A note on bifurcation from the essential spectrum

Abstract

In this paper we study a semilinear elliptic equation in all ℝN. This equation depends on a parameter λ, and we obtain, for small λ < 0, solutins which are small in H 1(ℝN). In this sense we have solutions bifurcating from the origin and, as the differential operator involved is the laplacian, we say that we have solutions bifurcating from the bottom of the essential spectrum of the laplacian. By a change of variables we transform the original bifurcation problem into a perturbation one. We adopt a variational procedure, looking for critical points of a suitable functional. We apply a recently developped reduction method, which allows to reduce the original variational problem in H1(ℝN) to a variational problem in a finite-dimensional manifold, and then we solve this last problem. In this way we are also able to manage the presence of critical nonlinearities, in the sense of Sobolev embedding.