Working on a connected, bounded subset $\Omega \subset \mathbb{R}^2$ with a smooth boundary, the following is true: for $(u, v, w) \in W^{1,2}(\Omega)\times W^{1,6}(\Omega) \times W^{1,2}(\Omega)$, the following integrals are bounded:
$$ (a) \int_\Omega v^2 \times \nabla u\cdot\nabla w\\ (b) \int_\Omega v^3 \times w\\ (c) \int_\Omega v^4, \quad \int_\Omega v^5\\ (d) \int_\Omega v^2 \times |\nabla u|^2\\ (e) \int_\Omega |\nabla u| \times v^2 \times |\nabla w|^2\\ $$
My solution: I use the following Sobolev embeddings: $$ W^{1,6}(\Omega) \subset C_B(\Omega) = \{v \in C^0(\Omega) : v \in L^\infty(\Omega)\}\\ L^p(\Omega) \subset L^q(\Omega) \qquad \forall 1 \leq q < p \leq \infty $$
$$ (a) \leq \|v\|_{0,\infty}^2\int_\Omega\nabla u \cdot \nabla w \leq \|v\|_{0,\infty}^2\|u\|_{1,2}\|w\|_{1, 2}\\ (b) \leq \|v\|_{0,6}^3\|w\|_{1, 2}\\ (c) \leq \|v\|_{0,3/2}, \|v\|_{0,6/5}\\ (d) \leq \|v\|_{0,\infty}^2 \|u\|_{1,2}^2\\ (e) ~\text{is possibly unbounded} $$
Thanks very much!