Solutions of Super-Linear Elliptic Equations and Their Morse Indices

We investigate the degenerate bi-harmonic equation
$$ \Delta_{m}^2 u=f(x,u)\quad \text{in} \ \ \Omega, \qquad u = \Delta u = 0\quad \text{on}\ \ \partial\Omega, $$
with $m\ge 2$, and also the degenerate tri-harmonic equation:
$$ -\Delta_{m}^3 u=f(x,u)\quad \text{in} \ \ \Omega,\qquad u = \frac{\partial u}{\partial\nu}= \frac{\partial^{2} u}{\partial\nu^{2}} = 0\quad \text{on}\ \ \partial\Omega, $$
where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $N>4$ or $N>6$ respectively, and $f \in\mathrm{C}^{1}(\Omega\times \mathbb{R})$ satisfies 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 estimates obtained in [1]–[4].