Classification of low energy sign-changing solutions of an almost critical problem

Abstract

In this paper we make the analysis of the blow up of low energy sign-changing solutions of a semilinear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball.

Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

“…We also stress that in view of the non-local nature of our problem the positive and negative parts are are not, in general, sub or super solutions to Problem (1.1) in their domain of definition, so it seems quite hard to overcome this difficulty by applying scaling arguments to u + ε , u − ε separately. On the other hand, as proved in [3] for the Laplacian, if the blow-up speeds of u + ε , u − ε are comparable then they must concentrate at two separate points. Therefore, in view of (ii), we believe that also for s ∈ (0, 1 2 ] the negative part concentrates at the center of the ball.…”

“…sign-changing solutions u ε to (1.1) such that u ε 2 s → 2S n 2s s , as ε → 0 + , we cannot establish by mere energetic arguments, neither by a Morse-index approach, the number of nodal components. In the local case it is well known that they possess exactly two nodal regions, since each nodal component carries the energy S n 2 1 (see [2,3]). In the fractional case we can only say that both the positive and the negative part globally carry the same energy S n 2s s , when ε → 0 + , but this does not hold true in general for each individual nodal component and causes many troubles when performing the asymptotic analysis.…”

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

“…As a consequence, the nodal regions of these solutions shrink to the blow-up point as ε goes to zero. We also mention the paper [10], where the authors study the blow-up of the low energy sign-changing solutions of problem (1.1) and they classify these solutions according to the concentration speeds of the positive and negative part. In particular, they obtain some qualitative results, such as symmetry or location of the concentration points when the domain is a ball.…”

New concentration phenomena for a class of radial fully nonlinear equations

“…Thus we have concentration of the positive and negative part at the same point. This is a new phenomenon, as compared with the classical Lane-Emden problem in which case, whenever the rate of blow-up of the positive and negative part is the same, the two nodal regions separate and the concentration points of the negative and positive part are different (see [3]).…”