Abstract. In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the four dimensional half sphere. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational structure, we prove some existence results like Bahri-Coron theorem. Furthermore, we consider the approximate subcritical problem and we construct some solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.
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.
This paper is concerned with the semi-linear elliptic problem involving nearly critical exponent (P ε ):where is a smooth bounded domain in R n , n ≥ 5, and ε is a positive real parameter. We show that, for ε small, (P ε ) has no sign-changing solutions with low energy which blow up at exactly three points. Moreover, we prove that (P ε ) has no bubble-tower sign-changing solutions.
MSC: 35J20; 35J60
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