Infinite concentration and oscillation estimates for supercritical semilinear elliptic equations in discs. I

TL;DR

This paper establishes infinite concentration and oscillation estimates for supercritical semilinear elliptic equations in discs, extending previous results to more general nonlinearities and classifying complex blow-up behaviors.

Contribution

It extends previous work to general supercritical nonlinearities, classifies infinite concentration behaviors, and introduces new phenomena for multiple exponential growth cases.

Findings

Classified all infinite concentration behaviors into two types.

Detected an infinite sequence of concentration points on blow-up solutions.

Described limit profiles, energy, and positions using Liouville equations.

Abstract

In our series of papers, we establish infinite concentration and oscillation estimates for supercritical semilinear elliptic equations in discs. Especially, we extend the previous result by the author (N. arXiv:2404.01634) to the general supercritical case. Our growth condition is related to the one introduced by Dupaigne-Farina (J. Eur. Math. Soc. 12: 855--882, 2010) and admits two types of supercritical nonlinearities, the Trudinger-Moser type growth $e^{u^p}$ with $p>2$ and the multiple exponential one $\exp{(\cdots(\exp{(u^m)})\cdots)}$ with $m>0$. In this first part, we carry out the analysis of infinite concentration phenomena on any blow-up solutions. As a result, we classify all the infinite concentration behaviors into two types which in particular shows a new behavior for the multiple exponential case. More precisely, we detect an infinite sequence of concentrating parts on any blow-up solutions via the scaling and pointwise techniques. The precise description of the limit profile, energy, and position of each concentration is given via the Liouville equation with two types of the energy recurrence formulas. This leads us to observe two types of supercritical behaviors. The behavior for the latter growth is new and can be understood as the limit case of the former one. Our concentration estimates lead to the analysis of infinite oscillation phenomena, including infinite oscillations of bifurcation diagrams, discussed in the second part.