Blowing up and multiplicity of solutions for a fourth-order equation with critical nonlinearity

Abstract

In this paper, we consider the following nonlinear elliptic problem : Δ2u=|u|8n−4u+μ|u|q−1u, in Ω, Δu = u = 0 on ∂Ω, where Ω is a bounded and smooth domain in Rn,n∈{5,6,7}, μ is a parameter and q ∈]4/(n−4),(12−n)/(n−4)[. We study the solutions which concentrate around two points of Ω. We prove that the concentration speeds are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω = (Ωα)α a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for μ > 0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Ω.