Abstract:
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.
