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

TL;DR

This paper analyzes infinite oscillation phenomena in supercritical semilinear elliptic equations in discs, providing detailed asymptotic descriptions and conditions for infinite oscillations, thereby establishing the existence of infinitely many solutions.

Contribution

It extends previous work by characterizing infinite oscillations and intersection properties of blow-up solutions for specific supercritical nonlinearities in discs.

Findings

Describes asymptotic shapes of blow-up solutions near the origin.

Identifies conditions for infinite oscillations in bifurcation diagrams.

Applies conditions to specific nonlinearities including exponential types.

Abstract

This paper is the latter part of our series concerning infinite concentration and oscillation phenomena on supercritical semilinear elliptic equations in discs. Our supercritical setting admits two types of nonlinearities, the Trudinger-Moser type $e^{u^p}$ with $p>2$ and the multiple exponential one $\exp{(\cdots(\exp{(u^m)}))}$ with $m>0$. In the first part, we accomplished the analysis of infinite concentration phenomena on any blow-up solutions. In this second part, we proceed to the study of infinite oscillation phenomena based on concentration estimates obtained in the first part. As a result, we provide a precise description of the asymptotic shapes of the graphs of blow-up solutions near the origin. Two types of oscillation and intersection properties are observed here depending on the choice of the growth. Moreover, it allows us to show several oscillation behaviors around singular solutions with a suitable asymptotic behavior. This leads to a natural sufficient condition for infinite oscillations of bifurcation diagrams which ensure the existence of infinitely many solutions. We successfully apply this condition to certain classes of nonlinearities including the two types mentioned above.