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We study two-dimensional (2D) matter-wave solitons in spinor Bose-Einsteincondensates (BECs) under the action of the spin-orbit coupling (SOC) andopposite signs of the self- and cross-interactions. Stable 2D two-componentsolitons of the mixed-mode (MM) type are found if the cross-interaction betweenthe components is attractive, while the self-interaction is repulsive in eachcomponent. Stable solitons of the semi-vortex type are formed in the oppositecase, under the action of competing self-attraction and cross-repulsion. Thesolitons exist with the total norm taking values below a collapse threshold.Further, in the case of the repulsive self-interaction and inter-componentattraction, stable 2D self-trapped modes, which may be considered as quantumdroplets (QDs), are created if the beyond-mean-field Lee-Huang-Yang (LHY) termsare added to the self-repulsion in the underlying system of coupledGross-Pitaevskii equations. Stable QDs of the MM type, of a large size with ananisotropic density profile, exist with arbitrarily large values of the norm,as the LHY terms eliminate the collapse. The effect of the SOC term oncharacteristics of the QDs is systematically studied. We also address theexistence and stability of QDs in the case of SOC with mixed Rashba andDresselhaus terms, which makes the density profile of the QD more isotropic.Thus, QDs in the spin-orbit-coupled binary BEC are for the first time studiedin the present work.

Two-dimensional solitons and quantum droplets supported by competing self- and cross-interactions in spin-orbit-coupled condensates