Existence of alternate steady states in a phosphorous cycling model (Q429090)

Summary: We analyze the positive solutions to the steady-state reaction diffusion equation with Dirichlet boundary conditions of the form: \(-\Delta u = \lambda[K - u + c(u^4/(1 + u^4))]\), \(x \in \Omega\), \(u = 0\), \(x \in \partial \Omega\). Here, \(\Delta u = \text{div}(\nabla u)\) is the Laplacian of \(u, 1/\lambda\) is the diffusion coefficient, \(K\) and \(c\) are positive constants, and \(\Omega \subset \mathbb R^N\) is a smooth bounded region with \(\partial \Omega\) in \(C^2\). This model describes the steady states of phosphorus cycling in stratified lakes. Also, it describes the colonization of barren soils in drylands by vegetation. In this paper, we discuss the existence of multiple positive solutions leading to the occurrence of an \(S\)-shaped bifurcation curve. We prove our results by the method of subsuper solutions.