Computation and analysis of global solution curves for super-critical equations

Philip Korman; Dieter S. Schmidt

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arXiv · 2512.14577

Computation and analysis of global solution curves for super-critical equations

Philip Korman·Dieter S. Schmidt

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Abstract

We study analytical and computational aspects for Dirichlet problem on the unit ball $B$: $|x|<1$ in $R^n$, modeled on the equation \[ Δu +λ\left(u^p+u^q \right)=0, \;\; \mbox{in $B$}, \;\; u=0 \s \mbox{on $\partial B$}, \] with a positive parameter $λ$, and $1 \frac{n}{n+2}$. This was already observed by I. Flores [6], who proved the existence of infinitely many ground state solutions. We study properties of infinitely many solution curves of this problem that are separated by these ground state solutions. We also study singular solutions (where $u(0)=\infty$), and again the Lin-Ni equation plays a special role. \medskip Super-critical equations are very challenging computationally: solutions exist only for very large $λ$, and curves of positive solutions make turns at very large values of $u(0)=||u||_{L^{\infty}}$. We overcome these difficulties by developing new results on singular solutions, and by using some delicate capabilities of {\em Mathematica} software.

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Philip Korman, Dieter S. Schmidt. 2025-12-16. Computation and analysis of global solution curves for super-critical equations. https://arxiv.org/abs/2512.14577

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