On the asymptotic properties of solutions to a differential equation in a case of bifurcation without eigenvalues

The semilinear equation Δu-λu + h(x)u^σ= 0 is studied on all of d-dimensional Euclidean space. In the bifurcation problem a non-trivial solution is sought for small A which tends to zero with A. The asymptotic dependence of the solution on A is examined. For fixed A = 1 the existence of non-degenerate non-trivial solutions is proved for generic measurable h(x) sufficiently near to a constant, provided (d) = 1 or 3. The two problems are seen to be interdependent. The bifurcation problem at A = 0 is particularly interesting as the linearised equation is of non-Fredholm type.