Set the $C^1$- functional $I_c(u)=E(u)-cP(u)$ defined for functions $H^1_T(\mathbb{R}^N)\cap L^4(\mathbb{R}^N)$ with values on $\mathbb{C}$ where $T>0$, $H^1_T(\mathbb{R}^N)$ is the sobolev subspace of $H^1(\mathbb{R}^N)$ composed by $T$- periodic functions, i.e. $u(x_1+T,...,x_N+T)=u(x_1,...,x_N)$ for all $(x_1,...,x_N)\in\mathbb{R}^N$, $0<c<\sqrt{2}$, and $$ E(u)=\int_{[0,T]^N} |\nabla u|^2\ dx\ +\ \int_{[0,T]^N} (1-|u|^2)^2\ dx $$ $$ P(u)= \int_{[0,T]^N} <iu_{x_1},u>\ dx $$ here $<,>$ denotes the scalar product of $\mathbb{C}$ seen as $\mathbb{R}^2$.

So I want to prove that $I_c$ is bounded from below on the set of funtions with $E(u)=\delta$ for some small fixed $\delta >0$. Any idea or approach is welcome! Thanks in advance.


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