Recall that Courant's Nodal domain gives an upper bound on the number of nodal domains of the $k$-th eigenfunction. I was wondering if it is possible to deduce some relation between the number of nodal domains of supersolutions to the eigenvalue problem.
For instance, let $u$ be a solution to $-\Delta u = \lambda u$ on some bounded domain with Dirichlet boundary conditions and let $v$ solve $-\Delta v \geq \lambda v$ with Dirichlet boundary conditions on the same domain then is the following true $$ \text{# of nodal domains of }v\leq \text{# of nodal domains of }u $$ under the assumption $v\leq u?$
