Bifurcation phenomena associated to the $p$-Laplace operator

Abstract:

We determine the structure of the set of the solutions $u$ of $- {(|{u_x}{|^{p - 2}}{u_x})_x} + f(u) = \lambda |u{|^{p - 2}}u$ on $(0, 1)$ such that $u(0) = u(1) = 0$, where $p > 1$ and $\lambda \in {\mathbf {R}}$. We prove that the solutions with $k$ zeros are unique when $1 < p \leqslant 2$ but may not be so when $p > 2$.