I am trying to check if the function $U(x,y)=6\ln x+\ln y$ is coercive. I know, that i need to check if $\lim_{\vert (x,y) \vert \rightarrow \infty} U(x,y) = +\infty$, and so far I have
$\lim_{(x,y) \rightarrow \infty} U(x,y) = +\infty$
$\lim_{ -(x,y) \rightarrow \infty} U(x,y) = -\infty$
Does this mean that the function is not coercive?
Thank you.
Best Answer
OK, but you need to stay in the domain of definition of $U$ which is $D=]0,+∞[×]0,+∞[$.
So you need $||(x,y)||\to+\infty$ only in $D$ for the norm you have chosen.
WARNING: It is different from $|x|\to+\infty\text{ AND }|y|\to+\infty$
For instance for $n\in\mathbb N^*$ and $n\to\infty$ we have $||(n,\frac 1{n^6})||\to+\infty$
$U(n,\frac 1{n^6})=6\ln(n)+\ln(\frac 1{n^6})=6\ln(n)-6\ln(n)=0$
Thus $U$ cannot be coercive.