The Lagrange multiplier method doesn't tell you what kind of critical point you've found. Usually, we would work around this by saying that the domain is compact and the objective function is smooth, so the minimum is attained at either a critical point or on the boundary. But in this case the domain is not compact, because it's not closed, and your objective function can't be smoothly extended to the closure of the domain, either.

So you need to do something slightly different, but in the same spirit. The idea is to notice that near $x=0,y=0$ or $z=0$, your objective function becomes very large. So the minimum cannot be in some small neighborhood of these points. So you can modify $x,y,z>0$ into $x,y,z \geq \delta$ where $\delta$ is some positive number. Then your domain becomes compact and you can use the argument I suggested above. If somehow the minimum is attained at $x=\delta,y=\delta$, or $z=\delta$, then just make $\delta$ smaller. Eventually the minimum cannot be there, since your objective function blows up to $+\infty$ near the boundary.