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

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