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.
This paper is concerned with the following nonlinear elliptic problem involving nearly critical exponent [Formula: see text]: [Formula: see text] in [Formula: see text], [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded smooth domain in [Formula: see text], [Formula: see text], [Formula: see text] is a small positive parameter. As [Formula: see text] goes to zero, we construct a solution with the shape of a tower of sign changing bubbles.
In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (Pε) : Δ2 u = |u|(8/(n-4))-εu, in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 5. We characterize the low energy sign-changing solutions (uε) of (Pε). We prove that (uε) are close to two bubbles with different signs and they have to blow up either at two different points with the same speed or at a critical point of the Robin function. Furthermore, we construct families of each kind of these solutions and we prove that the bubble-tower solutions exist in our case.
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