Boundedness and regularity for a class of solutions of a functional-differential system (Q1863450)

This paper is devoted to the study of properties of a class of solutions \((u,\psi_u)\in W_{2,p}^{1,q}(\Omega ,\nu ,\mu)\times L^q(\Omega)\) of a functional-differential system of fourth order, \[ b(x,u,\psi)=0, \qquad \langle {\mathcal A}u,v\rangle=0, \] where \(\mathcal A\) is the differential operator, \[ {\mathcal A}=\sum\limits_{|\alpha |=1}^2(-1)^{|\alpha |} a_{\alpha }(x,\delta)+a(x,u\psi_{u}), \] which acts in Banach space connected with weighted functions \(\nu \), \(\mu \) of Guglielmino-Nicolosi kind, \(\delta_2u=\{D^{\alpha }u:|\alpha |\leq 2\}\), \(\psi_u\in L^q(\Omega)\) and \(b(x,\eta ,\zeta)\), \(a_{\alpha }(x,\xi)\), \(a(x,\eta ,\zeta)\) are Carathéodory functions satisfying some conditions, e.g., the uniform monotonicity condition and degenerate ellipticity condition. By using suitable test functions, it is possible to apply Moser's iterative method to prove boundedness and Hölder continuity of solutions \(u(x)\) of the differential part of the system. Then using a ``Lipschitz'' property one can obtain the same results for \(\psi_u(x)\).