A bifurcation phenomenon for the critical Laplace and p-Laplace equation in the ball: part 2

In this paper we continue the discussion started in Dalbono, F., Franca, M., Sfecci, A.: A bifurcation phenomenon for the critical Laplace and p-Laplace equation in the ball. J. Math. Anal. Appl. 559, 130482 (2026), https://doi.org/10.1016/j.jmaa.2026.130482 concerning a bifurcation phenomenon for radial solutions of the following nonlinear eigenvalue problem: \begin{equation*} \begin{cases} \Delta_p u(x) + \la \K(|x|) \,u(x) \, |u(x)|^{q-2} =0\,, \\ u(x)>0\,, & \quad |x|<1\,,\\ u(x)=0\,, & \quad |x|=1\,, \end{cases} \end{equation*} where $q= \frac{np}{n-p}$ is the critical exponent and $x \in \R^n$. Our main purpose is to remove the restriction in the range of parameters, i.e. $\frac{2n}{n+2} \le p \le 2$, and to consider the whole range $p>1$. The proofs rely on a dynamical systems approach and the main technical contribution is the construction of an unstable manifold in a non-smooth context.