Concentration phenomena for a fourth-order equation on ℝⁿ

Mohamed Ben Ayed and Abdelbaki Selmi

Vol. 242 (2009), No. 1, 1–32

DOI: 10.2140/pjm.2009.242.1

Abstract

We consider the problem Δ²u = K(y)|u|^8∕(n−4)u in ℝⁿ with u,Δu → 0 as |y|→∞, where K is a bounded and continuous function on ℝⁿ, n ≥ 5. Our aim is to construct infinitely many solutions which concentrate around k points, k ≥ 2, under some appropriate conditions on K. Moreover we prove that there is no solution which concentrates at one point.

Keywords

sign-changing solution, critical Sobolev exponent, biharmonic operator

Mathematical Subject Classification 2000

Primary: 35J60, 35J65

Secondary: 58E05

Milestones

Received: 10 July 2008

Accepted: 3 April 2009

Published: 1 September 2009