Biharmonic elliptic problems with second Hessian and gradient nonlinearities

We establish the existence of a solution to the following problem: Δ2u−α|∇u|r=μS2(D2u)+λfinΩ,u=0=∂u∂non∂Ω,$$\begin{equation*}\hskip7pc \def\eqcellsep{&}\begin{array}{ll}\Delta ^2u-\alpha |\nabla u|^r=\mu S_2(D^2 u)+\lambda f & \mbox{in } \Omega , \\[3pt] u=0=\displaystyle \frac{\partial u}{\partial n} &\mbox{on } \partial \Omega , \end{array} \end{equation*}$$where Ω⊂RN,N=2,3$\Omega \subset {\mathbb {R}}^N,\, N=2,3$, is a smooth and bounded domain and S2(D2u)(x)=∑1≤i 0,λ>0$ f\in L^1(\Omega ),\, \,\alpha >0,\,\lambda >0$ and 0≤μ≤1$0\le \mu \le 1$ are parameters. Moreover, we assume that r≥1$r\ge 1$ if N=2$N=2$ and 1≤r≤6$1\le r\le 6$ if N=3$N=3$. We use variational arguments and an iterative technique to prove our results.