Bifurcation at isolated singular points for a degenerate elliptic eigenvalue problem

We consider bifurcation from the line of trivial solutions for a nonlinear eigenvalue problem on a bounded open subset, ΩΩ, of RNRN with N≥3N≥3, containing 00. The leading term is a degenerate elliptic operator of the form L(u)=∇⋅A∇uL(u)=∇⋅A∇u where A∈C(Ω¯) with A>0A>0 on Ω¯∖{0} and limx→0A(x)|x|2∈(0,∞). Solutions should satisfy u=0u=0 on ∂Ω∂Ω and the energy associated with LL should be finite: ∫ΩA|∇u|2dx<∞∫ΩA|∇u|2dx<∞. The nonlinear terms are of lower order, depending only on uu and ∇u∇u. Under our hypotheses the associated Nemytskii operators are not Fréchet differentiable at the trivial solution u=0u=0.