Closed $p$-Elastic Curves in Spheres of ${\mathbb{L}}^3$

For every $p\in \mathbb{R}$, we study $p$-elastic curves in the hyperbolic plane ${\mathbb{H}}^2$ and in the de Sitter $2$-space ${\mathbb{H}}_1^2$. We analyze the existence of closed $p$-elastic curves with nonconstant curvature showing that in the hyperbolic plane ${\mathbb{H}}^2$ these curves exist provided that $p>1$, while in the de Sitter $2$-space ${\mathbb{H}}_1^2$ the restriction $p<0$ must be satisfied.