Sign-changing multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

We construct families of sign-changing solutions for the four-dimensional Brezis–Nirenberg problem $$-\mathrm{\Delta }u=u^3+\epsilon u\text{in }\mathrm{\Omega },u=0\text{on }\mathrm{\partial }\mathrm{\Omega },$$ as $\epsilon \to 0^{+}$. A Lyapunov–Schmidt reduction shows that the location and relative scales of the bubbles are governed by a signed Green–Robin interaction matrix. We formulate an abstract existence criterion in terms of a simple positive eigenvalue admitting a positive eigenvector and a stable critical set. We then apply it to a positive–negative pair in a general domain and to several symmetric multi-peak configurations, including alternating regular polygons, orthogonal polygons, one central peak surrounded by peaks of the opposite sign, and aligned three-, four-, and five-peak patterns. For the two-peak solution we also prove that it has exactly two nodal domains and, under a natural balance condition and connectedness of the boundary, that the closure of its nodal set meets the boundary.