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

We investigate in this work families $(u_{\epsilon }{)}_{\epsilon >0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type: $${\mathrm{\Delta }}_g{u}_{\epsilon }+h_{\epsilon }{u}_{\epsilon }=\mathrm{\mid }{u}_{\epsilon }{\mathrm{\mid }}^{p_{\epsilon }-2}{u}_{\epsilon }$$ in a closed manifold $(M,g)$, where $h_{\epsilon }$ converges to $h$ in $C^1(M)$. Assuming that $(u_{\epsilon }{)}_{\epsilon >0}$ blows-up as \emph{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 $u_{\epsilon }$.