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!.


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