Equation of motion with respect to the metric.

Question

I am trying to compute the equation of motion by varying the action with respect to the metric. Unfortunately I am stuck; I know what the answer is, but it is not what I get. I'll describe first what I have done, firstly I have computed the Ricci scalar from the conformal gauge $$ ds^2=-e^{2\omega}dx^-dx^+$$ which is $R=8e^{-2\omega}\partial_+\partial_-\omega$. The action is $$I=\int d^2x\sqrt{-h}(\Phi^2R+\lambda(\partial\Phi)^2-U(\Phi^2/d^2)) $$ where, $\Phi^2$ is the dilation, and $U$is an arbitrary potential. To highlight the metric dependence of the action I have written as $$I=\int d^2x\sqrt{-h}(\Phi^2h^{\alpha\beta}R_{\alpha\beta}+\lambda h^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi-U(\Phi^2/d^2)). $$ Then I have the variation is $$ \delta I=\int d^2x\left[\delta(\sqrt{-h})(\Phi^2h^{\alpha\beta}R_{\alpha\beta}+\lambda h^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi-U(\Phi^2/d^2))+\sqrt{-h}(\Phi^2R_{\alpha\beta}\delta(h^{\alpha\beta})+\lambda \partial_{\alpha}\Phi\partial_{\beta}\Phi\delta(h^{\alpha\beta})) \right]$$ where I have used that the boundary is static, and I have that the variation of $\delta(\sqrt{-h})=-\frac{1}{2}\sqrt{-h}h_{\alpha\beta}\delta(h^{\alpha\beta}).$ Plugging that in, $$-\frac{1}{2}h_{\alpha\beta}(\Phi^2h^{\alpha\beta}R_{\alpha\beta}+\lambda h^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi-U(\Phi^2/d^2))+\Phi^2R_{\alpha\beta}+\lambda \partial_{\alpha}\Phi\partial_{\beta}\Phi=0, $$ then I could replace $h^{\alpha\beta}h_{\alpha\beta}=\delta^{\alpha}_{\beta}$. But the problem is that I know that the answer is supposed to be just $$ -e^{2\omega}\partial_+\left(e^{-2\omega}\partial_+\Phi^2 \right)=0,$$ and I don't see how I should get that.