Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents

In this paper we study the existence of nontrivial solutions for the boundary value problem {−Δu−λu−u|u|2⁎−2=0inΩu=0on∂Ω when Ω⊂Rn is a bounded domain, n ⩾ 3, 2⁎=[Formula presented] is the critical exponent for the Sobolev embedding H0 1(Ω)⊂Lp(Ω), λ is a real parameter. We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods] λj ⁎, λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ∈]λj ⁎, λj[, the number of nontrivial solutions is at least twice the multiplicity of λj. The same kind of results holds also when Ω is a compact Riemannian manifold of dimension n ⩾ 3, without boundary and Δ is the relative Laplace-Beltrami operator