I possesses a critical value bα. Moreover, b can be characterized as

b= inf γ Γ max u Q I ( γ ( u ) ) .

Thus, we prove that I has at least one nontrivial critical point. We denote by u ˜ a critical point of I such that I( u ˜ )=b. We claim that b is bounded. In fact, by (iii) of Lemma 3.1, we have

b max 0 t 1 I(t u 0 ),

and by (g3),

I ( t u 0 ) = t 2 ( 1 2 P + u 0 2 1 2 P u 0 2 ) Ω G ( x , t u 0 ) d x t 2 u 0 2 Ω G ( x , t u 0 ) d x t 2 u 0 2 + C 1 = C 2 t 2 + C 2

for some constant C 2 >0. Since 0t1, b is bounded:

b< C ˜ .
(3.1)

We claim that u ˜ is bounded. In fact, by contradiction, Δ 2 u ˜ +cΔ u ˜ =g(x, u ˜ ) and for any K>0, max Ω | u ˜ (x)|>K imply that

b=I( u ˜ )= 1 2 ( P + u ˜ 2 P u ˜ 2 ) Ω G(x, u ˜ )dx

is not bounded, which is absurd to the fact that b=I( u ˜ ) is bounded. Thus, u ˜ is bounded, so (1.1) has at least one bounded weak solution. Thus, we prove Theorem 1.1. □

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Acknowledgements

This work (Tacksun Jung) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2010-0023985).

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Correspondence to Q-Heung Choi.

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Authors’ contributions

TJ carried out the studies for the existence of weak solutions of the fourth order elliptic boundary value problem, participated in the sequence alignment and drafted the manuscript. QC participated in the sequence alignment and drafted the manuscript.

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Jung, T., Choi, QH. Fourth order elliptic boundary value problem with nonlinear term decaying at the origin. J Inequal Appl 2013, 432 (2013). https://doi.org/10.1186/1029-242X-2013-432

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Keywords

  • fourth order elliptic boundary value problem
  • nonlinear term decaying at the origin
  • bounded nonlinear term
  • variational method
  • generalized mountain pass theorem
  • ( PS ) c condition