On a fourth order elliptic equation with supercritical exponent
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On a fourth order elliptic equation with supercritical exponent Kamal Ould Bouh* * Correspondence: [email protected]; [email protected] Department of Mathematics, Taibah University, P.O. Box 30002, Almadinah Almunawwarah, Kingdom of Saudi Arabia
Abstract This paper is concerned with the semi-linear elliptic problem involving nearly critical exponent (Pε ): 2 u = |u|8/(n–4)+ε u in , u = u = 0 on ∂, where is a smooth bounded domain in Rn , n ≥ 5, and ε is a positive real parameter. We show that, for ε small, (Pε ) has no sign-changing solutions with low energy which blow up at exactly three points. Moreover, we prove that (Pε ) has no bubble-tower sign-changing solutions. MSC: 35J20; 35J60 Keywords: nonlinear problem; critical exponent; sign-changing solutions; bubble-tower solution
1 Introduction and results We consider the following semi-linear elliptic problem with supercritical nonlinearity:
(Pε )
⎧ ⎨ u = |u|p–+ε u in , ⎩u = u = on ∂,
where is a smooth bounded domain in Rn , n ≥ , ε is a positive real parameter and n p + = n– is the critical Sobolev exponent for the embedding of H () ∩ H () into Lp+ (). When the biharmonic operator in (Pε ) is replaced by the Laplacian operator, there are many works devoted to the study of the counterpart of (Pε ); see for example [–], and the references therein. When ε < , many works have been devoted to the study of the solutions of (Pε ) see for example [–]. In the critical case, this problem is not compact, that is, when ε = it corresponds exactly to the limiting case of the Sobolev embedding H () ∩ H () into Lp+ (), and thus we lose the compact embedding. In fact, van Der Vorst showed in [] that (P ) has no positive solutions if is a starshaped domain. Whereas Ebobisse and Ould Ahmedou proved in [] that (P ) has a positive solution provided that some homology group of is non-trivial. This topological condition is sufficient, but not necessary, as examples of contractible domains on which a positive solution exists show []. In the supercritical case, ε > , the problem (Pε ) becomes more delicate since we lose the Sobolev embedding which is an important point to overcome. The problem (Pε ) was studied in [] where the authors show that there is no one-bubble solution to the problem ©2014 Ould Bouh; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Ould Bouh Advances in Difference Equations 2014, 2014:319 http://www.advancesindifferenceequations.com/content/2014/1/319
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and there is a one-bubble solution to the slightly subcritical case under some suitable conditions. However, we proved in [] that (Pε ) has no sign-changing solutions which blow up exactly at two points. In this work we will show the non-existence of sign-changing solutio
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