Infinitely many global continua bifurcating from a single solution of an elliptic problem with concave-convex nonlinearity

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We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form align* aligned -Δu &= fλ(|x|,u,|∇ u|) &&in Ω, u &= 0 &&on ∂Ω, aligned align* on an annulus Ωⁿ, with a concave-convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: fλ(|x|,u,|∇ u|)=λ|u|q⁻²u + |u|ᵖ⁻²u with 1<q<2<p. Although the trivial solution u₀≡0 is nondegenerate if λ=0 we prove that (λ₀,u₀)=(0,0) is a bifurcation point. In fact, the bifurcation scenario is very singular: We show that there are infinitely many global continua of radial solutions Cⱼ±¹(Ω), j₀ which bifurcate from the trivial branch R×{0} at (λ₀,u₀)=(0,0) and consist of solutions having precisely j nodal annuli. A detailed study of these continua shows that they accumulate at R≥₀×{0} so that every (λ,0) with λ≥0 is a bifurcation point. Moreover, adding a point at infinity to C¹(Ω) they also accumulate at R×{∞}, so there is bifurcation from infinity at every λ.