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Concentration and oscillation analysis of positive solutions to semilinear elliptic equations with exponential growth in a disc. II
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 31 July 2025, pp. 1-28
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- Article
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This paper is the latter part of a series of our studies on the concentration and oscillation analysis of semilinear elliptic equations with exponential growth
$e^{u^p}$. In the first one [17], we completed the concentration analysis of blow-up positive solutions in the supercritical case p > 2 via a scaling approach. As a result, we detected infinite sequences of concentrating parts with precise quantification. In the present paper, we proceed to our second aim, the oscillation analysis. Especially, we deduce an infinite oscillation estimate directly from the previous infinite concentration ones. This allows us to investigate intersection properties between blow-up solutions and singular functions. Consequently, we show that the intersection number between blow-up and singular solutions diverges to infinity. This leads to a proof of infinite oscillations of bifurcation diagrams, which ensures the existence of infinitely many solutions. Finally, we also remark on infinite concentration and oscillation phenomena in the limit cases 
$p\to2^+$ and 
$p\to \infty$.
