I am learning PDE by myself now. I am considering converted the problem to bounded domain to use the strong maximum principle.
Using the $\lim u(x)=0$, then exists $\epsilon$ and $N$, such that if $x>N$, $|u(x)|<\epsilon$. Now consider the domain $1<|x|<N$. By maximum principle, $u$ get its maximum on the boundary. if it get its maximum on $|x|=1$, then we are done. Otherwise, if $u$ get its maximum on $|x|=N$, and suppose it is $p$. Then exists $N_0>N$ such that if $x>N_0$, $|u(x)|<\epsilon/2$. So considering the domain $1<|x|<N_0$, then $p$ is a maximum in the domain, so by maximum principle, $u$ is constant. So $u$ attains its maximum on $|x|=1$. So it is proved.
I have been thinking about it for nearly 2 hours and I feel something is wrong with my proof in using the limit. I really need some help about the proof.
Can anyone help me? I will be very grateful and thanks so much!