Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities
Calanchi M., Tarsi C
TL;DR
The paper extends symmetry-breaking phenomena to biharmonic problems with weighted exponential nonlinearities in dimension four. It establishes a weighted Adams–type inequality and identifies a sharp critical threshold $σ_α = 32π^2(1+α/4)$ dictating finiteness, then proves sufficiency and optimality of this threshold. By comparing full and radial maximization problems, it shows that for large weight exponents $α$ symmetry is broken, with nonradial maximizers arising for truncated functionals $F_m$ and under Navier/Dirichlet conditions. These results generalize second-order Hénon-type insights to higher-order PDEs and highlight the intricate balance between weight, nonlinearity, and symmetry at critical growth.
Abstract
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{equation*} F_m(u) = \int_B |x|^α\left( e^{σ|u|^2} - \sum_{k=0}^m \frac{σ^k}{k!} |u|^{2k} \right) dx, \end{equation*} defined on the unit ball \( B \subset \mathbb{R}^4 \), where $m\in \mathbb N_0$ \( α> 0 \), \( σ>0\) 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. %, 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 Hénon equation) to the biharmonic context and offer new insights into the interplay between weights, nonlinearity, and symmetry in higher-order PDEs.