Is there any reference to obtain an upper bound for $$ \int_{\Omega}v_{t}^2(v^2+1/v^2)dx $$ where $v\in H^{2}(\Omega)\cap H_{0}^1(\Omega) \setminus \{0\}$, $v_{t}\in H_{0}^1(\Omega) \setminus \{0\}$ in terms of $||\Delta v||_{2}$ or $||\nabla v_{t}||_{2}$? ($\Omega$ is open bounded in $R^N, N>1$).
Thanks in advance
This expression can very well be infinite, for example if $v=0$ on a set of positive measure and $v_t>0$ on this set. Even if you want $v$ and $v_t$ to be strictly positive, you will not get such a bound.
Let $v_t\in H^1_0(\Omega)$, $v\in H^2(\Omega)\cap H^1_0(\Omega)$ be strictly positive, and let $v_\epsilon=\epsilon v$. Then $$ \int_\Omega v_t^2\left(v_\epsilon^2+\frac 1{v_\epsilon^2}\right)\geq \epsilon^{-2}\|v_t\|_2^2\to \infty,\;\epsilon\to 0, $$ while $\|\Delta v_\epsilon\|_2=\epsilon\|\Delta\|_2\to 0$ (and $\|\nabla v_t\|$ stays constant in $\epsilon$).