On a phosphorus cycling model with nonlinear boundary conditions

Abstract

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.