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

Let $\mathrm{\Omega }$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(\mathrm{\Omega })$-solutions of the equation $-{\mathrm{\Delta }}_{p}u=\mu \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{r-2}u$ parameterized by $\mu$ are in bijection with properly normalized critical points of the $0$-homogeneous Rayleigh type quotient $R_{\alpha }(u)=\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_p^p\mathrm{/}(\mathrm{\parallel }u{\mathrm{\parallel }}_q^{\alpha p}\mathrm{\parallel }u{\mathrm{\parallel }}_r^{p-\alpha p})$ 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 $\alpha =(r-p)\mathrm{/}(r-q)$. In the subhomogeneous case $q<r\le p$ and under additional assumptions on $\mathrm{\Omega }$, 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 $R_{\alpha }$.