Spinorial Yamabe-type equations and the B\"ar-Hijazi-Lott invariant

We consider on a closed Riemannian spin manifold (Mⁿ,g,σ) the spinorial Yamabe type equation Dgφ=λ|φ|2/n-1φ, where φ is a spinor field and λ is a positive constant. For a normalized solution φ of this equation we find a positive lower bound for λ². As an application we obtain an explicit lower bound of the B\"ar-Hijazi-Lott invariant for some spin manifolds with positive scalar curvature.