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

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 $1<p<\frac{n+2}{n-2}<q$, 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 $λ$, 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.

💡 Research Summary

This paper presents a rigorous investigation into the analytical and computational properties of the global solution curves for super-critical elliptic partial differential equations (PDEs). The study focuses on the Dirichlet problem within the unit ball $B \subset \mathbb{R}^n$, governed by the equation $\Delta u + \lambda(u^p + u^q) = 0$, where the exponent $q$ exceeds the critical Sobolev exponent $\frac{n+2}{n-2}$. The super-critical regime is notoriously difficult to analyze because the loss of compactness in Sobolev embeddings prevents the application of standard variational methods used in sub-critical problems.

A central theme of the research is the identification of the Lin-Ni equation ($q = 2p - 1$) as a pivotal structural element. Building upon previous findings by I. Flores, the authors demonstrate that the existence of infinitely many ground state solutions plays a fundamental role in partitioning the solution curves. The study explores the intricate properties of these solution curves, which are separated by the aforementioned ground state solutions, providing a clearer picture of the bifurcation-like behavior inherent in the system.

Furthermore, the paper delves into the analysis of singular solutions, where the solution blows up at the origin ($u(0) = \infty$). By developing new analytical results regarding these singular solutions, the authors establish a bridge to understand the behavior of regular solutions in extreme regimes. This is particularly important because the super-critical nature of the equation leads to significant computational challenges: solutions only emerge for very large values of the parameter $\lambda$, and the solution curves exhibit complex “turning” phenomena at extremely high $L^\infty$ norms.

To overcome these formidable computational hurdles, the authors employ a sophisticated combination of advanced analytical insights and high-precision computational techniques using Mathematica. The research successfully maps the global structure of the solution curves, navigating through the numerical instabilities caused by the large-scale oscillations and the extreme values of $u(0)$. Ultimately, this work provides a comprehensive framework for understanding the global dynamics of super-critical non-linear elliptic equations, bridging the gap between theoretical singularity analysis and practical numerical computation.