In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is and the convex nonlinear term is with . By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as , here the explicit expression of is provided.
MSC: 35J35, 35J40, 35J65.
Keywords:biharmonic equations; concave-convex nonlinearities; weight functions
In recent years, there has been extensive attention on semilinear second-order elliptic equations,
here Ω is a bounded smooth domain in (), and λ is a positive parameter; see [1-8] and the references therein. As is sublinear, say, , , the monotone iteration scheme or the method of sub-solutions and super-solutions are effective; see . As is superlinear, for example, , , variational methods are applicable; see . In contrast with the pure sublinear case and the pure superlinear case, in  Ambrosetti et al. considered problem (1.1) when is, roughly, the sum of a sublinear and a superlinear term. To be precise, they considered the following problem:
with . They proved that problem (1.2) admits at least two positive solutions for λ sufficiently small. In , Sun and Li considered a similar problem:
with , the authors studied the value of Λ, the supremum of the set λ, related to the existence and multiplicity of positive solutions and established uniform lower bounds for Λ. In , Wu considered the subcritical case of problem (1.2) with replaced by , here is a sign-changing function, and he showed that problem (1.2) has at least two positive solutions as λ is small enough.
Some interesting generalizations of (1.2) have been provided in the framework of quasi-linear elliptic equations or systems, semilinear second-order elliptic systems or fourth-order elliptic equations. More recently, the semilinear fourth-order elliptic equations have been studied by many authors, we refer the reader to [11-13] and the references therein. Motivated by some work in [6,8,13], we deal with the following semilinear biharmonic elliptic equation:
For convenience and simplicity, we introduce some notations. The norm of u in is denoted by , the norm of u in is denoted by ; is denoted by , endowed with the norm ; S denotes the best Sobolev constant for the embedding of in (see ); to be precise, for all .
Now we define
It is well known that the weak solutions of problem (1.3) are the critical points of the energy functional (see Rabinowitz ).
Similarly to the method used in Tarantello , we split into three parts:
Note that all solutions of (1.3) are clearly in the Nehari manifold, . Hence, our approach to solve problem (1.3) is to analyze the structure of , and then to deal with the minimization problems for on and applying the direct variational method.
The following is our main result.
The paper is organized as follows: in Section 2, we give some lemmas; in Section 3, we prove Theorem 1.1.
In this section, we prove several lemmas.
By (2.1)-(2.2), the Sobolev inequality, and the Hölder inequality, we get
Hence, by (2.5) the desired conclusion yields. □
By the Sobolev inequality, we get
The proof is completed. □
Then we have the following lemma.
From (2.8) and
(iv) By Case II of item (i). □
Associated with (2.9), we consider the energy functional
and the minimization problem
It is easy to verify that , for small and for large. Hence, is reached at a unique such that and . To prove the continuity of , assume that in . It is easy to verify that is bounded. If a subsequence of converges to , it follows from (2.10) that and then . Finally the continuous map from the unit sphere of to , , is inverse to the retraction . □
Proof From Lemma 2.5, we know that . Since for and t large, we obtain . The manifold separates into two components. The component containing the origin also contains a small ball around the origin. Moreover, for all u in this component, because , . Then each has to cross and . Since the embedding is compact (see ), it is easy to prove that is a critical value of K and a positive solution corresponding to c. □
With the help of Lemma 2.6, we have the following result.
Next, we will use the idea of Tarantello  to get the following results.
Proof In view of Lemma 2.8, there exist and a differentiable functional such that , for all and we have (2.12). By use of , we have . In combination with the continuity of the functions and , we get as ϵ sufficiently small, this implies that . □
3 Proof of Theorem 1.1
Proof (i) By Lemma 2.7(ii) and the Ekeland variational principle , there exists a minimizing sequence such that
Taking n large, from Lemma 2.7(i) and (3.1), we have
Now, we will show that
and consequently by (3.9),
Then by (3.4), the Hölder inequality, Sobolev inequality and (3.9)-(3.10), we obtain
Thus, we get from (3.8) that
Hence, by (3.7) it follows that
(ii) Similar to the arguments in (i), by Lemma 2.9 and Lemma 2.2, we can prove (ii). □
First, we claim that
If not, by (3.14) we conclude that
which is a contradiction. Since and , by Lemma 2.3 we may assume that is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we find that is one positive solution of problem (1.3). In addition, by Lemma 2.7,
In addition, from Lemma 2.4(ii)-(iii), we have . Since and , by Lemma 2.3 we may assume that is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we see that is one positive solution of problem (1.3). □
Proof of Theorem 1.1 It is an immediate consequence of Theorems 3.1 and 3.2. □
The authors declare that they have no competing interests.
The authors read and approved the final manuscript.
This work is partly supported by NNSF (11101404, 11201204, 11361053) of China and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNU-LKQN-11-5).
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