ABSTRACT
As a next step, we use the fact that (9) admits a variational setting, since A is a Freche´t derivative of the following functional in H1ρ(IR
N ) ∩ Lp+1ρ (IRN ):
G(f) = − 12 ∫ ρ|∇f |2 + 12(p−1)
∫ ρf2 + 1p+1
∫ ρ|f |p+1. (18)
Variational approaches for functionals in weighted spaces of functions in IRN
go back to Kurtz [260]. Later, Weissler [405] applied the variational approach to the elliptic equation (9) establishing the existence of a countable sequence of similarity patterns (cf. also Escobedo-Kavian [115] for an analogous elliptic problem with absorption). However, the author in [405] made a comment that he did not know whether this variational countable sequence coincided with that obtained simultaneously in [404] by ODE methods for radially symmetric solutions. We show that this is not the case and the variational/fibering family of solutions of (9) is incomparably wider than the ODE (radial) one. In Section 2.4, we use the ideas of the fibering method [329] based on
Lusternik-Schnirel’man (L-S) category theory of calculus of variations [252] to show that there exists a countable family of global p-bifurcation branches originated at the critical exponents p = pl, (4). Thus, we begin our study of (2) with bifurcation theory for (9).
