@WillieWong Thank you very much for the comments. I need to take a close look at those constants.

@WillieWong If $n=1$ and $\Omega = [0,1]$, the system can be reduced to $u_1'' = C u_1$, where $C = C_0 C_1$, which is a standard ODE. The constant $C_0$ is positive but it can be taken arbitrarily small, where as $C_1$ is negative.

@WillieWong I believe that this may work as well, but I don't know any good literature on bi-laplacian equations. It would be great if you have any suggestions?

@IgorKhavkine I believe that is the case. In fact, I have convinced that the RHS of the first equation is of the form $ a^{ij} \partial_{ij} u_0$ (Summation convention is used here, $a^{ij}$ are independent of $u_0, u_1$). So I want to see what happens if $i,j$ are fixed first.

@MichaelRenardy Actually, I'm looking for some nonzero solutions. It doesn't have to be unique though. On the other hand, one can pretty much put any conditions on $C_0$ and $C_1$, expect for them to be $0$. I suspect the smallness would help. But I don't know at this point.

@AHusain Let's fix $i,j$ for the moment. But it is ok to assume that it is the determinant of Hessian if that helps.