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

Abstract

Let $Ω$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(Ω)$-solutions of the equation $-Δ_p u = μ|u|^{q-2}u + |u|^{r-2}u$ parameterized by $μ$ are in bijection with properly normalized critical points of the $0$-homogeneous Rayleigh type quotient $R_α(u)=\|\nabla u\|_p^p/ (\|u\|_q^{αp} \|u\|_r^{p-αp})$ parameterized by $α$. We study this bijection and properties of $R_α$ 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 $R_α$ exclusively with $α= (r-p)/(r-q)$. In the subhomogeneous case $q<r \leq p$ and under additional assumptions on $Ω$, the ground state level of $R_α$ is simple and isolated, and minimizers of $R_α$ 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 $R_α$.