I am reading a paper about the Harmonic Map Flow onto the sphere $\mathbb S^2$ and at some point in the paper the authors reach the following equation $$L_W(\phi) + h = 0, \quad \phi \cdot W = 0 \quad \text{ in } \mathbb R^2,$$ where $h$ is a certain function independent of $\phi$ and $L_W$ is an elliptic operator depending on a function $W$. More precisely, if we consider the definition of harmonic maps onto the sphere $$\Delta W + |\nabla W|^2 W = 0,$$ $L_W$ is just the associated linearized operator around the harmonic map $W$: $$L_W(\phi) = \Delta \phi + |\nabla W|^2 \phi + 2 (\nabla \phi \cdot \nabla W) W.$$ Now they consider the finite set of function $Z_i$ (of which they provide an explicit form) such that $L_W(Z_i) = 0$ for all $i$ and $\{Z_i\}$ form a basis of $\text{ker }L_W$. Then they claim the following "If there exists $\phi(y, t)$ is a solution of the system above with sufficient space decay, then necessarily $\langle h, Z_i \rangle_{L^2(\mathbb R^2)} = 0$ for every $i$. [sic]", for $\langle \cdot, \cdot\rangle_{L^2(\mathbb R^2)}$ the scalar product in $L^2(\mathbb R^2)$.
My question is now the following, why do they add the space decay condition ? This statement seems to be exactly the Fredholm alternative which states that there exists a solution if and only if $h\perp_{L^2(\mathbb R^2)} \text{ker }L_W$. However, as far as I know, this theorem doesn't allow us to say anything about the asymptotic behavior of the solution, am I correct ? Or it because we have the extra orthogonality condition $\phi \cdot W = 0$ ? So to summarize, is there a more powerful version of Fredholm alternative of which I am unaware or am I missing something?