As a part of a Lax-Milgram problem, I need to show a particular operator is coercive, but I am having trouble finding the appropriate bounds.
I need to show that there is some $\alpha >0$ such that $|B(u,u)|\geq \alpha ||u||_{H_0^1(\Omega)}^2$ for all $u \in H_0^1(\Omega)$, where $\Omega$ is bounded in $\mathbb{R}^2$, $K>1$, and
$$ B(u, u) = \int_{\Omega} (\nabla u)^2 - u_xu-u_yu+Ku^2 dA $$
So far, I have that $$B(u, u) \geq ||u||_{H_0^1}^2 + K||u||_{L^2}^2 - \int_{\Omega} u(u_x+u_y)dA$$.
After this, I tried to bound $\int_{\Omega} u(u_x+u_y)dA$ above...But could not get anything that would be fruitful in the above inequality.
Using the inequality $\lvert ab \rvert \le a^2/2 + b^2/2$, we see $$\int_\Omega u(u_x + u_y) dA \le \int_\Omega (u^2 + \frac 12 u_x^2 + \frac 1 2 u_y^2) \, dA \le \| u \|_{L^2(\Omega)} + \frac 1 2 \|u \|_{H^1_0(\Omega)}.$$ Thus $$B[u,u] \ge \frac{1}{2} \| u \|_{H^1_0(\Omega)} + (K-1) \|u\|_{L^2(\Omega)} \ge C \|u\|_{H^1(\Omega)}$$ where $C = \min\{1/2, K-1\} > 0$ since $K > 1$.