Solution
= Solution
Taking Fourier transforms gives
$$
(1+|\xi|^4)\widehat u(\xi)=\widehat f(\xi).
$$
Thus
$$
\widehat u(\xi)=\frac{\widehat f(\xi)}{1+|\xi|^4}.
$$
The multiplier and all its polynomially weighted <derivatives> have at most <polynomial> growth, so this defines a tempered distribution. In fact it maps $H^s$ continuously into $H^{s+4}$. Since the multiplier never vanishes, any homogeneous tempered solution has zero Fourier transform; uniqueness follows.