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 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|^α ( eσ |u|² - ∑ ₖ₌₀ᵐ σ ᵏ/k! |u|²ᵏ ) dx, aligned defined on the unit ball B ⊂ R⁴ , where m∈ N₀ α > 0 , σ >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 α , 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.