Boundedly nonhomogeneous elliptic and parabolic equations

This paper considers elliptic equations of the form
\begin{equation*} 0=F(u_{x^ix^j},u_{x^i},u,1,x) \tag{</nomathmode><mathmode>$*$} \end{equation*}
</mathmode><nomathmode> and parabolic equations of the form
\begin{equation*} u_t=F(u_{x^ix^j},u_{x^i},u,1,t,x), \tag{</nomathmode><mathmode>$**$} \end{equation*}
</mathmode><nomathmode> where $F(u_{ij},u_i,u,\beta,x)$ and $F(u_{ij},u_i,u,\beta,t,x)$ are positive homogeneous functions of the first order of homogeneity with respect to $(u_{ij},u_i,u,\beta)$, convex upwards with respect $u_{ij}$ and satisfying a uniform condition of strict ellipticity. Under certain smoothness conditions on $F$ and boundedness from above of the second derivatives of $F$ with respect to $(u_{ij},u_i,u)$, solvability is established for these equations of a problem over the whole space, of the Dirichlet problem in a domain with a sufficiently regular boundary (for the equation ($*$)), and of the Cauchy problem and the first boundary value problem (for equation ($**$)). Solutions are sought in the classes $C^{2+\alpha}$, and their existence is proved with the aid of internal a priori estimates in $C^{2+\alpha}$.
Bibliography: 29 titles.