Abstract
In this paper, we consider semilinear biharmonic equations with concaveconvex 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; concaveconvex nonlinearities; weight functions1 Introduction
In recent years, there has been extensive attention on semilinear secondorder elliptic equations,
here Ω is a bounded smooth domain in (), and λ is a positive parameter; see [18] and the references therein. As is sublinear, say, , , the monotone iteration scheme or the method of subsolutions and supersolutions are effective; see [9]. As is superlinear, for example, , , variational methods are applicable; see [10]. In contrast with the pure sublinear case and the pure superlinear case, in [2] 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 [6], 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 [8], Wu considered the subcritical case of problem (1.2) with replaced by , here is a signchanging 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 quasilinear elliptic equations or systems, semilinear secondorder elliptic systems or fourthorder elliptic equations. More recently, the semilinear fourthorder elliptic equations have been studied by many authors, we refer the reader to [1113] and the references therein. Motivated by some work in [6,8,13], we deal with the following semilinear biharmonic elliptic equation:
where Ω is a bounded smooth domain in (), ( for and for ), is a parameter, is a positive or signchanging weight function and is a positive weight function.
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 [14]); 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 [15]).
Next, we consider the Nehari minimization problem: for ,
Similarly to the method used in Tarantello [16], 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.
Theorem 1.1Letwith, then problem (1.3) has at least two positive solutions for any.
The paper is organized as follows: in Section 2, we give some lemmas; in Section 3, we prove Theorem 1.1.
2 Preliminaries
In this section, we prove several lemmas.
Lemma 2.1For (whereis given in Theorem 1.1), we have.
Proof Suppose that for all . If , then we have
and
By (2.1)(2.2), the Sobolev inequality, and the Hölder inequality, we get
and
where . Thus, using (2.3) and (2.4), we have
Hence, by (2.5) the desired conclusion yields. □
Proof From , it is easy to see that
By the Sobolev inequality, we get
In addition,
The proof is completed. □
By Lemma 2.1, for we write and define
The following lemma shows that the minimizers on are ‘usually’ critical points for .
Lemma 2.3For, ifis a local minimizer foron, thenin.
Proof If is a local minimizer for on , then is a solution of the optimization problem
Hence, by the theory of Lagrange multipliers, there exists such that
Thus,
From and Lemma 2.1, we have and . So, by (2.6)(2.7) we get in . □
Then we have the following lemma.
(i) there is a uniquesuch thatand;
(ii) is a continuous function for nonzerou;
(iv) if, then there is a uniquesuch thatand.
Then we have , as , is concave and reaches its maximum at . Moreover,
There is a unique such that and . Now,
and
and
From (2.8) and
there exist unique and such that ,
and
Similar to the argument in Case I above, we have , , and
(ii) By the uniqueness of and the external property of , we find that is continuous function of .
(iii) For , let . By item (i), there is a unique such that , that is, . Since , we have , which implies
Conversely, let such that . Then . Therefore,
(iv) By Case II of item (i). □
By and changes sign in Ω, we have is an open set in . Without loss of generality, we may assume that Θ is a domain in . Consider the following biharmonic equation:
Associated with (2.9), we consider the energy functional
and the minimization problem
where . Now we prove that problem (2.9) has a positive solution such that .
Lemma 2.5For any, there exists a uniquesuch that. The maximum offoris reached at, the map
is continuous and the induced continuous mapdefines a homeomorphism of the unit sphere ofwith.
Proof For any given , consider the function , . Clearly,
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 . □
Define
Lemma 2.6is a critical value ofK.
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 [14]), 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.
Lemma 2.7 (i) For, there existssuch that
(ii) is coercive and bounded below onfor all.
Proof (i) Let be a positive solution of problem (2.9) such that . Then
Set as defined by Lemma 2.4(iv). Hence, and
This implies
(ii) For , we have . Then by the Hölder, Sobolev, and Young inequalities,
Next, we will use the idea of Tarantello [16] to get the following results.
Lemma 2.8Forand any given, there existand a differentiable functionalsuch that, the functionand
Since and by Lemma 2.1, we obtain
we can get the desired results applying the implicit function theorem at the point . □
Lemma 2.9Forand any given, there existand a differentiable functionalsuch that, the functionand
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
Firstly, we provide the existence of minimizing sequences for on and as λ is sufficiently small.
(i) there exists a minimizing sequencesuch that
(ii) there exists a minimizing sequencesuch that
Proof (i) By Lemma 2.7(ii) and the Ekeland variational principle [17], there exists a minimizing sequence such that
and
Taking n large, from Lemma 2.7(i) and (3.1), we have
This implies
that is,
Now, we will show that
Exactly as in Lemma 2.8 we may apply suitable functionals to and obtain
Hence, if and small, substituting in (3.6) and applying (3.2), we have
Dividing by and passing to the limit as we derive
Since
for a suitable positive constant .
Next, we show that is bounded away from zero. Arguing by contradiction, assume that
and consequently by (3.9),
Then by (3.4), the Hölder inequality, Sobolev inequality and (3.9)(3.10), we obtain
moreover, , which contradicts (3.5).
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). □
Now, we establish the existence of a local minimum for on .
Theorem 3.1Let, then the functionalhas a minimizerinand it satisfies
(ii) is a positive solution of problem (1.3);
Proof By Proposition 3.1(i), there is a minimizing sequence for on such that
Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence and such that
and
First, we claim that
If not, by (3.14) we conclude that
In combination with (3.11)(3.14), it is easy to verify that is a nontrivial weak solution of problem (1.3).
Now we prove that strongly in . Supposing the contrary, then and so
this contradicts . Hence, strongly in . This implies
Moreover, we have . In fact, if , by Lemma 2.4, there exist unique and such that and , we get . Since
and
there exists such that . By Lemma 2.4,
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,
Next, we establish the existence of a local minimum for on .
Theorem 3.2Let, then the functionalhas a minimizerinand it satisfies
(ii) is a positive solution of problem (1.3).
Proof By Proposition 3.1(ii), there is a minimizing sequence for on such that
Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence and such that
and
Connecting with Lemma 2.2, it is easy to see that is a nontrivial weak solution of problem (1.3).
Next we prove that strongly in . Supposing the contrary, then and so
this contradicts . Hence, strongly in . This implies
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. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Acknowledgements
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 (NWNULKQN115).
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