Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm $$ \|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}. $$

Suppose $K(x) \in C^1\left(\mathbf{R}^3\right), K(x)$ and $\nabla K(x)$ are bounded in $\mathbf{R}^3, K_{x_2}(x)$ is non-negative but not identically zero ($K_{x_2}(x)$ denotes the partial derivative of $K(x)$ in $x_2$-direction).

Then the only solution of $$ -\Delta u=K(x) u^5 \text { in } \mathbf{R}^3 $$ in $E$ is the trivial solution $u \equiv 0$.

Proof: Let $u$ be any solution in $E$. Multiply the equation $$ -\Delta u=K(x) u^5 \text { in } \mathbf{R}^3 $$ by $u_{x_2}$ and integrate by parts. We obtain $$ \int_{\mathbf{R}^3} K_{x_2}(x) u(x)^6=0. $$

The hypotheses on $K(x)$ imply that $u$ is identically zero in an open set. My question is how to understand the unique continuation result to show $u \equiv 0$.