Bifurcation in weighted Sobolev spaces

When P(x, ∂) is a second order linear elliptic differential operator on RN, many bifurcation problems P(x, ∂)u − λu + f(x, u) = 0 cannot be formulated as a functional equation from W2,p:=W2,p(RN) to Lp:=Lp(RN) irrespective of p in [1, ∞], either because the Nemystskii operator f (u) := f(x, u) does not map W2,p to Lp due to the growth of f as |x| → ∞ or because, while well defined, f is not Fréchet differentiable. Far from being pathological, the latter may happen even when f is C∞. In this paper, we show that all these difficulties may often be circumvented by replacing the spaces W2,p and Lp by weighted spaces Wω2,p and Lωp where ω is an 'admissible' weight and p in (1, ∞), p > N/2. Even though the admissibility of ω depends in part upon f, this still yields a bifurcation theorem in W2,p due to the inclusion Wω2,p ↪ W2,p. In addition, this approach can be fine tuned to discuss bifurcation in some degenerate elliptic problems after a suitable change of the variables x and u. The problem −|x|4Δu + (Q(x) − λ)u − g(u) = 0 is treated as an example