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

Abstract

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form{−Δu=fλ(|x|,u,|∇u|)in Ω,u=0on ∂Ω, on an annulus Ω⊂RN, 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±⊂R×C1(Ω‾), j∈N0 that bifurcate from the trivial branch R×{0} at (λ0,u0)=(0,0) and consist of solutions having precisely j+1 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∈R.