# Does my functional satisfy the Palais Smale condition?

Consider the functional $$I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$ where $$u:\mathbb{R}^2 \to \mathbb{C}$$ is a $$H^1$$-function ($$H^1=$$ Sobolev space), $$\Omega \subset \mathbb{R}^2$$ bounded, $$c\in (0,\sqrt{2})$$ and $$\langle , \rangle$$ stands for the scalar product of $$\mathbb{C} \equiv \mathbb{R}^2$$. I have proved that $$I$$ is coercive and weakly lower semicontinuous and $$C^1$$.

I wonder if $$I$$ satisfies the Palais Smale condition at level $$c$$ $$(PS)_c$$, that is, assume that $$x_n$$ is a sequence s.t $$I(x_n)\to c$$ and $$I ' (x_n)\to 0$$, then there exists a converging subsequence of $$x_n$$ in $$H^1(\Omega)$$.

Any idea is welcome. Thank you in advance!.