We investigate a class of fourth-order elliptic problems involving exponential-type nonlinearities and spatial weights of Hénon type. Motivated by the symmetry-breaking phenomena observed in semilinear second-order problems—such as those governed by the Hénon equation—we consider weighted functionals of the form \(\begin{aligned} F_m(u) = \int _B |x|^\alpha \left( e^{\sigma |u|^2} - \sum _{k=0}^m \frac{\sigma ^k}{k!} |u|^{2k} \right) dx, \end{aligned}\) defined on the unit ball \( B \subset \mathbb {R}^4 \) , where \(m\in \mathbb N_0\) \( \alpha > 0 \) , \( \sigma >0\) are suitable parameters. 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 \( \alpha \) , radial symmetry of maximizers is broken. These results extend classical findings in the second-order setting (e.g., Trudinger–Moser-type functionals and the weighted Hénon equation) to the biharmonic context and offer new insights into the interplay between weights, nonlinearity, and symmetry in higher-order PDEs.
