Bifurcation Theory for a Class of Periodic Superlinear Problems

Both the paper and the candidate solution carry out a Lyapunov–Schmidt reduction at (σ_k,0), expand the reduced problem to third order on the two-dimensional kernel, impose the same structural hypothesis (H) on the cubic coefficients, and then use a scaling/normal-form analysis plus the implicit function theorem to obtain exactly four C^1 branches for λ just below σ_k, each with exactly 2k simple zeros. The paper implements this via a cubic algebraic system C_k(x,y)=0; the candidate casts the cubic as a gradient of a quartic potential V_4 and studies the limit equation v−∇V_4(v)=0. Aside from presentation, the logic and conclusions coincide. The candidate’s extra positivity condition a_k±2b_k>0 is stronger than what the paper explicitly assumes (the paper uses a nonvanishing condition corresponding to (3.28)), but it does not conflict with the main conclusions under the examples treated.