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)$.

\textbf{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.

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.

I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional $$ E(u):= \int_{R^N} | \nabla u(x)|^2 dx + \int_{R^N} (u^2-1)^2dx.$$ 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)$.

\textbf{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.

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)$.

\textbf{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.