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

Abstract

In this paper we show that the number of radial positive solutions of the following critical problem $${\Delta }_{p}u(x)+\lambda K(|x|)u(x)|u(x){|}^{q-2}=0,$$ $$u(x)>0|x|<1,$$ $$u(x)=0|x|=1,$$ 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.

41 pages, 5 figures