One-bubble nodal blow-up for asymptotically critical stationary Schr\"odinger-type equations

We investigate in this work families (u_ε)ε >0 of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schr\"odinger equations of the following type: Δg u_ε + h_ε u_ε = |uε|p_ε-2 u_ε in a closed manifold $(M,g)$, where h_ε converges to h in C¹(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.