On the profile of sign-changing solutions of an almost critical problem in the ball

Abstract

We study the existence and the profile of sign-changing solutions to the slightly subcritical problem − Δ u = | u | 2 * − 2 − ε u i n ℬ , u = 0 o n ∂ ℬ , where ℬ is the unit ball in ℝN, N⩾3, 2*=2N/(N−2) and ε>0 is a small parameter. Using a Lyapunov–Schmidt reduction, we discover two new nonradial solutions having three bubbles with different nodal structures. An interesting feature is that the solutions are obtained as a local minimum and a local saddle point of a reduced function, hence they do not have a global min–max description.