Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities

Abstract

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 \begin{equation*} 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{equation*} 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. %, i.e., the supremum of the functional is strictly larger when taken over the full space compared to the radial subspace. 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.