= Solution
Interpret the square as $[0,1]^2$, use equal spacing $h=1/J$ in both directions and retain interior indices $1\leq m,n\leq J-1$. Boundary values contribute to the right-hand side and do not change the interior coefficient <matrix>. Products of the one-dimensional sine modes used in the <discrete sine transform>
$$
v_{pq}(m,n)=\sin(p\pi m/J)\sin(q\pi n/J)
$$
form a <basis>, and substitution into the <five-point Laplacian> gives
$$
\Delta_hv_{pq}=-\Lambda_{pq}v_{pq},\qquad
\Lambda_{pq}=\frac4{h^2}\left[
\sin^2\left(\frac{p\pi}{2J}\right)+
\sin^2\left(\frac{q\pi}{2J}\right)\right].
$$
The interior <matrix> is $\Delta_h+\lambda I$, so it is nonsingular exactly when
$$
\boxed{\lambda\notin\{\Lambda_{pq}:1\leq p,q\leq J-1\}.}
$$
These are the <discrete Helmholtz resonance> values, including their possible multiplicities. A sufficient condition is $\lambda<\Lambda_{11}=8h^{-2}\sin^2(\pi/(2J))$, so in particular every nonpositive $\lambda$ is allowed. That is not a necessary restriction: positive values between, or above, the finitely many discrete <eigenvalues> are nonsingular as well. With homogeneous <Dirichlet boundary conditions> a nonresonant system has only the zero solution; at a resonance the corresponding modes of the <discrete sine transform> give nonzero homogeneous solutions.
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