Stability and intersection properties of solutions to the nonlinear biharmonic equation

TL;DR

This paper investigates the stability and intersection properties of radially symmetric solutions to a nonlinear biharmonic equation, identifying a critical exponent that determines stability and proving solutions do not intersect in the supercritical regime.

Contribution

It establishes a critical exponent for stability and proves non-intersection of solutions in the supercritical case for the nonlinear biharmonic equation.

Findings

Existence of a critical exponent p_c depending on dimension.

Solutions are linearly unstable if p < p_c.

Solutions do not intersect if p ≥ p_c.

Abstract

We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $\Delta^2 \phi = \phi^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions are linearly unstable if $p<p_c$ and linearly stable if $p\geq p_c$. Then, we focus on the supercritical case $p\geq p_c$ and we show that the graphs of no two solutions intersect one another.