Nirenberg-Gagliardo interpolation inequality and regularity of solutions of nonlinear higher order equations (Q1380878)

The authors are interested in quasilinear elliptic equations in divergence form \[ \sum_{|\alpha|\leq m} (-1)^{|\alpha|} D^\alpha A_\alpha (x,u,\dots,D^m u)=0, \qquad m>1, \tag{1} \] where \(A_\alpha(x,\xi)\) satisfy an ellipticity condition of the form \[ \sum_{|\alpha |=m} A_\alpha(x,\xi)\xi_\alpha\geq C'\sum_{|\alpha|=m}|\xi_\alpha|^p - C''\sum_{|\beta|<m} |\xi_\beta|^{p(\beta)}-f(x). \tag{2} \] They establish an analog of the Nirenberg-Gagliardo inequality for the weighted case. Moreover, they study the regularity of the solutions of (1). Under the assumption that \(A_\alpha(x,\xi)\) are Carathéodory functions and some additional assumptions on the weight functions, they prove local and global boundedness and Hölder continuity of the solutions \(u\).