A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball

TL;DR

This paper investigates the bifurcation behavior of positive radial solutions to a critical p-Laplace equation in a ball, revealing conditions for existence, non-existence, and multiplicity of solutions based on the properties of the coefficient function.

Contribution

It introduces a bifurcation analysis for the critical p-Laplace problem, including the novel result of a second solution in the classical Laplace case, using dynamical systems methods.

Findings

One solution exists for all positive mbda if K is steep at zero.

No solutions for small mbda when K is flat at zero.

Two solutions appear for large \u03bb when K is flat at zero.

Abstract

In this paper we show that the number of radial positive solutions of the following critical problem $$ \Delta_p u(x) + \lambda K(|x|) \,u(x) \, |u(x)|^{q-2} =0\,,$$ $$ u(x)>0 \quad |x|<1,$$ $$ u(x)=0 \quad |x|=1,$$ where $q= \frac{np}{n-p}$, $\frac{2n}{n+2} \le p \le 2$ and $x \in \mathbb{R}^n$, undergoes a bifurcation phenomenon. Namely, the problem admits one solution for any $\lambda>0$ if $K$ is steep enough at $0$, while it admits no solutions for $\lambda$ small and two solutions for $\lambda$ large if $K$ is too flat at $0$. The existence of the second solution is new, even in the classical Laplace case. The proofs use Fowler transformation and dynamical systems tools.