Electron. J. Diff. Equ., Vol. 2014 (2014), No. 77, pp. 1-13.

Sign-changing solutions of a fourth-order elliptic equation with supercritical exponent

Kamal Ould Bouh

Abstract:
In this article we study the nonlinear elliptic problem involving nearly critical exponent
$$\displaylines{ \Delta^2 u = |u|^{8/(n-4)+\varepsilon}u\quad\text{in } \Omega, \cr \Delta u=u = 0\quad \text{on } \partial \Omega, }$$
where $\Omega $ is a smooth bounded domain in $\mathbb{R}^n $ with $n \geq 5 $, and $\varepsilon$ is a positive parameter. We show that, for $\varepsilon$ small, there is no sign-changing solution with low energy which blow up at exactly two points. Moreover, we prove that this problem has no bubble-tower sign-changing solutions.

Submitted November 15, 2013. Published March 19, 2014.
Math Subject Classifications: 35J20, 35J60.
Key Words: Sign-changing solutions; critical exponent; bubble-tower solution.

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Kamal Ould Bouh
Department of Mathematics
Taibah University, P.O. Box: 30002
Almadinah Almunawwarah, Saudi Arabia
email: hbouh@taibahu.edu.sa, kamal_bouh@yahoo.fr

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