Solutions of super-linear elliptic equations and their Morse indices

TL;DR

This paper studies solutions to degenerate bi- and tri-harmonic equations, providing explicit bounds for solutions based on Morse index, extending previous estimates in the context of nonlinear elliptic PDEs.

Contribution

It introduces new explicit bounds for solutions of degenerate higher-order elliptic equations using Morse index, expanding the understanding of solution behavior in these complex equations.

Findings

Established $L^{p}$ bounds for solutions.

Derived $L^{ abla}$ bounds for solutions.

Extended previous estimates to higher-order degenerate equations.

Abstract

We investigate here the degenerate bi-harmonic equation: $$\Delta_{m}^2 u=f(x,u)\; \;\;\mbox{in} \O,\quad u = \Delta u = 0\quad \mbox{on }\; \p\Omega,$$ with $m\ge 2,$ and also the degenerate tri-harmonic equation: $$ -\Delta_{m}^3 u=f(x,u)\;\;\; \mbox{in} \O,\quad u = \frac{\p u}{\p \nu} = \frac{\p^{2} u}{\p\nu^{2}} = 0\quad \mbox{on }\; \p\Omega,$$ where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $N>4$ or $N>6$ resp, and $f \in \mathrm{C}^{1}(\Omega\times \mathbb{R})$ satisfying suitable m-superlinear and subcritical growth conditions. Our main purpose is to establish $L^{p}$ and $L^{\infty}$ explicit bounds for weak solutions via the Morse index. Our results extend previous explicit estimate obtained in \cite{c, HHF, hyf, lec}.