γ are positive parameters. Such steady state equations arise in population dynamics encapsulating assumptions regarding the patch/matrix interfaces such as patch preference and movement behavior. In this paper, we will discuss the exact bifurcation diagram and stability properties for such a steady state model.

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Please cite this article in press as: R. Dhanya et al., Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl. (2015), http://dx. AbstractWe study positive radial solutions tois a class of non-decreasing functions satisfying lim s→∞ f (s) s = ∞ (superlinear) and f (0) < 0 (semipositone). We consider solutions, u, such that u → 0 as |x| → ∞, and which also satisfy the nonlinear boundary conditon ∂u ∂η +c(u)u = 0 when |x| = r 0 , where ∂ ∂η is the outward normal derivative, andc ∈ C ([0, ∞), (0, ∞)). We will establish the existence of a positive radial solution for small values of the parameter λ. We also establish a similar result for the case when u satisfies the Dirichlet boundary conditon (u = 0) for |x| = r 0 . We establish our results via variational methods, namely using the Mountain Pass Lemma.

We prove the existence of positive radial solutions to a class of semipositone p-Laplacian problems on the exterior of a ball subject to Dirichlet and nonlinear boundary conditions. Using variational methods we prove the existence of a solution, and then use a priori estimates to prove the positivity of the solution.

Solutions to systems of differential equations which model disease transmission are of particular use and importance to epidemiologists who wish to study effective means to slow and prevent the spread of disease. In this paper, we examine a system that models two related diseases within a population, which is of particular importance to those studying co-infection and partial cross-immunity phenomena. Criteria for stability of equilibria are improved upon from previous research by Long, Vaidya, and Brandeau (2008).