One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations

We investigate in this work families (u_ϵ)_ϵ >0 of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type: Δ_g u_ϵ + h_ϵ u_ϵ = |u_ϵ|^p_ϵ-2 u_ϵ in a closed manifold (M,g), where h_ϵ converges to h in C^1(M). Assuming that (u_ϵ)_ϵ >0 blows-up as a single sign-changing bubble, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between h, the geometry of (M,g) and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of u_ϵ.