| Abstract: |
| Let \( B \subset \mathbb{R}^4 \) be the unit ball. We discuss different maximization problems for exponential-type nonlinearities with weights of H\`{e}non type,
$$
F_m(u)=\int_{B}
\left(e^{\sigma u^{2}}-\sum_{k=0}^m \frac{\sigma^k u^{2k}}{k!}\right)|x|^\alpha\, \quad m\geq 0,
$$
by varying $u$ in $H_{\mathcal N}^2 (B)=H_{0}^1(B)\cap H^2(B)$, in $H_0^2(B)$ and in their radial subspaces. It is well known that, as a consequence of Adams` type inequalities, all the suprema are finite if $\sigma \leq 32\pi^2$. We first prove that, in the radial framework, the finiteness of the suprema is enlarged up to $\sigma_\alpha=32\pi^2(1+\frac{\alpha}4)$. Then, for $\sigma \leq 32\pi^2$ and $m\geq 1$, we prove that radial symmetry of maximizers in the whole spaces $H_{\mathcal N}^2 (B)$ and $H_0^2 (B)$ is broken for large values of the weight exponent $\alpha$.
These results extend classical ones in the second-order setting
to the biharmonic context in the limiting dimension $N=4$.
This is a joint work with Marta Calanchi (Universit\`{a} degli Studi di Milano, Italy). |
|