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

TL;DR

This paper explores the relationship between solutions of p-Laplace equations with polynomial nonlinearities and critical points of a Rayleigh quotient, providing new insights into solution existence, characterization, and properties across different parameter regimes.

Contribution

It establishes a bijection between solutions of p-Laplace equations and critical points of a Rayleigh quotient, analyzing their properties for various parameter relations.

Findings

Degenerate solutions correspond to the inflection point of the energy functional.

Ground state levels are simple and isolated in the subhomogeneous case.

No sign-changing critical points near the ground state in the superhomogeneous case.

Abstract

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 $R_\alpha$.