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

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 $R_{\alpha }(u)=\Vert \nabla u{\Vert }_p^p/(\Vert u{\Vert }_q^{\alpha p}\Vert u{\Vert }_r^{p-\alpha p})$ 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 $q<r\le p$ 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 $R_{\alpha }$.

40 pages, 5 figure