Abstract
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.
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1 Introduction and main result
This paper is concerned with the following biharmonic equations:
where is the biharmonic operator, , , , , and . There are many results for biharmonic equations, but most of them are on bounded domains; see [1]–[5]. In addition, biharmonic equations on unbounded domains also have captured a lot of interest; see [6]–[11] and the references therein. Many of these papers are devoted to the study of the existence and multiplicity of solutions for problem (1). In [6], [7], [9], [11], the authors considered the superlinear case; one considered the sublinear case in [8]–[10]. 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 [12]–[14] and the references therein.
More precisely, we make the following assumptions:
(V): and , and there exists a constant such that
where denotes the Lebesgue measure in ;
(F1): , such that for all and . Moreover, there exists with such that
and
where , are defined in (3);
(F2): there exists with such that
(F3): there exist two constants θ, satisfying and such that
where .
Before stating our result, we denote . The main result of this paper is the following theorem.
Theorem 1.1
Suppose that (V), (F1)-(F3) are satisfied. with. In addition, for any real number:
(I1): Ifandwith
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.
Remark 1.2
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.
2 Variational setting and proof of the main result
Now we establish the variational setting for our problem (1). Let
equipped with the inner product
and the norm
Lemma 2.1
([15])
Under assumptions (V), the embeddingis compact for any, whereif, if.
Clearly, E is continuously embedded into and from Lemma 2.1, there exist and such that
Now, on E we define the following functional:
By a standard argument, it is easy to verify that and
for all .
Lemma 2.2
, and this is achieved by somewith, whereis given in (2).
Proof
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.
Theorem 2.3
([16])
Let E be a real Banach space, with dual space, and suppose thatsatisfies
for some, andwith. Letbe characterized by
whereis the set of continuous paths joining 0 and e, then there exists a sequencesuch that
Lemma 2.4
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.
Proof
For any , it follows from the conditions (F1) and (F2) that there exist and such that
Thus, from (3), (6), and the Sobolev inequality, we have, for all ,
which implies that
Take and define
It is easy to prove that there exists such that
Then it follows from (7) that there exists such that, for every , there exists such that . □
Lemma 2.5
For any real number, assume that (F1), (F2) are satisfied, andwith. Letbe as in Lemma 2.4. Then we have the following results:
-
(i)
If and , then there exists with such that for all ;
-
(ii)
if , then there exists with such that for all .
Proof
-
(i)
In the case , since , we can choose a nonnegative function with
Therefore, from (F2) and Fatou’s lemma, we have
So, if as , then there exists with such that .
-
(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. □
Next, we define
where . Then by Theorem 2.3 and Lemmas 2.4 and 2.5, there exists a sequence such that
Lemma 2.6
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.
Proof
For n large enough, from (F2), (3), the Hölder inequality, and Lemma 2.4, we have
which implies that is bounded in E since . □
Lemma 2.7
For any real number, assume that (V) and (F1)-(F2) are satisfied, andwith. Letbe as in Lemma 2.4. Then for every, there existssuch that
andis a nontrivial solution of problem (1).
Proof
Since with , we can choose a function such that
By (9), for , we have
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
From Lemmas 2.1 and 2.6, there exists a constant such that, up to a subsequence,
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 . □
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Liu, B. Solutions of biharmonic equations with mixed nonlinearity. Bound Value Probl 2014, 238 (2014). https://doi.org/10.1186/s13661-014-0238-8
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DOI: https://doi.org/10.1186/s13661-014-0238-8