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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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M1">View MathML</a>

(1)
where is the biharmonic operator and Ω is a bounded smooth domain in ().

In problem (1), let , then we get the following Dirichlet problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M2">View MathML</a>

(2)
where and . We let () denote the eigenvalues of −△ in .

Thus, fourth-order problems with 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 solutions when , and by the global bifurcation method. In [[3]], Tarantello found a negative solution when by a degree argument. For problem (1) when , 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 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 of a subcritical (polynomial) growth, say,

(SCP): there exist positive constants and and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M3">View MathML</a>

(3)

where denotes the critical Sobolev exponent. One of the main reasons to assume this condition (SCP) is that they can use the Sobolev compact embedding (). 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 :

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M4">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M5">View MathML</a>

(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 and ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M6">View MathML</a>

(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 . We will also use the symmetric mountain pass theorem to get infinitely many solutions for problem (1) in the general case when nonlinearity f is odd.

Let us now state our results. In this paper, we always assume that . The conditions imposed on are as follows:

(H1): for all , ;

(H2): uniformly for , where is a constant;

(H3): uniformly for ;

(H4): is nondecreasing in for any .

Let be the eigenvalues of and be the eigenfunction corresponding to . Let denote the eigenspace associated to . In fact, . Throughout this paper, we denote by the norm, in and the norm of u in will be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M7">View MathML</a>

(7)
We also define .

Theorem 1.1

Letand assume thatfhas the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H1)-(H4). If, then problem (1) has at least two nontrivial solutions.

Theorem 1.2

Letand assume thatfhas the improved subcritical polynomial growth on Ω (condition (SCPI)), is odd intand satisfies (H3) and (H4). If, then problem (1) has infinitely many nontrivial solutions.

In the case of , we have . 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,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M8">View MathML</a>

(8)
So, we now define the subcritical (exponential) growth in this case as follows:

(SCE): f has subcritical (exponential) growth on Ω, i.e., uniformly on for all .

When 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

Letand assume thatfhas the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)-(H4). If, then problem (1) has at least two nontrivial solutions.

Theorem 1.4

Letand assume thatfhas the subcritical exponential growth on Ω (condition (SCE)), is odd intand satisfies (H3) and (H4). If, then problem (1) has infinitely many nontrivial solutions.

Preliminaries and auxiliary lemmas

Definition 2.1

Let be a real Banach space with its dual space and . For , we say that I satisfies the condition if for any sequence with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M9">View MathML</a>

(9)
there is a subsequence such that converges strongly in E. Also, we say that I satisfies the condition if for any sequence with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M10">View MathML</a>

(10)
there is a subsequence such that 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 thatsatisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M11">View MathML</a>

(11)
for some, andwith. Letbe characterized by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M12">View MathML</a>

(12)
whereis the set of continuous paths joining 0 and . Then there exists a sequencesuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M13">View MathML</a>

(13)

Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M14">View MathML</a>

(14)
where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M15">View MathML</a>

(15)

Define a functional by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M16">View MathML</a>

(16)
where , then .

Lemma 2.1

Letandbe a-eigenfunction withand assume that (H2), (H3) and (SCPI) hold. If, then:

(i) There existsuch thatfor allwith.

(ii) as.

Proof

By (SCPI), (H2) and (H3), for any , there exist , and such that for all ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M17">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M18">View MathML</a>

(18)
Choose such that . By (4), the Poincaré inequality and the Sobolev inequality , we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M19">View MathML</a>

(19)
So, part (i) is proved if we choose small enough.

On the other hand, from (5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M20">View MathML</a>

(20)
Thus part (ii) is proved. □

Lemma 2.2

(see [[12]])

Letbe a bounded domain. Then there exists a constantsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M21">View MathML</a>

(21)
and this inequality is sharp.

Lemma 2.3

Letandbe a-eigenfunction withand assume that (H2), (H3) and (SCE) hold. If, then:

(i) There existsuch thatfor allwith.

(ii) as.

Proof

By (SCE), (H2) and (H3), for any , there exist , , , and such that for all ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M22">View MathML</a>

(22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M23">View MathML</a>

(23)
Choose such that . By (6), the Holder inequality and Lemma 2.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M24">View MathML</a>

(24)
where is sufficiently close to 1, and . So, part (i) is proved if we choose small enough.

On the other hand, from (7) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M25">View MathML</a>

(25)
Thus part (ii) is proved. □

Lemma 2.4

For the functionalIdefined by (3), if condition (H4) holds, and for anywith

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M26">View MathML</a>

(26)
then there is a subsequence, still denoted by, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M27">View MathML</a>

(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 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M28">View MathML</a>

(28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M29">View MathML</a>

(29)
Clearly, (9) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M30">View MathML</a>

(30)

To complete our proof, we first need to verify that is bounded in E. Assume as . Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M31">View MathML</a>

(31)
Since is bounded in E, it is possible to extract a subsequence (denoted also by ) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M32">View MathML</a>

(32)
where , and .

We claim that if as , then . In fact, we set , . Obviously, by (11), a.e. in , noticing condition (H3), then for any given , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M33">View MathML</a>

(33)
From (10), (11) and (12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M34">View MathML</a>

(34)
Noticing that in and can be chosen large enough, so and in Ω. However, if , then and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M35">View MathML</a>

(35)
By as and in view of (11), we observe that , then it follows from Lemma 2.4 and (8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M36">View MathML</a>

(36)
Clearly, (13) and (14) are contradictory. So is bounded in E.

Next, we prove that has a convergence subsequence. In fact, we can suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M37">View MathML</a>

(37)
Now, since f has the improved subcritical growth on Ω, for every , we can find a constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M38">View MathML</a>

(38)
then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M39">View MathML</a>

(39)
Similarly, since in E, . Since is arbitrary, we can conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M40">View MathML</a>

(40)
By (10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M41">View MathML</a>

(41)
From (15) and (16), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M42">View MathML</a>

(42)
So we have in E which means that satisfies . Thus, from the strong maximum principle, we obtain that the functional has a positive critical point , i.e., is a positive solution of problem (1). Similarly, we also obtain a negative solution for problem (1). □

Proof of Theorem 1.2

It follows from the assumptions that I is even. Obviously, and . By the proof of Theorem 1.1, we easily prove that satisfies condition (). 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 , . For all , by (SCPI), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M43">View MathML</a>

(43)
where is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M44">View MathML</a>

(44)
Choose so that the coefficient of in the above formula is . Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M45">View MathML</a>

(45)
for . Since as , as . Choose k so that . Consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M46">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M47">View MathML</a>

(47)
So, for any , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M48">View MathML</a>

(48)
Hence, for every finite dimension subspace , there exists such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M49">View MathML</a>

(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 hold. So, we only need to verify condition . Similar to the previous part of the proof of Theorem 1.1, we easily know that sequence is bounded in E. Next, we prove that has a convergence subsequence. Without loss of generality, suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M50">View MathML</a>

(50)
Now, since has the subcritical exponential growth (SCE) on Ω, we can find a constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M51">View MathML</a>

(51)
Thus, by the Adams-type inequality (see Lemma 2.2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/162/mathml/M52">View MathML</a>

(52)
Similar to the last proof of Theorem 1.1, we have in E, which means that satisfies . Thus, from the strong maximum principle, we obtain that the functional has a positive critical point , i.e., is a positive solution of problem (1). Similarly, we also obtain a negative solution 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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