# Is the functional bounded from below?

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 $$(x_1,...,x_N)\in\mathbb{R}^N$$, $$0, and $$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} \ 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.