Explicit \(L^\infty\)-norm estimates via Morse index for the bi-harmonic and tri-harmonic semilinear problems (Q1740374)

The authors of this article derive explicit $L^p$ and $L^\infty$ estimates for solutions of some polyharmonic elliptic equations using the Morse index. Precisely, they consider $(-\Delta)^ku=f(x,u)$ in a bounded domain $\Omega\subset \mathbb{R}^N$ with $N>2k$ having a smooth boundary. The boundary conditions under considerations are either Dirichlet or Navier boundary conditions. Here, $f$ is a $C^1(\overline{\Omega}\times \mathbb{R})$ function.