In Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, it is known that the conical singularity arises at the center of a spherically symmetric body (r = 0) in the case where the parameter αH4 characterizing the deviation from the Horndeski Lagrangian L4 approaches a non-zero constant as r 0. We derive spherically symmetric solutions around the center in full GLPV theories and show that the GLPV Lagrangian L5 does not modify the divergent property of the Ricci scalar R induced by the non-zero αH4. Provided that αH4 = 0, curvature scalar quantities can remain finite at r = 0 even in the presence of L5 beyond the Horndeski domain. For the theories in which the scalar field φ is directly coupled to R, we also obtain spherically symmetric solutions inside/outside the body to study whether the fifth force mediated by φ can be screened by non-linear field self-interactions. We find that there is one specific model of GLPV theories in which the effect of L5 vanishes in the equations of motion. We also show that, depending on the sign of a L5-dependent term in the field equation, the model can be compatible with solar-system constraints under the Vainshtein mechanism or it is plagued by the problem of a divergence of the field derivative in high-density regions.
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