# Kernel of perturbation of biharmonic operator

Suppose we have a linear fourth order operator defined on $\mathbb{R}^{2n}$ with $n\geq2$ of the form: $$\mathcal{L}(f)=\Delta^{2}f+\sum_{i,j=1}^{2n}P_{ij}(x)\partial_{i}\partial_{j}f$$ with $P_{ij}(x)$ smooth (even analytc) functions s.t. $\forall i,j$ $$\lim_{|x|\rightarrow +\infty}|x|^{2n}|P_{ij}(x)|<C\qquad C>0$$ Suppose moreover we know that there is no nontrivial bounded $f\in C^{4}(\mathbb{R}^{2n})$ s.t. $$\mathcal{L}(f)=0$$ Now we take the ball $B_{R}(0)\subset \mathbb{R^{2n}}$ and we consider the problem $$(*):\begin{cases} \mathcal{L}(f)=0 & \textrm {on }B_{R}(0)\\f=0 & \textrm {on }\partial B_{R}(0)\\ \Delta f=0 &\textrm {on }\partial B_{R}(0)\\ \end{cases}$$ My question is the following: is there a sufficiently big $R$ s.t. the only solution to the problem $(*)$ is the constant $0$ function? Or is there a reference for this kind of problem?

Does the answer change if i modify the second boundary condition and the problem becomes $$(*'):\begin{cases} \mathcal{L}(f)=0 & \textrm {on }B_{R}(0)\\f=0 & \textrm {on }\partial B_{R}(0)\\ \partial_{\nu} f=0 &\textrm {on }\partial B_{R}(0)\\ \end{cases}$$