Biharmonic elliptic problems involving the 2nd Hessian operator

Abstract

In this paper we will study the equation

$$\begin{aligned} \Delta ^2 u=S_2(D^2u),\quad \Omega \subset \mathbb {R}^N, \end{aligned}$$

with $N=3,$ where $S_2(D^2u)(x)=\sum _{1\le i<j\le {N}}\lambda _i(x)\lambda _j(x)$, being $\lambda _i,$ the solutions to the equation

$$\begin{aligned} \mathrm{det}\left( \lambda I-D^2u(x)\right) =0, \end{aligned}$$

$i=1,\dots ,N,$ and $\Omega$ is a bounded domain with smooth boundary. We deal with several boundary conditions looking for the appropriate framework to get existence and multiplicity of nontrivial solutions. This kind of equation is related to some models of growth, and for this reason it is natural to study the effect of zero order local reaction terms of the type $F_{\lambda }(x,u)=\lambda |u|^{p-1}u$, with $\lambda \in \mathbb {R}$, $\lambda >0$, and $0<p<\infty$, and also the solvability of the boundary problems with a source term $f$ satisfying some integrability hypotheses.