I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional
$$ E(u):= \frac{1}{2}\int\_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int\_{R^N} (u^2-1)^2dx.$$ There has been a lot of work on this equation and in particular on the DeGiorgi conjecture. My question is related to whether any of the solutions have finite energy. So here is my exact question. Lets take $N=9$ and suppose $ x=(x',x\_9)$.


**Question.** Does there exist a function $u$ with $ -1 <u<1$ with a $ u\_{x\_9}>0$ and $\lim\_{x\_9 \rightarrow \infty} u(x',x\_9)=1$ and $ \lim\_{x\_9 \rightarrow -\infty} u(x',x\_9)=-1$. Furthermore $ E(u)<\infty$. 


The reason I ask this is I see some results about 'finite energy solutions' yet they just impose growth on the energy in terms of $B\_R$ (ball radius $R$ centered at the origin) so I thought if the full energy is finite maybe something is trivial (but I just can't see it..) 


Thanks for the comments. 



Finite energy solution for Allen -Cahn equation
Sketch of an argument:


For a fixed $x'$, let $f(x')$ denote the measure of the set $\{ u(x',x\_9) \in (-1/2,1/2) \}$. 


Note that for fixed $x'$ you have 
$$ \int |\nabla u(x',x\_9)|^2 d x\_9 \geq \int |\partial\_{x\_9} u(x',x\_9)|^2 d x\_9 \geq 1 / f(x') $$
(on the ends of the interval (it is an interval by monotonicity) defining $f(x')$ the function takes values $-1/2$ and $1/2$ respectively, and so the minimizer is the linear function with slope $1/f(x')$.)


For fixed $x'$ you also have 
$$ \int (u^2 - 1)^2 dx\_9 \geq \frac9{16} f(x') $$


Finiteness of energy requires both be integrable in $x'$, which is not possible. 


