Abstract 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 aligned Fₘ (u) = B |x|^ (e^ |u|² - ₊=₀ᵐ ᵏk! |u|^2k) dx, aligned defined on the unit ball B R⁴, where m N₀ > 0, >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, 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.
Calanchi et al. (Fri,) studied this question.