Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined as
$$F(x)= \int_0^{2\pi}G(x,\theta) (\cos \theta, \sin \theta) d\theta, $$ is surjective. Does there exists a continuous positive function $g:\mathbb{R} \rightarrow \mathbb{R}$ with $\lim_{t\rightarrow \infty}g(t)=\infty$ such that
$$H(x)= \int_0^{2\pi}g(G(x,\theta)) (\cos \theta, \sin \theta) d\theta $$
is not surjective?