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The interplay of isospin asymmetry and finite temperature in nuclei plays an important role on properties of nuclear excitations and weak interaction rates in stellar environment. Recently a fully self‐consistent microscopic framework, based on Hartree‐Fock plus random phase approximation using Skyrme functionals, has been introduced for description of excitations and weak‐interaction cross sections at finite temperature. Another self‐consistent framework involving nuclei at finite temperature has also been developed within relativistic mean field theory using effective Lagrangians with density dependent meson‐nucleon vertex functions. Nuclear excitations are studied using finite temperature random phase approximation for the range of temperatures T = 0–2 MeV, as well as in nuclei far from stability. In the focus of research are the structure properties of exotic modes of excitation (e.g. pygmy dipole resonances) and charge‐exchange modes (e.g. Gamow‐Teller resonances and forbidden transitions). It is shown that finite temperature effects include novel low‐energy multipole excitations and modifications of the Gamow‐Teller transition spectra. Using a representative set of Skyrme functionals, as well as covariant energy density functional with DD‐ME2 parameterization, both theory frameworks have been applied in calculations of electron‐capture cross sections relevant in the stage of supernova precollapse.

This article provides an insight into several open problems in the quest for novel modes of excitation in nuclei with isospin asymmetry, deformation and finite temperature characteristic in stellar environment. Major unsolved problems include the nature of pygmy dipole resonances, the quest for various multipole and spin-isospin excitations both in neutron-rich and proton drip-line nuclei mainly driven by loosely bound nucleons, excitations in unstable deformed nuclei and evolution of their properties with the shape phase transition. Exotic modes of excitation in nuclei at finite temperatures characteristic for supernova evolution present open problems with possible impact in modeling astrophysically relevant weak interaction rates. All these issues challenge self-consistent many body theory frameworks at the frontiers of on-going research, including nuclear energy density functionals, both phenomenological and constrained by the strong interaction physics of QCD, models based on low-momentum two-nucleon interaction V_{low-k} and correlated realistic nucleon-nucleon interaction V_{UCOM}, supplemented by three-body force, as well as two-nucleon and three-nucleon interactions derived from the chiral effective field theory. Joined theoretical and experimental efforts, including research with radioactive isotope beams, are needed to provide insight into dynamical properties of nuclei away from the valley of stability, involving the interplay of isospin asymmetry, deformation and finite temperature., 14 pages, submitted to the Journal of Physics G special issue on Open Problems in Nuclear Structure (OPeNST)