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On a fourth order elliptic equation with supercritical exponent
Advances in Difference Equations volume 2014, Article number: 319 (2014)
This paper is concerned with the semi-linear elliptic problem involving nearly critical exponent : in Ω, on ∂ Ω, where Ω is a smooth bounded domain in , , and ε is a positive real parameter. We show that, for ε small, has no sign-changing solutions with low energy which blow up at exactly three points. Moreover, we prove that has no bubble-tower sign-changing solutions.
1 Introduction and results
We consider the following semi-linear elliptic problem with supercritical nonlinearity:
where Ω is a smooth bounded domain in , , ε is a positive real parameter and is the critical Sobolev exponent for the embedding of into .
When the biharmonic operator in is replaced by the Laplacian operator, there are many works devoted to the study of the counterpart of ; see for example [1–6], and the references therein.
When , many works have been devoted to the study of the solutions of see for example [7–9]. In the critical case, this problem is not compact, that is, when it corresponds exactly to the limiting case of the Sobolev embedding into , and thus we lose the compact embedding. In fact, van Der Vorst showed in  that has no positive solutions if Ω is a starshaped domain. Whereas Ebobisse and Ould Ahmedou proved in  that has a positive solution provided that some homology group of Ω is non-trivial. This topological condition is sufficient, but not necessary, as examples of contractible domains Ω on which a positive solution exists show .
In the supercritical case, , the problem becomes more delicate since we lose the Sobolev embedding which is an important point to overcome. The problem was studied in  where the authors show that there is no one-bubble solution to the problem and there is a one-bubble solution to the slightly subcritical case under some suitable conditions. However, we proved in  that has no sign-changing solutions which blow up exactly at two points. In this work we will show the non-existence of sign-changing solutions of having three concentration points.
We note that problem has a variational structure. The related functional is
J satisfies the Palais-Smale condition in the subcritical case, while this condition fails in the critical case. Such a failure is due to the functions
is chosen so that is the family of solutions of the following problem:
When we study problem (1.2) in a bounded smooth domain Ω, we need to introduce the function which is the projection of on . It satisfies
These functions are almost positive solutions of (1.2).
We denote by G the Green’s function defined by, ,
where is the Dirac mass at x and , with is the area of the unit sphere of . We denote by H the regular part of G, that is,
For , with , we denote by the matrix defined by
and let be its least eigenvalue.
The space is equipped with the norm and its corresponding inner product defined by
Now, we are able to state our result.
Theorem 1.1 Let Ω be any smooth bounded domain in , . If 0 is a regular value of , then there exists , such that, for each , problem has no sign-changing solutions which satisfy
with is bounded and
The second result deals with the phenomenon of bubble-tower solutions for the biharmonic problem with supercritical exponent. We will give a generalization of the result found in . More precisely, we have the following.
Theorem 1.2 Let Ω be any smooth bounded domain in , . There exists , such that, for each , problem has no solutions of the form
where , , , for each , is bounded and as , , , for , and if , , where .
The proof of our results will be by contradiction. Thus, throughout this paper we will assume that there exist solutions of which satisfy (1.5) or (1.6). In Section 2, we will obtain some information as regards such which allows us to develop Section 3 which deals with some useful estimates to the proof of our theorems. Finally, in Section 4, we combine these estimates to obtain a contradiction. Hence the proof of our results follows.
2 Preliminary results
In this section, we assume that there exist solutions of which satisfy
with is bounded, , , and as , , , for . Arguing as in  and , we see that for satisfying (2.1), there is a unique way to choose , , , and v such that
where E denotes the subspace of defined by
Here, denotes the j th component of and in the sequel, in order to simplify the notations, we set and . We always assume that (which satisfies (2.1)) is written as in (2.2) and (2.3) holds. From (2.1), it is easy to see that the following remark holds.
Lemma 2.1 
Let satisfying the assumption of the theorems. occurring in (2.2) satisfies
We recall the following estimate:
3 Some useful estimates
As usual in this type of problems, we first deal with the v-part of , in order to show that it is negligible with respect to the concentration phenomenon.
Lemma 3.1 The function v defined in (2.2), satisfies the following estimate:
where for and for , is defined by
Proof The proof is the same as that of Lemma 3.1 of , so we omit it. □
Now, we state the crucial points in the proof of our theorems.
Proposition 3.2 Assume that and let , and be the variables defined in (2.2) with and . We have
where with and , are positive constants.
It suffices to prove the proposition for . Multiplying by and integrating on Ω, we obtain
Using , we derive
where and satisfies
For the other term of (3.3), we have
Concerning the last integral, it can be written as
where for .
Observe that, for , we have , thus
But for , we have
For the other integral in (3.8), using [16, 17], and Remark 2.2, we get
It remains to estimate the second integral of (3.7). We have
Now, using Remark 2.2 and , we have
where for ,
Therefore, combining (3.3)-(3.15), and Lemma 3.1, the proof of Proposition 3.2 follows. □
Proposition 3.3 Let . We have the following estimate:
where and .
Proof The proof is similar to the proof of Proposition 3.2. But there exist some integrals which have different estimates. We will focus in those integrals. In fact, (3.3), (3.7)-(3.12) are also true if we change by . It remains to deal with the other equations. Following , we get
The proof of Proposition 3.3 is thereby completed. □
4 Proof of the theorems
Proof of Theorem 1.1
Arguing by contradiction, let us assume that problem has solutions as stated in Theorem 1.1. Recall that is written as
with orthogonal to each and their derivatives with respect to and , where denotes the k th component of (see (2.2) and (2.3)). For simplicity, we will write , , and . From Proposition 3.2, for each , with , . We have
Furthermore, an easy computation shows that
On the other hand, following the proof of Proposition 3.3, we have, for each ,
We distinguish many cases depending on the set
and we will prove that all these cases cannot occur.
We remark that if we derive and as .
Furthermore, the behavior of depends on the set Ϝ. In fact we have, assuming that ,
First we start by proving the following crucial lemmas.
Remark 4.1 Ordering the ’s: , adding , and using (4.2), it is easy to derive a contradiction if we have .
Lemma 4.2 Let . Then there exists a positive constant such that
Proof The proof will be by contradiction.
Proof of (i). Assume that . In this case, we have
which implies that . Using Remark 4.1, we derive a contradiction. In the same way, we prove that . Hence the proof of Claim (i) is completed.
Proof of (ii). Assume that . By Claim (i), we have . Four cases may occur.
Case 1. or . Using (4.5), implies that
By Claim (i) and , we obtain . By Remark 4.1, this case cannot occur.
Case 2. , , and . In this case, it is easy to obtain . Using Remark 4.1, we derive a contradiction.
Case 3. , , , and . In this case, we see that is bounded and . Hence, we derive that , which implies that for . Thus
Then by Remark 4.1, we get a contradiction.
Case 4. , , and . In this case, it is easy to get .
Using the formula , we deduce that , which implies that . Hence by Remark 4.1, we derive a contradiction and Claim (ii) is thereby completed.
Proof of (iii). Without loss of generality, we can assume that . First, as in the proof of Claim (i), we get . Now assume that , which implies
Two cases may occur.
Case 1. or . Using , we obtain
and we derive a contradiction from .
Case 2. and . Let such that . Using Claim (ii) and the fact that , we derive that , which implies that , , and is bounded. Using (4.3) for , we get
Since is bounded and , we derive that
which implies that
By Remark 4.1, we get a contradiction. □
Lemma 4.3 There exists a positive constant such that
Proof Without loss of generality, we can assume that .
Proof of (i). Assume that . First we claim that . In fact, arguing by contradiction we assume that , we get , , and . Hence, . From , we obtain
Let be the outward normal vector at . Since , , and are of the same order, we have (see  and )
Using , we get , which implies that . From , we derive a contradiction. Hence our claim is proved.
Thus there exists a positive constant c so that . Now, since we have assumed that , Lemma 4.2 implies that . Finally, using Remark 4.1, we get a contradiction and the proof of Claim (i) follows.
Proof of (ii). Assume that . Note that Claim (i) and imply that (4.9) holds.
Now, following the proof of (i), we obtain a contradiction. □
We turn now to the proof of Theorem 1.1. By the previous lemmas, we know that and are of the same order, and , for where c is a positive constant.
Hence, implies that (4.9) holds. Furthermore, for implies that
We denote by the eigenvector associated to whose norm is 1. We point out that we can choose so that all their components are positive (see  and ).
Let , , and . From (4.11), we have
The scalar product of (4.12) by gives
Since the components of are positive and , are of the same order, there exists a positive constant c, such that . Hence, we get
We deduce from (4.3) and (4.11) that
Observe that Λ may be written in the form
Using (4.15), we get
Since for and , the matrix is bounded.
Furthermore, we have , which implies that
Let us consider the equality
and derivative it with respect to ; we obtain
The scalar product with gives
Using (4.18), we obtain
Hence, we derive a contradiction from (4.14), (4.20), and the fact that 0 is a regular value of ρ. Thus the proof of our theorem follows.
Proof of Theorem 1.2
Arguing by contradiction, let us assume that problem has solutions as stated in Theorem 1.2. From Section 2, these solutions have to satisfy (2.2) and (2.3).
As in the proof of Proposition 3.2, we have, for each ,
Observe that, if , we have is bounded (by the assumption) which implies that
where c is a positive constant. Using (4.21), easy computations show that
Thus, using (4.22), can be written as
and for ,
The proof will depend on the value of l which is defined in the theorem.
Case 1. . From the definition of l we get . Now using (4.1) and , we derive that
Now, using (4.23) and , we derive the estimate of and by induction we get
Finally, using (4.22), (4.23), (4.24), and we obtain
which gives a contradiction.
Case 2. . Using (4.1), an easy computation implies that
Then from , , (4.1), (4.25), and the fact that and (since ), we obtain
Now using and (4.26) we get (4.23) and as before, (4.24) is satisfied. Hence we also derive a contradiction from .
Case 3. . Recall that in this case we have assumed that . This implies that
Hence, using , the definition of l and (4.1) we obtain the first part of (4.23). The second part follows from and the first one. Finally, as before we derive a contradiction from .
Hence, our theorem is proved.
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The author gratefully acknowledges the Deanship of Scientific Research at Taibah University on material and moral support, in particular by financing this research project.
The author declares that they have no competing interests.
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Ould Bouh, K. On a fourth order elliptic equation with supercritical exponent. Adv Differ Equ 2014, 319 (2014). https://doi.org/10.1186/1687-1847-2014-319
- nonlinear problem
- critical exponent
- sign-changing solutions
- bubble-tower solution