High order nonlinear elliptic equations in the subcoercive case

We consider nonlinear elliptic equations and systems of the form $div^t A(x,D^s u)=f(x)$ under structure conditions provide coerciveness and monotonicity in pair with degree of Laplacian $\Delta^{(s-t)/2}u$. In the nonstrictly divergent case $s\ne t$, the estimate for $D^{s-1}u$ is established for such equations even under degenerate structure conditions. But this is not enough to have a weak solution (solution in the sense of integral identity). We introduce the notion of a subweak solution, which can have only $s-1$ derivatives. Our definition is quite similar to the notion of generalized pseudomonotonicity of Browder and Hess. We will discuss the existence and uniqueness results for such solutions, as well as a way to extend the qualitative properties of weak solutions to the subweak solutions.