Branch continuation inside the essential spectrum for the nonlinear Schrodinger equation

We consider the nonlinear stationary Schrodinger equation -Delta u - lambda u = Q(x)vertical bar u vertical bar(p-2) u, in R-N in the case where N >= 3, p is a superlinear, subcritical exponent, Q is a bounded, nonnegative and nontrivial weight function with compact support in R-N and lambda is an element of R is a parameter. Under further restrictions either on the exponent p or on the shape of Q, we establish the existence of a continuous branch C of nontrivial solutions to this equation which intersects {lambda} x L-s(R-N) for every lambda is an element of (-infinity, lambda(Q)) and s > 2N/N-1. Here, lambda(Q) > 0 is an explicit positive constant which only depends on N and diam(supp Q). In particular, the set of values lambda along the branch enters the essential spectrum of the operator -Delta.