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Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth
Boundary Value Problemsvolume 2014, Article number: 162 (2014)
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.
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, fourth-order problems with have been studied by many authors. In [], 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 [] proved the existence of solutions when , and by the global bifurcation method. In [], Tarantello found a negative solution when by a degree argument. For problem (1) when , Micheletti and Pistoia [] proved that there exist two or three solutions for a more general nonlinearity g by the variational method. Xu and Zhang [] discussed the problem when f satisfies the local superlinearity and sublinearity. Zhang [] proved the existence of solutions for a more general nonlinearity under some weaker assumptions. Zhang and Li [] proved the existence of multiple nontrivial solutions by means of Morse theory and local linking. An and Liu [] and Liu and Wang [] also obtained the existence result for nontrivial solutions when f is asymptotically linear at positive infinity.
(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 :
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 [] 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
We also define .
Letand assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)) and satisfies (H1)-(H4). If, then problem (1) has at least two nontrivial solutions.
Letand assume that f has the improved subcritical polynomial growth on Ω (condition (SCPI)), is odd in t and 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 []) for the fourth-order 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 .
Letand assume that f has the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)-(H4). If, then problem (1) has at least two nontrivial solutions.
Letand assume that f has the subcritical exponential growth on Ω (condition (SCE)), is odd in t and satisfies (H3) and (H4). If, then problem (1) has infinitely many nontrivial solutions.
Preliminaries and auxiliary lemmas
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 []).
Let E be a real Banach space and suppose thatsatisfies the condition
for some, andwith. Letbe characterized by
whereis the set of continuous paths joining 0 and . Then there exists a sequencesuch that
Consider the following problem:
Define a functional by
where , then .
Letandbe a-eigenfunction withand assume that (H2), (H3) and (SCPI) hold. If, then:
There exist such that for all with .
By (SCPI), (H2) and (H3), 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. □
Letbe a bounded domain. Then there exists a constantsuch that
and this inequality is sharp.
Letandbe a-eigenfunction withand assume that (H2), (H3) and (SCE) hold. If, then:
There exist such that for all with .
By (SCE), (H2) and (H3), 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. □
For the functional I defined by (3), if condition (H4) holds, and for anywith
then there is a subsequence, still denoted by, such that
This lemma is essentially due to []. 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 that
To 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 (H3), then for any given , we have
Noticing that in and can be chosen large enough, so and in Ω. However, if , then and consequently
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
Similarly, since in E, . Since is arbitrary, we can conclude that
By (10), we have
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 [–].
Step 1. We claim that condition (i) holds in Theorem 9.12 (see []). 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 []). By (H3), 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 Adams-type 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. □
Lazer AC, McKenna PJ: Large amplitude periodic oscillation in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 1990, 32: 537-578. 10.1137/1032120
Lazer AC, McKenna PJ: Global bifurcation and a theorem of Tarantello. J. Math. Anal. Appl. 1994, 181: 648-655. 10.1006/jmaa.1994.1049
Tarantello G: A note on a semilinear elliptic problem. Differ. Integral Equ. 1992, 5: 561-566.
Micheletti AM, Pistoia A: Multiplicity solutions for a fourth order semilinear elliptic problems. Nonlinear Anal. TMA 1998, 31: 895-908. 10.1016/S0362-546X(97)00446-X
Xu GX, Zhang JH: Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity. J. Math. Anal. Appl. 2003, 281: 633-640. 10.1016/S0022-247X(03)00170-7
Zhang JH: Existence results for some fourth-order nonlinear elliptic problems. Nonlinear Anal. TMA 2001, 45: 29-36. 10.1016/S0362-546X(99)00328-4
Zhang JH, Li SJ: Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. Nonlinear Anal. TMA 2005, 60: 221-230. 10.1016/j.na.2004.07.047
An YK, Liu RY: Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation. Nonlinear Anal. TMA 2008, 68: 3325-3331. 10.1016/j.na.2007.03.028
Liu Y, Wang ZP: Biharmonic equations with asymptotically linear nonlinearities. Acta Math. Sci. 2007, 27: 549-560. 10.1016/S0252-9602(07)60055-1
Lam N, Lu GZ: N -Laplacian equations in with subcritical and critical growth without the Ambrosetti-Rabinowitz condition. Adv. Nonlinear Stud. 2013, 13: 289-308.
Liu ZL, Wang ZQ: On the Ambrosetti-Rabinowitz superlinear condition. Adv. Nonlinear Stud. 2004, 4: 563-574.
Ruf B, Sani F:Sharp Adams-type inequalities in . Trans. Am. Math. Soc. 2013, 365: 645-670. 10.1090/S0002-9947-2012-05561-9
Costa DG, Miyagaki OH: Nontrivial solutions for perturbations of the p -Laplacian on unbounded domains. J. Math. Anal. Appl. 1995, 193: 737-755. 10.1006/jmaa.1995.1264
Zhou HS: Existence of asymptotically linear Dirichlet problem. Nonlinear Anal. TMA 2001, 44: 909-918. 10.1016/S0362-546X(99)00314-4
Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7
Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.
Li GB, Zhou HS: Multiple solutions to p -Laplacian problems with asymptotic nonlinearity as at infinity. J. Lond. Math. Soc. 2002, 65: 123-138. 10.1112/S0024610701002708
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).
The authors declare that they have no competing interests.
The authors read and approved the final manuscript.