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This paper discusses the unique continuation principal of the solutions ofthe following perturbed fourth order elliptic differential operator$\mathcal{L}_{A,q}u=0$, where \[ \mathcal{L}_{A,q}(x,D)\ =\\sum_{j=1}^nD^4_{x_j} + \sum_{j=1}^n A_jD_{x_j} + q, \qquad (A, q) \inW^{1,\infty}(\Omega,\mathbb{C}^n) \times L^{\infty}(\Omega,\mathbb{C}) \] whoseprincipal term is not given by some integer power of the Laplacian operator. Wederive some suitable Carleman estimates which is the main tool to prove theunique continuation principle. As a by-product, we also deduce some stabilityestimate and prove the strong unique continuation principle in $2$-dimension.

Unique continuation for a non bi-Laplacian fourth order elliptic operator