On a phosphorus cycling model with nonlinear boundary conditions

On a phosphorus cycling model

Résumé

In this paper we study the existence and multiplicity of positive
solutions for a phosphorus cycling model with nonlinear boundary
conditions, namely
$$
\left\{\begin{array}{ll}
-\Delta u = \lambda \Big( k-u+c\frac{u^{4}}{1+u^{4}}\Big) =: \lambda f(u), & x\in \Omega,\\
\mathbf{n}.\nabla u+ a(u) u=0 , & x\in\partial \Omega,\\
\end{array}\right.
$$
where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N},$
$\Delta$ is the Laplacian operator, $1/\lambda>0$ is a the diffusion
coefficient, $k$ and $c$ are positive parameters and $a: [0,\infty)
\rightarrow (0,\infty)$ is nondecreasing $C^{1}$ function. This
model describes the steady states of phosphorus cycling in
stratified lakes. Also, it describes the colonization of barren
soils in drylands by vegetation. We prove our results by the method
of
sub-- and supersolutions.