I need help in obtaining the following estimate which am stuck in calculating it in details:
$$ \big| ((v^2+2v\langle u\rangle-v, (-\Delta)^{-1}v))\big| $$ $$ \leqslant \dfrac{3c_0}{8}\int_{\Omega}(v^4+v^2\langle u \rangle^2)\,dx +c\|v\|_{L^2(\Omega)}^2+c', $$ for $c_0>0$ and $v=u-\langle u\rangle$ where $\langle u\rangle=\displaystyle \dfrac{1}{Vol(\Omega)}\int_{\Omega} u\,dx$ and $((\cdot, \cdot))$ the scalar product in $L^2(\Omega)$.