In this paper, we study the existence and energetic stability of equilibrium solutions to the Landau-de Gennes energy functional subject to various anchoring conditions. Building on the work of Park et al. (Calc Variat Partial Differ Equ 56(41):1–15, 2017), which established the energetic stability of a uniaxial solution with homeotropic anchoring for 1D variation perturbation when the anisotropic coefficient \(L \in (-\frac{3}{2}, 0],\) we extend the analysis to three-dimensional cases. Specifically, we demonstrate the 3D energetic stability in the optimal range of \(L \in [-1, 0]\) under the nonnegative physical hypothesis. The key techniques involve the decomposition of squared terms and the application of integration by parts for cross-terms. We also prove, for the first time, the existence of a smooth biaxial solution under the planar anchoring condition using the metric geometry method. By applying the maximum principle and exploiting its minimality in a certain sense, we derive properties of the biaxial solution. We also establish a sufficient condition for its energetic stability when \(L < 0,\) which relies on the negativity of a specific integral, consistent with our numerical results.
