Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912776953266176 |
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| author | M., Calanchi C, Tarsi |
| author_facet | M., Calanchi C, Tarsi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17611 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities M., Calanchi C, Tarsi Analysis of PDEs 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. |
| title | Symmetry Breaking in Biharmonic Equations with Weighted Exponential Nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.17611 |