Nonsingular black holes and spherically symmetric objects in nonlinear electrodynamics with a scalar field

In general relativity with vector and scalar fields given by the Lagrangian $\mathcal{L}(F,\ensuremath{\phi},X)$, where $F$ is a Maxwell term and $X$ is a kinetic term of the scalar field $\ensuremath{\phi}$, we study the linear stability of static and spherically symmetric objects without curvature singularities at their centers. We show that the background solutions are generally described by either purely electrically or magnetically charged objects with a nontrivial scalar-field profile. In theories with the Lagrangian $\stackrel{\texttildelow{}}{\mathcal{L}}(F)+K(\ensuremath{\phi},X)$, which correspond to nonlinear electrodynamics with a $k$-essence scalar field, angular Laplacian instabilities induced by vector-field perturbations exclude all the regular spherically symmetric solutions including nonsingular black holes. In theories described by the Lagrangian $\mathcal{L}=X+\ensuremath{\mu}(\ensuremath{\phi}){F}^{n}$, where $\ensuremath{\mu}$ is a function of $\ensuremath{\phi}$ and $n$ is a constant, the absence of angular Laplacian instabilities of spherically symmetric objects requires that $n>1/2$, under which nonsingular black holes with apparent horizons are not present. However, for some particular ranges of $n$, there are horizonless compact objects with neither ghosts nor Laplacian instabilities in the small-scale limit. In theories given by $\mathcal{L}=X\ensuremath{\kappa}(F)$, where $\ensuremath{\kappa}$ is a function of $F$, regular spherically symmetric objects are prone to Laplacian instabilities either around the center or at spatial infinity. Thus, in our theoretical framework, we do not find any example of linearly stable nonsingular black holes.