- Open Access
Solutions of biharmonic equations with mixed nonlinearity
© Liu; licensee Springer. 2014
- Received: 16 August 2014
- Accepted: 2 November 2014
- Published: 8 November 2014
In this paper, we study the following biharmonic equations with mixed nonlinearity: , , , where , , , and is a parameter. The existence of multiple solutions is obtained via variational methods. Some recent results are improved and extended.
MSC: 35J35, 35J60.
- biharmonic equations
- mixed nonlinearity
- variational methods
where is the biharmonic operator, , , , , and . There are many results for biharmonic equations, but most of them are on bounded domains; see –. In addition, biharmonic equations on unbounded domains also have captured a lot of interest; see – and the references therein. Many of these papers are devoted to the study of the existence and multiplicity of solutions for problem (1). In , , , , the authors considered the superlinear case; one considered the sublinear case in –. However, there are not many works focused on the asymptotically linear case. Motivated by the above facts, in the present paper, we shall study problem (1) with mixed nonlinearity, that is, a combination of superlinear and sublinear terms, or asymptotically linear and sublinear terms. So, the aim of the present paper is to unify and generalize the results of the above papers to a more general case. To the best of our knowledge, there have been no works concerning this case up to now, hence this is an interesting and new research problem. For related results, we refer the readers to – and the references therein.
More precisely, we make the following assumptions:
where denotes the Lebesgue measure in ;
where , are defined in (3);
Before stating our result, we denote . The main result of this paper is the following theorem.
Suppose that (V), (F1)-(F3) are satisfied. with. In addition, for any real number:
then there existssuch that, for every, problem (1) has at least two solutions;
(I2): If, then there existssuch that, for every, problem (1) has at least two solutions.
It is easy to check that is asymptotically linear at infinity in u when and is superlinear at infinity in u when . Together with and , we see easily that our nonlinearity is a more general mixed nonlinearity, that is, a combination of sublinear, superlinear, and asymptotically linear terms. Therefore, our result unifies and sharply improves some recent results.
Under assumptions (V), the embeddingis compact for any, whereif, if.
for all .
, and this is achieved by somewith, whereis given in (2).
By Lemma 2.1 and standard arguments, it is easy to prove this lemma, so we omit the proof here. □
Next, we give a useful theorem. It is the variant version of the mountain pass theorem, which allows us to find a sequence.
For any real number, assume that (F1) and (F2) are satisfied, andwith. Then there existssuch that, for every, there exist two positive constants ρ, η such that.
Then it follows from (7) that there exists such that, for every , there exists such that . □
If and , then there exists with such that for all ;
if , then there exists with such that for all .
- (i)In the case , since , we can choose a nonnegative function with
- (ii)In the case , since with , we can choose a nonnegative function such that . Thus, from (F2) and Fatou’s lemma, we have
So, if as , then there exists with such that . This completes the proof. □
For any real number, assume that (V) and (F1)-(F3) are satisfied, andwith. Letbe as in Lemma 2.4. Thendefined by (8) is bounded in E for all.
which implies that is bounded in E since . □
andis a nontrivial solution of problem (1).
Hence, . By Ekeland’s variational principle, there exists a minimizing sequence such that and as . Hence, Lemma 2.1 implies that there exists such that and . □
Proof of Theorem 1.1
By using a standard procedure, we can prove that in E. Moreover, and is another nontrivial solution of problem (1). Therefore, combining with Lemma 2.7, we can prove that problem (1) has at least two nontrivial solutions satisfying and . □
The author thanks the referees and the editors for their helpful comments and suggestions.
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