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

TL;DR

This paper investigates the solution structure of super-critical elliptic equations on the unit ball, focusing on solution curves, singular solutions, and the special role of the Lin-Ni equation, using analytical methods and computational tools.

Contribution

It provides new insights into the properties of solution curves and singular solutions for super-critical equations, highlighting the significance of the Lin-Ni equation and employing advanced computational techniques.

Findings

Existence of infinitely many ground state solutions.

Characterization of solution curves with turns at large solution norms.

Development of computational methods to handle super-critical regimes.

Abstract

We study analytical and computational aspects for Dirichlet problem on the unit ball $B$: $|x|<1$ in $R^n$, modeled on the equation \[ \Delta u +\lambda \left(u^p+u^q \right)=0, \;\; \mbox{in $B$}, \;\; u=0 \s \mbox{on $\partial B$}, \] with a positive parameter $\lambda$, and $1<p<\frac{n+2}{n-2}<q$, where $\frac{n+2}{n-2}$ is the critical power. It turns out that a special role is played by the Lin-Ni equation [18], where $q=2p-1$ and $p>\frac{n}{n+2}$. 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 $\lambda$, 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.