On a biharmonic equation involving slightly supercritical exponent

Abstract

We consider the biharmonic equation with supercritical nonlinearity (Pε) : ∆ 2 u = K|u| 8/(n−4)+ε u in Ω , ∆u = u = 0 on ∂Ω , where Ω is a smooth bounded domain in R n , n ≥ 5 , K is a C 3 positive function, and ε is a positive real parameter. In contrast with the subcritical case, we prove the nonexistence of sign-changing solutions of (Pε) that blow up at two near points. We also show that (Pε) has no bubble-tower sign-changing solutions.

Sign-changing bubble tower solutions for a Paneitz-type problem

“…Ayed and Mehdi [4] got that the supercritical problem (1.3) has no solutions which concentrate around a point of Ω as ε → 0. The case K is a nonconstant function, it was proved [7] that for ε small, (1.3) has no sign-changing solutions that blow up at two near points and also has no bubble-tower sign-changing solutions.…”

Sign-changing bubble tower solutions for a Paneitz-type problem

“…Ayed et al [4] got that the supercritical problem (1.1) has no solutions which concentrate around a point of Ω as ε → 0. The case K is a nonconstant function, it was proved [7] that for ε small, (1.3) has no sign-changing solutions that blow up at two near points and also has no bubble-tower sign-changing solutions.…”