Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

TL;DR

This paper investigates multiple positive radial solutions of a semilinear elliptic problem with Neumann boundary conditions in a ball, showing the existence of arbitrarily many solutions as parameters vary, and providing explicit bounds and numerical insights.

Contribution

It demonstrates the existence of infinitely many positive solutions for large exponents or domain sizes, partially confirming a conjecture and analyzing solution properties with explicit bounds and numerical illustrations.

Findings

Existence of arbitrarily many positive solutions as p→∞ or R→∞.

Explicit lower bounds for p and R for multiple solutions.

Numerical simulations supporting theoretical results and conjectures.

Abstract

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem \[ \begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases} \] that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\to\infty$ (in particular, for supercritical exponents) or as $R \to \infty$ for any fixed value of $p>2$, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for $p$ and $R$ so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.