On a fourth order elliptic equation with supercritical exponent
© Ould Bouh; licensee Springer. 2014
Received: 26 August 2014
Accepted: 1 December 2014
Published: 30 December 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
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 , 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.
These functions are almost positive solutions of (1.2).
and let be its least eigenvalue.
Now, we are able to state our result.
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.
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
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 
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.
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.
where with and , are positive constants.
where for .
Therefore, combining (3.3)-(3.15), and Lemma 3.1, the proof of Proposition 3.2 follows. □
where and .
The proof of Proposition 3.3 is thereby completed. □
4 Proof of the theorems
Proof of Theorem 1.1
and we will prove that all these cases cannot occur.
We remark that if we derive and as .
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 .
Proof The proof will be by contradiction.
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.
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.
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.
Two cases may occur.
and we derive a contradiction from .
By Remark 4.1, we get a contradiction. □
Proof Without loss of generality, we can assume that .
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.
Since for and , the matrix is bounded.
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).
The proof will depend on the value of l which is defined in the theorem.
which gives a contradiction.
Now using and (4.26) we get (4.23) and as before, (4.24) is satisfied. Hence we also derive a contradiction from .
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.
The author gratefully acknowledges the Deanship of Scientific Research at Taibah University on material and moral support, in particular by financing this research project.
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