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

TL;DR

This work develops a unified variational framework linking $p$-Laplace equations with polynomial nonlinearities to normalized critical points of a $0$-homogeneous Rayleigh-type quotient $R_\alpha(u)$ on $W_0^{1,p}(\Omega)$. By establishing a bijection between solutions of $-\Delta_p u=\mu|u|^{q-2}u+|u|^{r-2}u$ and normalized critical points of $R_\alpha$, the authors obtain a comprehensive spectrum $\lambda_k(\alpha)$ of variational eigenvalues for $\alpha>\alpha_0$, together with detailed analytic properties (bounds, monotonicity, continuity) and the translation level framework $\mu_\alpha$ that connects to the original nonlocal problems. They derive sharp results in subhomogeneous ($q<r\le p$) and superhomogeneous ($p<q<r$) regimes, including simple, isolated ground states and absence of nearby sign-changing critical points, and identify a degenerate, inflection-type family of solutions at $\alpha^*=(r-p)/(r-q)$ in the convex–concave setting. The findings illuminate energy landscapes, multiplicity, and bifurcation behavior for generalized nonlinear eigenvalue problems and offer a robust pathway to classify solutions via the index $k$ of $\lambda_k(\alpha)$. Overall, the work advances a rigorous, parameterized variational approach to nonlocal $p$-Laplacian problems with polynomial nonlinearities and clarifies the interplay between normalization, energy structure, and solution sign patterns.

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 $R_α$.

Figures (5)

• Figure 1: Schematically depicted results of Propositions \ref{['prop:omega-to-zero']}--\ref{['prop:boundedness:concon']}, where $\mu_\alpha$ stands for $\underline{\mu}\space_\alpha^k$ or $\overline{\mu}\space_\alpha^k$ for some $k \in \mathbb{N}$.
• Figure 2: Schematically depicted results of Propositions \ref{['prop:omega-to-zero:u']}--\ref{['prop:omega-to-infty:u']}, where $u_\alpha$ stands for $\underline{u}_\alpha^k$ or $\overline{u}_\alpha^k$ for some $k \in \mathbb{N}$.
• Figure 3: Schematically depicted bifurcation diagrams for the problems \ref{['eq:Pconconx']}, \ref{['eq:Pconconx2']}, where $\mu_\alpha$ and $u_\alpha$ stand for $\underline{\mu}\space_\alpha^k$ or $\overline{\mu}\space_\alpha^k$, and $\underline{u}_\alpha^k$ or $\overline{u}_\alpha^k$ for some $k \in \mathbb{N}$, respectively.
• Figure 4: Schematically depicted behavior of $t \mapsto E_{\mu_\alpha}(tu)$ for some $u \in W_0^{1,p}(\Omega) \setminus \{0\}$ and various values of $\mu_\alpha$. A curve with the critical point of inflection type is distinguished.
• Figure 5: Schematically depicted results of Propositions \ref{['prop:omega:2']}, \ref{['prop:omega:u2']}, where $\nu_\alpha$, $u_\alpha$ stand for $\underline{\nu}_\alpha^k$, $\underline{u}_\alpha^k$ or $\overline{\nu}_\alpha^k$, $\overline{u}_\alpha^k$ for some $k \in \mathbb{N}$.

Theorems & Definitions (99)

• Lemma 1.1
• proof
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Remark 2.4