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 fourthorder 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; Adamstype inequality; subcritical polynomial growth; subcritical exponential growthIntroduction
Consider the following Navier boundary value problem:
where is the biharmonic operator and Ω is a bounded smooth domain in ().In problem (1), let , then we get the following Dirichlet problem:
where and . We let () denote the eigenvalues of −△ in .Thus, fourthorder 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 fourthorder 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
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 :
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
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 , 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:
(H_{1}): for all , ;
(H_{2}): uniformly for , where is a constant;
(H_{3}): uniformly for ;
(H_{4}): 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
We also define .Theorem 1.1
Letand assume thatfhas the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H_{1})(H_{4}). 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 (H_{3}) and (H_{4}). 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 fourthorder derivative, namely,
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 (H_{1})(H_{4}). 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 (H_{3}) and (H_{4}). 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
there is a subsequence such that converges strongly in E. Also, we say that I satisfies the condition if for any sequence with 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
for some, andwith. Letbe characterized bywhereis the set of continuous paths joining 0 and . Then there exists a sequencesuch thatConsider the following problem:
whereDefine a functional by
where , then .Lemma 2.1
Letandbe aeigenfunction withand assume that (H_{2}), (H_{3}) and (SCPI) hold. If, then:
(i) There existsuch thatfor allwith.
(ii) as.
Proof
By (SCPI), (H_{2}) and (H_{3}), for any , there exist , and such that for all ,
Choose such that . By (4), the Poincaré inequality and the Sobolev inequality , we get So, part (i) is proved if we choose small enough.On the other hand, from (5) we have
Thus part (ii) is proved. □Lemma 2.2
(see [[12]])
Letbe a bounded domain. Then there exists a constantsuch that
and this inequality is sharp.Lemma 2.3
Letandbe aeigenfunction withand assume that (H_{2}), (H_{3}) and (SCE) hold. If, then:
(i) There existsuch thatfor allwith.
(ii) as.
Proof
By (SCE), (H_{2}) and (H_{3}), for any , there exist , , , and such that for all ,
Choose such that . By (6), the Holder inequality and Lemma 2.2, we get 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
Thus part (ii) is proved. □Lemma 2.4
For the functionalIdefined by (3), if condition (H_{4}) holds, and for anywith
then there is a subsequence, still denoted by, such thatProof
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
Clearly, (9) implies thatTo complete our proof, we first need to verify that is bounded in E. Assume as . Let
Since is bounded in E, it is possible to extract a subsequence (denoted also by ) such that where , and .We claim that if as , then . In fact, we set , . Obviously, by (11), a.e. in , noticing condition (H_{3}), then for any given , we have
From (10), (11) and (12), we obtain Noticing that in and can be chosen large enough, so and in Ω. However, if , then and consequently By as and in view of (11), we observe that , then it follows from Lemma 2.4 and (8) that 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
Now, since f has the improved subcritical growth on Ω, for every , we can find a constant such that then Similarly, since in E, . Since is arbitrary, we can conclude that By (10), we have From (15) and (16), we obtain 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
where is defined by Choose so that the coefficient of in the above formula is . Therefore for . Since as , as . Choose k so that . Consequently Hence, our claim holds.Step 2. We claim that condition (ii) holds in Theorem 9.12 (see [[16]]). By (H_{3}), there exists large enough M such that
So, for any , we have Hence, for every finite dimension subspace , there exists such that 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
Now, since has the subcritical exponential growth (SCE) on Ω, we can find a constant such that Thus, by the Adamstype inequality (see Lemma 2.2), 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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