Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth
© Pei and Zhang; licensee Springer. 2014
Received: 11 February 2014
Accepted: 17 June 2014
Published: 12 July 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.
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.
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 .
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.
(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
We have the following version of the mountain pass theorem (see []).
There exist such that for all with .
There exist such that for all with .
This lemma is essentially due to []. We omit it here. □
Proofs of the main results
Proof of Theorem 1.1
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 [–].
Proof of Theorem 1.3
Proof of Theorem 1.4
Combining the proof of Theorem 1.2 and Theorem 1.3, we easily prove it. □
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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