Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities

We investigate a class of fourth-order elliptic problems involving exponential- type nonlinearities and spatial weights of H´enon type. Motivated by the symmetry- breaking phenomena observed in semilinear second-order problems – such as those governed by the H´enon equation – we consider weighted functionals of the form Fm(u) = Z B |x|α eσ|u|2 − mX k=0 σk k! |u|2k ! dx, defined on the unit ball B ⊂ R4, where m ∈ N0 α > 0, σ > 0 are suitable pa- rameters. We first establish an Adams-type inequality with weight, characterizing the sharp threshold for the boundedness of F on the unit sphere of the biharmonic Sobolev space. Then, we prove that for large values of the weight exponent α, ra- dial symmetry of maximizers is broken. These results extend classical findings in the second-order setting (e.g., Trudinger–Moser-type functionals and the weighted H´enon equation) to the biharmonic context and offer new insights into the interplay between weights, nonlinearity, and symmetry in higher-order PDEs.