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

Abstract: We construct families of sign-changing solutions for the four-dimensional Brezis--Nirenberg problem [ -Δu=u^3+\varepsilon u\quad\text{in }Ω,\qquad u=0\quad\text{on }\partialΩ, ] as $\varepsilon\to0<sup>+$. 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.