Lowest eigenvalues and formally self-adjoint fourth order elliptic differential operators

Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $m \geq 1$. Let $\eta$ be a smooth $(0,1)$-tensor field on $M$. The divergence of $\eta$ is defined as $\text{div}_g(\eta):=g^{ij}(\nabla \eta)_{ij}$. Now let $\Delta_g$ be a differential operator on $M$ that is given on functions by $\Delta_g u = \text{div}_g \nabla u$. We will call $\Delta_g$ the Laplace-Beltrami operator. With this definition in place, it is not difficult to produce an example of a formally self-adjoint elliptic differential operator on $M$ that has a sign-changing eigenfunction that is associated with the operator's lowest eigenvalue. Indeed, let $\lambda_2$ be the second lowest eigenvalue of $-\Delta_g$, and let $L_g$ be a differential operator on $M$ that is given on functions by $L_g u = \Delta^2_g u + \lambda_2 \Delta u$. Then $L_g$ will possess a sign-changing eigenfunction that is associated with $L_g$'s lowest eigenvalue.. The question that remains is given a smooth, closed manifold $M$ of dimension $m \geq 1$, how rare are formally self-adjoint elliptic differential operators on $M$ that have sign-changing eigenfunctions that are associated with the operators' lowest eigenvalues. In this paper, we will see that if $A = T -\lambda g$, where $T$ is a smooth, symmetric, negative semi-definite $(0,2)$-tensor field on $M$, then $P_g$, the differential operator on $M$ given on functions by $P_gu=,\Delta_g^2 - \text{div}_g(A(\nabla u)^\sharp)$, will have the property that it possesses a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator. This suggests that on any smooth, closed manifold of dimension $m \geq 1$ there exists a lot of formally self-adjoint fourth order elliptic differential operators on the manifold that possess sign-changing eigenfunctions that are associated with the lowest eigenvalues of the operators.