Existence of biological populations stabilized by diffusion (Q1355046)

The existence of solutions to the differential equation \[ (p(x)y'(x))'+ \mu p(x)f(x,y(x))= 0 \] with boundary conditions \(\lim_{x\to 0+} p(x)y'(x)=0\), \(y'(1)= g(y(1))\) is studied on the interval \([0,1]\); here \(\mu\) is a positive parameter, \(p\in C[0,1]\) and \(p>0\) on \([0,1]\); \(f\in C[(0,1]\times [0,\infty)]\); \(g\in C[0,\infty)\), the range of \(g\) contains \((-\infty,0)\) and there exists a constant \(k>0\) such that \(g(z)>0\) for \(z<k\) and \(g(z)<0\) for \(z>k\). The boundary condition at \(x=1\) in this nonlinear diffusion model corresponds to nonlinear migration across the boundary, at \(x=1\) we have diffusion into the region \(0<x<1\) if \(g(y(1))>0\) and diffusion out of this region if \(g(y(1))<0\). A-priori estimates and the topological transversality theorem are used to show that, under suitable hypotheses and for sufficiently small \(\mu\), there exists at least one nonnegative solution that is not identically zero.