We study the spectrum of a (p, q)-biharmonic system in a bounded domain \(\Omega\) . Under appropriate different conditions, we prove that the system either has at least one nondecreasing sequence of positive eigenvalues, or has at least one nonincreasing sequence of negative eigenvalues, or has both at least one nondecreasing sequence of positive eigenvalues and at least one nonincreasing sequence of negative eigenvalues. Our results cover respectively the cases when the weight functions in the system are positive definite, negative definite, or indefinite. We also study the property of an eigenfunction \((u_1, v_1)\) associated with the principal eigenvalue of the system, and under some suitable conditions, we prove that \((u_1, v_1)\) is not semitrivial—that is, neither \(u_1\) nor \(v_1\) can be identically equal to zero in \(\Omega\) .
