What we are calling Emden-Fowler-Hénon (EFH) type equations are elliptic partial differential equations that have the main structure as $-\Delta u=h(x)|u|^{p-1}u$, usually considering $h(x)$ a radial function and with $p$ a subcritical exponent in the sense of Sobolev embeddings, that is, $1 < p < 2^*-1$. Taking $h(x) \equiv 1$, the resulting equation is called Lane-Emden equation which is used in astrophysics to model stellar structures. If we consider $h(x) = |x|^\alpha$, we obtain the so called Hénon equation, proposed in [1] as a perturbation of Lane-Emden equation to study the stability of spherical stellar systems. In mathematics, the Hénon equation plays many important roles in the theory of qualitative analysis of PDE, for instance, by providing a nontrivial example to theorems and working as a guide to develop new tools to fortify the theory. Significant developments have been made by [2, 3, 4, 5] to understand the qualitative properties of the Hénon equation and to set a process to analyze EFH type equations. In this setting, we propose an EFH type equation considering $h(x) = (4|x|(1 − |x|))^\alpha$, which is a weight that presents a behavior unexplored in literature and, we believe, necessary to comprehend the full picture of EFH type equations. We are also interested in understanding how well the developed technics apply to this new configuration.

1. Hénon, M. (1974). Numerical Experiments on the Stability of Spherical Stellar Systems. Symposium - International Astronomical Union, 62, 259-259. https://doi.org/10.1017/s0074180900070662
2. Smets, D., Willem, M. & Su, J. Non-radial ground states for the Hénon equation. Communications In Contemporary Mathematics. 4, 467-480 (2002,8)
3. Cao, D., Peng, S. & Yan, S. Asymptotic behaviour of ground state solutions for the Hénon equation. IMA Journal Of Applied Mathematics. 74, 468-480 (2008,12)
4. Silva, W. & Santos, E. Asymptotic profile and Morse index of the radial solutions of the Hénon equation. Journal Of Differential Equations. 287 pp. 212-235 (2021,6)
5. Mercuri, C. & Santos, E. Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations. Nonlinearity. 32, 4445-4464 (2019,10).