Research

# Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

Ruichang Pei12* and Jihui Zhang2

Author Affiliations

1 School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, P.R. China

2 School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing 210097, P.R. China

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Boundary Value Problems 2014, 2014:162  doi:10.1186/s13661-014-0162-y

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/162

 Received: 11 February 2014 Accepted: 17 June 2014 Published: 12 July 2014

© 2014 Pei and Zhang; licensee Springer.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

### Abstract

The main purpose of this paper is to establish the existence of two nontrivial solutions and the existence of infinitely many solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using a suitable version of the mountain pass theorem and the symmetric mountain pass theorem.

##### Keywords:
mountain pass theorem; Adams-type inequality; subcritical polynomial growth; subcritical exponential growth

### Introduction

Consider the following Navier boundary value problem:

{ 2 u ( x ) + c u = f ( x , u ) , in  Ω ; u = u = 0 , in  Ω , (1)
where 2 is the biharmonic operator and Ω is a bounded smooth domain in R N ( N 4 ).

In problem (1), let f ( x , u ) = b [ ( u + 1 ) + 1 ] , then we get the following Dirichlet problem:

{ 2 u ( x ) + c u = b [ ( u + 1 ) + 1 ] , in  Ω ; u = u = 0 , in  Ω , (2)
where u + = max { u , 0 } and b R . We let λ k ( k = 1 , 2 , ) denote the eigenvalues of −△ in H 0 1 ( Ω ) .

Thus, fourth-order problems with N > 4 have been studied by many authors. In [[1]], Lazer and McKenna pointed out that this type of nonlinearity furnishes a model to study traveling waves in suspension bridges. Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied. For problem (2), Lazer and McKenna [[2]] proved the existence of 2 k 1 solutions when N = 1 , and b > λ k ( λ k c ) by the global bifurcation method. In [[3]], Tarantello found a negative solution when b λ 1 ( λ 1 c ) by a degree argument. For problem (1) when f ( x , u ) = b g ( x , u ) , Micheletti and Pistoia [[4]] proved that there exist two or three solutions for a more general nonlinearity g by the variational method. Xu and Zhang [[5]] discussed the problem when f satisfies the local superlinearity and sublinearity. Zhang [[6]] proved the existence of solutions for a more general nonlinearity f ( x , u ) under some weaker assumptions. Zhang and Li [[7]] proved the existence of multiple nontrivial solutions by means of Morse theory and local linking. An and Liu [[8]] and Liu and Wang [[9]] also obtained the existence result for nontrivial solutions when f is asymptotically linear at positive infinity.

We noticed that almost all of works (see [[4]–[9]]) mentioned above involve the nonlinear term f ( x , u ) of a subcritical (polynomial) growth, say,

(SCP): there exist positive constants c 1 and c 2 and q 0 ( 1 , p 1 ) such that

| f ( x , t ) | c 1 + c 2 | t | q 0 for all  t R  and  x Ω , (3)

where p = 2 N / ( N 4 ) denotes the critical Sobolev exponent. One of the main reasons to assume this condition (SCP) is that they can use the Sobolev compact embedding H 2 ( Ω ) H 0 1 ( Ω ) L q ( Ω ) ( 1 q < p ). At that time, it is easy to see that seeking a weak solution of problem (1) is equivalent to finding a nonzero critical points of the following functional on H 2 ( Ω ) H 0 1 ( Ω ) :

I ( u ) = 1 2 Ω ( | u | 2 c | u | 2 ) d x Ω F ( x , u ) d x , where  F ( x , u ) = 0 u f ( x , t ) d t . (4)

In this paper, stimulated by Lam and Lu [[10]], our first main results will be to study problem (1) in the improved subcritical polynomial growth

(SCPI): lim t f ( x , t ) | t | p 1 = 0 (5)
which is much weaker than (SCP). Note that in this case, we do not have the Sobolev compact embedding anymore. Our work is to study problem (1) when nonlinearity f does not satisfy the (AR) condition, i.e., for some θ > 2 and γ > 0 ,
0 < θ F ( x , t ) f ( x , t ) t for all  | t | γ  and  x Ω . (6)
In fact, this condition was studied by Liu and Wang in [[11]] in the case of Laplacian by the Nehari manifold approach. However, we will use a suitable version of the mountain pass theorem to get the nontrivial solution to problem (1) in the general case N > 4 . We will also use the symmetric mountain pass theorem to get infinitely many solutions for problem (1) in the general case N > 4 when nonlinearity f is odd.

Let us now state our results. In this paper, we always assume that f ( x , t ) C ( Ω ¯ × R ) . The conditions imposed on f ( x , t ) are as follows:

(H1): f ( x , t ) t 0 for all x Ω , t R ;

(H2): lim | t | 0 f ( x , t ) t = f 0 uniformly for x Ω , where f 0 is a constant;

(H3): lim | t | f ( x , t ) t = + uniformly for x Ω ;

(H4): f ( x , t ) | t | is nondecreasing in t R for any x Ω .

Let 0 < μ 1 < μ 2 < < μ k < be the eigenvalues of ( 2 c , H 2 ( Ω ) H 0 1 ( Ω ) ) and φ 1 ( x ) > 0 be the eigenfunction corresponding to μ 1 . Let E μ k denote the eigenspace associated to μ k . In fact, μ k = λ k ( λ k c ) . Throughout this paper, we denote by | | p the L p ( Ω ) norm, c < λ 1 in 2 c and the norm of u in H 2 ( Ω ) H 0 1 ( Ω ) will be defined by

u : = ( Ω ( | u | 2 c | u | 2 ) d x ) 1 2 . (7)
We also define E = H 2 ( Ω ) H 0 1 ( Ω ) .

#### Theorem 1.1

Let N > 4 and assume thatfhas the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H1)-(H4). If f 0 < μ 1 , then problem (1) has at least two nontrivial solutions.

#### Theorem 1.2

Let N > 4 and assume thatfhas the improved subcritical polynomial growth on Ω (condition (SCPI)), is odd intand satisfies (H3) and (H4). If f ( x , 0 ) = 0 , then problem (1) has infinitely many nontrivial solutions.

In the case of N = 4 , we have p = + . So it is necessary to introduce the definition of the subcritical (exponential) growth in this case. By the improved Adams inequality (see [[12]]) for the fourth-order derivative, namely,

sup u E , u 1 Ω e 32 π 2 u 2 d x C | Ω | . (8)
So, we now define the subcritical (exponential) growth in this case as follows:

(SCE): f has subcritical (exponential) growth on Ω, i.e., lim t | f ( x , t ) | exp ( α t 2 ) = 0 uniformly on x Ω for all α > 0 .

When N = 4 and f has the subcritical (exponential) growth (SCE), our work is still to study problem (1) without the (AR) condition. Our results are as follows.

#### Theorem 1.3

Let N = 4 and assume thatfhas the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)-(H4). If f 0 < μ 1 , then problem (1) has at least two nontrivial solutions.

#### Theorem 1.4

Let N = 4 and assume thatfhas the subcritical exponential growth on Ω (condition (SCE)), is odd intand satisfies (H3) and (H4). If f ( x , 0 ) = 0 , then problem (1) has infinitely many nontrivial solutions.

### Preliminaries and auxiliary lemmas

#### Definition 2.1

Let ( E , E ) be a real Banach space with its dual space ( E , E ) and I C 1 ( E , R ) . For c R , we say that I satisfies the ( PS ) c condition if for any sequence { x n } E with

I ( x n ) c , D I ( x n ) 0 in  E , (9)
there is a subsequence { x n k } such that { x n k } converges strongly in E. Also, we say that I satisfies the ( C ) c condition if for any sequence { x n } E with
I ( x n ) c , D I ( x n ) E ( 1 + x n E ) 0 , (10)
there is a subsequence { x n k } such that { x n k } converges strongly in E.

We have the following version of the mountain pass theorem (see [[13]]).

#### Proposition 2.1

LetEbe a real Banach space and suppose that I C 1 ( E , R ) satisfies the condition

max { I ( 0 ) , I ( u 1 ) } α < β inf u = ρ I ( u ) (11)
for some α < β , ρ > 0 and u 1 E with u 1 > ρ . Let c β be characterized by
c = inf γ Γ max 0 t 1 I ( γ ( t ) ) , (12)
where Γ = { γ C ( [ 0 , 1 ] , E ) , γ ( 0 ) = 0 , γ ( 1 ) = u 1 } is the set of continuous paths joining 0 and  u 1 . Then there exists a sequence { u n } E such that
I ( u n ) c β and ( 1 + u n ) I ( u n ) E 0 as  n . (13)

Consider the following problem:

{ 2 u + c u = f + ( x , u ) , x Ω , u | Ω = u | Ω = 0 , (14)
where
f + ( x , t ) = { f ( x , t ) , t > 0 , 0 , t 0 . (15)

Define a functional I + : E R by

I + ( u ) = 1 2 Ω ( | u | 2 c | u | 2 ) d x Ω F + ( x , u ) d x , (16)
where F + ( x , t ) = 0 t f + ( x , s ) d s , then I + C 1 ( E , R ) .

#### Lemma 2.1

Let N > 4 and φ 1 > 0 be a μ 1 -eigenfunction with φ 1 = 1 and assume that (H2), (H3) and (SCPI) hold. If f 0 < μ 1 , then:

(i) There exist ρ , α > 0 such that I + ( u ) α for all u E with u = ρ .

(ii) I + ( t φ 1 ) as t + .

#### Proof

By (SCPI), (H2) and (H3), for any ε > 0 , there exist A 1 = A 1 ( ε ) , B 1 = B 1 ( ε ) and l > 2 μ 1 such that for all ( x , s ) Ω × R ,

F + ( x , s ) 1 2 ( f 0 + ε ) s 2 + A 1 s p , (17)
F + ( x , s ) 1 2 l s 2 B 1 . (18)
Choose ε > 0 such that ( f 0 + ε ) < μ 1 . By (4), the Poincaré inequality and the Sobolev inequality | u | p p K u p , we get
I + ( u ) 1 2 u 2 f 0 + ε 2 | u | 2 2 A 1 | u | p p 1 2 ( 1 f 0 + ε μ 1 ) u 2 A 1 K u p . (19)
So, part (i) is proved if we choose u = ρ > 0 small enough.

On the other hand, from (5) we have

I + ( t φ 1 ) 1 2 ( 1 l μ 1 ) t 2 + B 1 | Ω | as  t . (20)
Thus part (ii) is proved. □

#### Lemma 2.2

(see [[12]])

Let Ω R 4 be a bounded domain. Then there exists a constant C > 0 such that

sup u E , u 1 Ω e 32 π 2 u 2 d x C | Ω | , (21)
and this inequality is sharp.

#### Lemma 2.3

Let N = 4 and φ 1 > 0 be a μ 1 -eigenfunction with φ 1 = 1 and assume that (H2), (H3) and (SCE) hold. If f 0 < μ 1 , then:

(i) There exist ρ , α > 0 such that I + ( u ) α for all u E with u = ρ .

(ii) I + ( t φ 1 ) as t + .

#### Proof

By (SCE), (H2) and (H3), for any ε > 0 , there exist A 1 = A 1 ( ε ) , B 1 = B 1 ( ε ) , κ > 0 , q > 2 and l > 2 μ 1 such that for all ( x , s ) Ω × R ,

F + ( x , s ) 1 2 ( f 0 + ε ) s 2 + A 1 exp ( κ | s | 2 ) s q , (22)
F + ( x , s ) 1 2 l s 2 B 1 . (23)
Choose ε > 0 such that ( f 0 + ε ) < μ 1 . By (6), the Holder inequality and Lemma 2.2, we get
I + ( u ) 1 2 u 2 f 0 + ε 2 | u | 2 2 A 1 Ω exp ( κ | u | 2 ) | u | q d x 1 2 ( 1 f 0 + ε μ 1 ) u 2 A 1 ( Ω exp ( κ r u 2 ( | u | u ) 2 ) d x ) 1 r ( Ω | u | r q d x ) 1 r 1 2 ( 1 f 0 + ε μ 1 ) u 2 C u q , (24)
where r > 1 is sufficiently close to 1, u σ and κ r σ 2 < 32 π 2 . So, part (i) is proved if we choose u = ρ > 0 small enough.

On the other hand, from (7) we have

I + ( t φ 1 ) 1 2 ( 1 l μ 1 ) | t | 2 + B 1 | Ω | as  t . (25)
Thus part (ii) is proved. □

#### Lemma 2.4

For the functionalIdefined by (3), if condition (H4) holds, and for any { u n } E with

I ( u n ) , u n 0 as  n , (26)
then there is a subsequence, still denoted by { u n } , such that
I ( t u n ) 1 + t 2 2 n + I ( u n ) for all  t R  and  n N . (27)

#### Proof

This lemma is essentially due to [[14]]. We omit it here. □

### Proofs of the main results

#### Proof of Theorem 1.1

By Lemma 2.1 and Proposition 2.1, there exists a sequence { u n } E such that

I + ( u n ) = 1 2 u n 2 Ω F + ( x , u n ) d x = c + o ( 1 ) , (28)
( 1 + u n ) I + ( u n ) E 0 as  n . (29)
Clearly, (9) implies that
I + ( u n ) , u n = u n 2 Ω f + ( x , u n ( x ) ) u n d x = o ( 1 ) . (30)

To complete our proof, we first need to verify that { u n } is bounded in E. Assume u n + as n . Let

s n = 2 c u n , w n = s n u n = 2 c u n u n . (31)
Since { w n } is bounded in E, it is possible to extract a subsequence (denoted also by { w n } ) such that
w n w 0 in  E , w n + w 0 + in  L 2 ( Ω ) , w n + ( x ) w 0 + ( x ) a.e.  x Ω , | w n + ( x ) | h ( x ) a.e.  x Ω , (32)
where w n + = max { w n , 0 } , w 0 E and h L 2 ( Ω ) .

We claim that if u n + as n + , then w + ( x ) 0 . In fact, we set Ω 1 = { x Ω : w + = 0 } , Ω 2 = { x Ω : w + > 0 } . Obviously, by (11), u n + + a.e. in Ω 2 , noticing condition (H3), then for any given K > 0 , we have

lim n + f ( x , u n + ) u n + ( w n + ( x ) ) 2 K w + ( x ) 2 for a.e.  x Ω 2 . (33)
From (10), (11) and (12), we obtain
4 c = lim n + w n 2 = lim n + Ω f ( x , u n + ) u n + ( w n + ) 2 d x Ω 2 lim n + f ( x , u n + ) u n + ( w n + ) 2 d x K Ω 2 ( w + ) 2 d x . (34)
Noticing that w + > 0 in Ω 2 and K > 0 can be chosen large enough, so | Ω 2 | = 0 and w + 0 in Ω. However, if w + 0 , then lim n + Ω F ( x , w n + ) d x = 0 and consequently
I + ( w n ) = 1 2 w n 2 + o ( 1 ) = 2 c + o ( 1 ) . (35)
By u n + as n + and in view of (11), we observe that s n 0 , then it follows from Lemma 2.4 and (8) that
I + ( w n ) = I + ( s n u n ) 1 + s n 2 2 n + I + ( u n ) c > 0 as  n + . (36)
Clearly, (13) and (14) are contradictory. So { u n } is bounded in E.

Next, we prove that { u n } has a convergence subsequence. In fact, we can suppose that

u n u in  E , u n u in  L q ( Ω ) , 1 q < p , u n ( x ) u ( x ) a.e.  x Ω . (37)
Now, since f has the improved subcritical growth on Ω, for every ε > 0 , we can find a constant C ( ε ) > 0 such that
f + ( x , s ) C ( ε ) + ε | s | p 1 , ( x , s ) Ω × R , (38)
then
| Ω f + ( x , u n ) ( u n u ) d x | C ( ε ) Ω | u n u | d x + ε Ω | u n u | | u n | p 1 d x C ( ε ) Ω | u n u | d x + ε ( Ω ( | u n | p 1 ) p p 1 d x ) p 1 p ( Ω | u n u | p ) 1 p C ( ε ) Ω | u n u | d x + ε C ( Ω ) . (39)
Similarly, since u n u in E, Ω | u n u | d x 0 . Since ε > 0 is arbitrary, we can conclude that
Ω ( f + ( x , u n ) f + ( x , u ) ) ( u n u ) d x 0 as  n . (40)
By (10), we have
I + ( u n ) I + ( u ) , ( u n u ) 0 as  n . (41)
From (15) and (16), we obtain
Ω [ | ( u n u ) | 2 c | ( u n u ) | 2 ] d x 0 as  n . (42)
So we have u n u in E which means that I + satisfies ( C ) c . Thus, from the strong maximum principle, we obtain that the functional I + has a positive critical point u 1 , i.e., u 1 is a positive solution of problem (1). Similarly, we also obtain a negative solution u 2 for problem (1). □

#### Proof of Theorem 1.2

It follows from the assumptions that I is even. Obviously, I C 1 ( E , R ) and I ( 0 ) = 0 . By the proof of Theorem 1.1, we easily prove that I ( u ) satisfies condition ( C ) c ( c > 0 ). Now, we can prove the theorem by using the symmetric mountain pass theorem in [[15]–[17]].

Step 1. We claim that condition (i) holds in Theorem 9.12 (see [[16]]). Let V 1 = E μ 1 E μ 2 E μ k , V 2 = E V 1 . For all u V 2 , by (SCPI), we have

I ( u ) = 1 2 Ω ( | u | 2 c | u | 2 ) d x Ω F ( x , u ) d x 1 2 Ω ( | u | 2 c | u | 2 ) d x c 3 Ω | u | p d x c 4 u 2 ( 1 2 c 5 λ k + 1 ( 1 a ) p / 2 u p 2 ) c 6 , (43)
where a ( 0 , 1 ) is defined by
1 p = a ( 1 2 1 N ) + ( 1 a ) 1 2 . (44)
Choose ρ = ρ ( k ) = u so that the coefficient of ρ 2 in the above formula is 1 4 . Therefore
I ( u ) 1 4 ρ 2 c 6 (45)
for u B ρ V 2 . Since λ k as k , ρ ( k ) as k . Choose k so that 1 4 ρ 2 > 2 c 6 . Consequently
I ( u ) 1 8 ρ 2 α . (46)
Hence, our claim holds.

Step 2. We claim that condition (ii) holds in Theorem 9.12 (see [[16]]). By (H3), there exists large enough M such that

F ( x , t ) M t 2 c 7 , x Ω , t R . (47)
So, for any u E { 0 } , we have
I ( t u ) = 1 2 t 2 Ω ( | u | 2 c | u | 2 ) d x Ω F ( x , t u ) d x 1 2 t 2 u 2 M t 2 Ω u 2 d x + c 7 | Ω | as  t + . (48)
Hence, for every finite dimension subspace E ˜ E , there exists R = R ( E ˜ ) such that
I ( u ) 0 , u E ˜ B R ( E ˜ ) (49)
and our claim holds. □

#### Proof of Theorem 1.3

By Lemma 2.3, the geometry conditions of the mountain pass theorem (see Proposition 2.1) for the functional I + hold. So, we only need to verify condition ( C ) c . Similar to the previous part of the proof of Theorem 1.1, we easily know that ( C ) c sequence { u n } is bounded in E. Next, we prove that { u n } has a convergence subsequence. Without loss of generality, suppose that

u n β , u n u in  E , u n u in  L q ( Ω ) , q 1 , u n ( x ) u ( x ) a.e.  x Ω . (50)
Now, since f + has the subcritical exponential growth (SCE) on Ω, we can find a constant C β > 0 such that
| f + ( x , t ) | C β exp ( 32 π 2 2 β 2 | t | 2 ) , ( x , t ) Ω × R . (51)
Thus, by the Adams-type inequality (see Lemma 2.2),
| Ω f + ( x , u n ) ( u n u ) d x | C ( Ω exp ( 32 π 2 β 2 | u n | 2 ) d x ) 1 2 | u n u | 2 C ( Ω exp ( 32 π 2 β 2 u n 2 | u n u n | 2 ) d x ) 1 2 | u n u | 2 C | u n u | 2 0 . (52)
Similar to the last proof of Theorem 1.1, we have u n u in E, which means that I + satisfies ( C ) c . Thus, from the strong maximum principle, we obtain that the functional I + has a positive critical point u 1 , i.e., u 1 is a positive solution of problem (1). Similarly, we also obtain a negative solution u 2 for problem (1). □

#### Proof of Theorem 1.4

Combining the proof of Theorem 1.2 and Theorem 1.3, we easily prove it. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors read and approved the final manuscript.

### Acknowledgements

This study was supported by the National NSF (Grant No. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C).

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