Biharmonic elliptic problems with second Hessian and gradient nonlinearities

We establish the existence of a solution to the following problem: (Formula presented.) where (Formula presented.), is a smooth and bounded domain and (Formula presented.), where (Formula presented.) is the ith eigenvalue of symmetric matrix (Formula presented.). We assume that (Formula presented.) and (Formula presented.) are parameters. Moreover, we assume that (Formula presented.) if (Formula presented.) and (Formula presented.) if (Formula presented.). We use variational arguments and an iterative technique to prove our results.