I'm looking for $H(\theta,\varphi)$ ( $ 0 \leq \theta \leq \pi$ and $0 \leq \varphi < 2 \pi$) a solution of the following equation
$$\Delta_{S} H(\theta,\varphi) = \sin^2 \theta \cos^2 \varphi := f(\theta,\varphi)$$
where $\Delta_{S}$ is the laplacian on the unit sphere defined by
$$\Delta_{S} H(\theta,\varphi) := \frac{\partial^2 H}{\partial^2 \theta} (\theta,\varphi) + \frac{1}{\tan \theta } \frac{\partial H}{\partial \theta} (\theta,\varphi) + \frac{1}{\sin^2 \theta} \frac{\partial H}{\partial \varphi^2} (\theta,\varphi).$$
I'm looking for a method to find solutions to equation on this type, I'm also interested on other kind of source term $f(\theta,\varphi)$ but always with the form of a trigonometric polynomial in $(\theta,\varphi)$.
I was thinking of writing $H(\theta,\varphi):= \cos^m(\theta) \sin^{n}(\theta) \cos^p(\varphi) \sin^q(\varphi)$ and try to find the good exponent $(n,m,p,q)$.
Edit
A decomposition in spherical harmonics $H(\theta,\varphi):= \sum_{l=0}^{+ \infty} \sum_{m=-l}^l C_{l,m} Y_{l,m}(\theta,\varphi)$ seems more appropriate to use as they form an hilbert basis made of eigenvalues of $\Delta_S$, as
$$- \Delta_S Y_{l,m} = l(l+1) Y_{m,l}.$$
However there is one issue. Writing the source term
$$f(\theta, \varphi) = \sin^2 \theta \cos^2 \varphi = \frac{1}{2}-\frac{\cos^2(\theta)}{2} + \frac{\sin^2 \theta \ \cos 2 \varphi}{2}$$
we see that the first spherical harmonic $Y_{00}(\theta,\varphi)=\frac{1}{\sqrt{4 \pi}}$ will appear in this decomposition. This harmonic belongs to the kernel of the operator $\Delta_{S}$ and thus it will not be possible to inverse the system
$$\Delta_S \Big(\sum_{l=0}^{+ \infty} \sum_{m=-l}^l C_{l,m} Y_{l,m}(\theta,\varphi) \Big) = f(\theta,\varphi).$$
Any suggestion or references on this kind of equation is welcomed.