= Solution
Take $\Omega=(0,\pi)$, $a_{11}=1$, $q=1$, $\psi=0$ and $f(x)=\sin x$. The coefficients are bounded and the operator is <uniformly elliptic>. If a <weak solution> $u\in H_0^1(0,\pi)$ existed, testing its <weak formulation> with $\varphi=\sin x\in H_0^1(0,\pi)$ would give
$$
\int_0^\pi u'\cos x-\int_0^\pi u\sin x=-\int_0^\pi\sin^2x.
$$
By <integration by parts> for $H_0^1$ functions, the first integral on the left is $\int_0^\pi u\sin x$, so the left side is zero. The right side equals $-\pi/2$, a contradiction. Thus
$$
\boxed{u''+u=\sin x,\quad u(0)=u(\pi)=0\text{ has no }H^1\text{ weak solution}.}
$$
The forcing is not orthogonal to the homogeneous zero-boundary <eigenfunction> $\sin x$. This is the obstruction described by the <Fredholm alternative for an elliptic Dirichlet problem>, proved here directly without assuming that theorem.
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