Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading coefficient matrix $A(x)$ is only positive definite, rather than strong elliptic type $$A(x)\xi\cdot\xi\geq\lambda|\xi|^2,\quad\forall\xi\in\mathbb{R}^{mn},\quad a.e. x\in\Omega$$


Any answer and reference will be appreciated!



Fredholm alternative result for general elliptic system?
In 1948 A.V. Bitsadze, 
“On unique solvability of the
 Dirichlet problem for elliptic partial
 differential equations,”Uspekhi Mat.
 Nauk [Russian Math. Surveys],3, No. 6,
 211–212,


constructed an elliptic equation with complex coefficients
$$
Lu=\frac{\partial^2 u}{\partial x^2}+2i \frac{\partial^2 u}{\partial x\partial y}+\frac{\partial^2 u}{\partial x^2}=0
$$
for which the Dirichlet problem in the unit circle $D=\{x^2+y^2<1\}$,
$$
Lu=0 \text{ in } D,\quad u|\_{\partial D}=0,
$$ 
is neither Fredholm, nor Noetherian. Namely, there are infinitely many solutions of this problem of the form $u(z)=f(z)(1-|z|^2)\,$ where $f$ is an analytic function in $\bar D$.


In the real form it is a uniformly elliptic system
$$
\frac{\partial^2 u\_1}{\partial x^2}-2 \frac{\partial^2 u\_2}{\partial x\partial y}-\frac{\partial^2 u\_1}{\partial y^2}=0,
$$
$$
\frac{\partial^2 u\_2}{\partial x^2}+2 \frac{\partial^2 u\_1}{\partial x\partial y}-\frac{\partial^2 u\_2}{\partial y^2}=0.
$$
The notion of a strong elliptic system was introduced exactly to get the case where the corresponding operators are still Noetherian.


