Multiplicity Results for p-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions
Academic Editor: Pavel Drábek
Received04 Dec 2008
Revised24 Jun 2009
Accepted07 Sept 2009
Published19 Oct 2009
The multiple results of positive solutions for the following quasilinear
elliptic equation: in on , are established. Here, is a bounded smooth domain in denotes the -Laplacian operator, is a positive real parameter, and are continuous functions on which are somewhere positive but which may change sign on . The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.
In this paper, we study the multiple results of positive solutions for the following quasilinear elliptic equation:
where , is the -Laplacian, is a bounded domain in with smooth boundary , , is the so-called critical Sobolev exponent and the weight functions are satisfying the following conditions:
() and ;() there exist and such that and for all . Without loss of generality, we assume that ,() and ;();() for all ;() there exists such that
For the weight functions , () has been studied extensively. Historically, the role played by such concave-convex nonlinearities in producing multiple solutions was investigated first in the work . They studied the following semilinear elliptic equation:
for and showed the existence of such that (1.2) admits at least two solutions for all and no solution for . Subsequently, in the work [2, 3], the corresponding quasilinear version has been studied
where and . They obtained results similar to the results of  above, but only for some ranges of the exponents and . We summarize their results in what follows.
Theorem 1.1 (see [2, 3]). Assume that either or . Then there exists such that (1.3) admits at least two solutions for all and no solution for .
It is possible to get complete multiplicity result for problem (1.3) if is taken to be a ball in . Prashanth and Sreenadh  have studied (1.3) in the unit ball in and obtained the following results.
Theorem 1.2 (see ). Let . Then there exists such that (1.3) admits at least two solutions for all and no solution for . Additionally, if , then (1.3) admits exactly two solutions for all small .
For , Tang  has studied the exact multiplicity about the following semilinear elliptic equation:
where and . We also mention his result below.
Theorem 1.3 (see ). There exists such that (1.4) admits exactly two solutions for , exactly one solution for and no solution for .
To proceed, we make some motivations of the present paper. Recently, in  the author has considered (1.2) with subcritical nonlinearity of concave-convex type, and is a continuous function which changes sign in , and showed the existence of such that (1.2) admits at least two solutions for all via the extraction of Palais-Smale sequences in the Nehair manifold. In a recent work , the author extended the results of  to the quasilinear case with the more general weight functions but also having subcritical nonlinearity of concave-convex type. In the present paper, we continue the study of  by considering critical nonlinearity of concave-convex type and sign-changing weight functions .
In this paper, we use a variational method involving the Nehari manifold to prove the multiplicity of positive solutions. The Nehari method has been used also in  to prove the existence of multiple for a singular elliptic problem. The existence of at least one solution can be obtained by using the same arguments as in the subcritical case . The existence of a second solution needs different arguments due to the lack of compactness of the Palais-Smale sequences. For what, we need addtional assumptions and to prove the compactness of the extraction of Palais-Smale sequences in the Nehari manifold (see Theorem 4.4). The multiplicity result is proved only for the parameter (see Theorem 1.5) but for all and . This is not the case in the papers referred [2, 3] where the multiplicity is global but not with the full range of , and with the weight functions . Finally, we mention a recent contribution on -Laplacian equation with changing sign nonlinearity by Figuereido et al.  which gives the global multiplicity but not with the full range of and . The method used in the paper by Figuereido et al. is similar to the method introduced in .
In order to represpent our main results, we need to define the following constant . Set
where is the Lebesgue measure of and is the best Sobolev constant (see (2.2)).
Theorem 1.4. Assume and hold. If , then () admits at least one positive solution for some .
Theorem 1.5. Assume that - and hold. If , then () admits at least two positive solutions for some .
This paper is organized as follows. In Section 2, we give some preliminaries and some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorems 1.4 and 1.5.
2. Preliminaries and Nehari Manifold
Throughout this paper, and will be assumed. The dual space of a Banach space will be denoted by . denotes the standard Sobolev space with the following norm:
with the norm is simply denoted by . We denote the norm in by and the norm in by . is the Lebesgue measure of . is a ball centered at with radius . denotes , denotes as and denotes as . , will denote various positive constants; the exact values of which are not important. is the best Sobolev embedding constant defined by
Definition 2.1. Let , be a Banach space and . (i) is a (PS)-sequence in for if and strongly in as (ii) We say that satisfies the (PS) condition if any (PS)-sequence in for has a convergent subsequence.
Associated with (), we consider the energy functional in , for each ,
It is well known that is of in and the solutions of () are the critical points of the energy functional (see Rabinowitz ).
As the energy functional is not bounded below on , it is useful to consider the functional on the Nehari manifold
Thus, if and only if
Note that contains every nonzero solution of (). Moreover, we have the following results.
Lemma 2.2. The energy functional is coercive and bounded below on .
Proof. If , then by , (2.5), and the Hölder inequality and the Sobolev embedding theorem we have
Thus, is coercive and bounded below on .
Then for ,
Similar to the method used in Tarantello , we split into three parts:
Then, we have the following results.
Lemma 2.3. Assume that is a local minimizer for on and . Then in .
Proof. Our proof is almost the same as that in Brown and Zhang [12, Theorem ] (or see Binding et al. ).
Lemma 2.4. One has the following. (i) If , then . (ii) If , then and . (iii) If , then .
Proof. The proof is immediate from (2.10) and (2.11).
Moreover, we have the following result.
Lemma 2.5. If , then where is the same as in (1.5).
Proof. Suppose otherwise that there exists such that . Then by (2.10) and (2.11), for , we have
Moreover, by , , and the Hölder inequality and the Sobolev embedding theorem, we have
which is a contradiction. Thus, we can conclude that if , we have .
Theorem 2.6. (i) If and , then one has and . (ii) If , then for some positive constant depending on , and .
Proof. (i) Let . By (2.10), we have
Therefore, from the definition of , , we can deduce that . (ii) Let . By (2.10), we have
Moreover, by and the Sobolev embedding theorem, we have
By(2.7) in the proof of Lemma 2.2, we have
Thus, if , then
for some positive constant . This completes the proof.
For each with , we write
Then the following lemma holds.
Lemma 2.7. Let . For each with , one has the following: (i) if , then there exists a unique such that and
(ii) if , then there exists unique such that , and
Proof. Fix with . Let
It is clear that , as . From
we can deduce that at , for and for . Then that achieves its maximum at is increasing for and decreasing for . Moreover,
We have . There exists a unique such that and . Now,
Then we have that . For , we have
Thus, . (ii) We have . By (2.29) and
there are unique and such that ,
We have , , and for each and for each . Thus,
This completes the proof.
First, we will use the idea of Tarantello  to get the following results.
Lemma 3.1. If , then for each , there exist and a differentiable function such that , the function and
for all .
Proof. For , define a function by
According to the implicit function theorem, there exist and a differentiable function such that ,
which is equivalent to
that is, .
Lemma 3.2. Let , then for each , there exist and a differentiable function such that , the function and
for all .
Proof. Similar to the argument in Lemma 3.1, there exist and a differentiable function such that and for all . Since
Thus, by the continuity of the function , we have
if sufficiently small, this implies that .
Proposition 3.3. (i) If , then there exists a (PS)-sequence in for . (ii) If , then there exists a (PS)-sequence in for .
Proof. (i) By Lemma 2.2 and the Ekeland variational principle , there exists a minimizing sequence such that
By and taking large, we have
From (2.7), (3.10), and the Hölder inequality, we deduce that
Consequently, and putting together (3.10), (3.11), and the Hölder inequality, we obtain
Now, we show that
Apply Lemma 3.1 with to obtain the functions for some , such that . Choose . Let with and let . We set . Since , we deduce from (3.9) that
and by the mean value theorem, we have
Since and (3.16) it follows that
if we let in (3.18) for a fixed , then by (3.12) we can find a constant , independent of , such that
The proof will be complete once we show that is uniformly bounded in . By (3.1), (3.12), , , and the Hölder inequality and the Sobolev embedding theorem, we have
We only need to show that
for some and large enough. We argue by contradiction. Assume that there exists a subsequence such that
By (3.23) and the fact that , we get
Moreover, by , , and the Hölder inequality and the Sobolev embedding theorem, we have
This implies which is a contradiction. We obtain
This completes the proof of (i). (ii) Similarly, by using Lemma 3.2, we can prove (ii). We will omit detailed proof here.
Now, we establish the existence of a local minimum for on .
Theorem 3.4. If , then has a minimizer in and it satisfies that (i); (ii) is a positive solution of () in for some .
Proof. By Proposition 3.3(i), there exists a minimizing sequence for on such that
Since is coercive on (see Lemma 2.2), we get that is bounded in . Going if necessary to a subsequence, we can assume that there exists such that
First, we claim that is a nontrivial solution of (). By (3.27) and (3.28), it is easy to see that is a solution of (). From and (2.6), we deduce that
Let in (3.29), by (3.27), (3.28), and , we get
Thus, is a nontrivial solution of (). Now we prove that strongly in and . By (3.29), if , then
In order to prove that , it suffices to recall that , by (3.31), and applying Fatou's lemma to get
This implies that and Let , then Brézis and Lieb lemma  implies that
Therefore, strongly in . Moreover, we have . On the contrary, if , then by Lemma 2.7, there are unique and such that and . In particular, we have . Since
there exists such that . By Lemma 2.7,
which is a contradiction. Since and , by Lemma 2.3 we may assume that is a nontrivial nonnegative solution of (). Moreover, from , then using the standard bootstrap argument (see, e.g., ) we obtain ; hence by applying regularity results [17, 18] we derive that for some and finally, by the Harnack inequality  we deduce that . This completes the proof.
Now, we begin the proof of Theorem 1.4. By Theorem 3.4, we obtain () that has a positive solution in for some .
Next, we will establish the existence of the second positive solution of () by proving that satisfies the (PS) condition.
Lemma 4.1. Assume that and hold. If is a (PS)-sequence for , then is bounded in .
Proof. We argue by contradiction. Assume that . Let . We may assume that in . This implies that strongly in for all and
Since is a (PS)-sequence for and , there hold
From (4.1)-(4.2), we can deduce that
Since and , (4.3) implies
which is contrary to the fact for all .
Lemma 4.2. Assume that and hold. If is a (PS)-sequence for with , then there exists a subsequence of converging weakly to a nontrivial solution of ().
Proof. Let be a (PS)-sequence for with . We know from Lemma 4.1 that is bounded in , and then there exists a subsequence of (still denoted by and such that
It is easy to see that and
Next we verify that . Arguing by contradiction, we assume . Setting
Since and is bounded, then by (4.6), we can deduce that
If , then we get , which contradicts with . Thus we conclude that . Furthermore, the Sobolev inequality implies that
Then as we have
which implies that
Hence, from (4.6) to (4.12) we get
This is a contradiction to . Therefore is a nontrivial solution of ().
Lemma 4.3. Assume that - and hold. Then for any , there exists such that
In particular, for all where is as in (1.5).
Proof. For convenience, we introduce the following notations:
From to , we know that there exists such that for all ,
Motivated by some ideas of selecting cut-off functions in [20, Lemma