On Rayleigh quotients connected to $p$-Laplace equations with polynomial nonlinearities

Let $\Omega$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(\Omega)$-solutions of the equation $-\Delta_p u = \mu |u|^{q-2}u + |u|^{r-2}u$ parameterized by $\mu$ are in bijection with properly normalized critical points of the $0$-homogeneous Rayleigh type quotient $R_\alpha(u)=\|\nabla u\|_p^p/ (\|u\|_q^{\alpha p} \|u\|_r^{p-\alpha p})$ parameterized by $\alpha$. We study this bijection and properties of $R_\alpha$ for various relations between $p,q,r$. In particular, for the generalized convex-concave problem (the case $q<p<r$) the bijection allows to provide the existence and characterization of all degenerate solutions corresponding to the inflection point of the fibred energy functional: they are critical points of $R_\alpha$ exclusively with $\alpha = (r-p)/(r-q)$. In the subhomogeneous case $q<r \leq p$ and under additional assumptions on $\Omega$, the ground state level of $R_\alpha$ is simple and isolated, and minimizers of $R_\alpha$ exhaust the whole set of sign-constant solutions of the corresponding equation. In the superhomogeneous case $p < q<r$, there are no sign-changing critical points in a vicinity of the ground state level of $R_\alpha$.