How to show the inf can be achieved by some nonnegative $u\in H^1(M)$?

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When I read some about Perelman's $\mathcal W$ function, I get stuck with the red line in the picture below.Seemly, I should to read the 8.2 Existence of minimizers of Evans' PDE. But I am not sure , so I ask this question.

Besides, I can find the 108 reference "Logarithmic Sobolev inequalities and the spectrum of Schrodinger operators". Where I can get it ? Or there are some paper which can replace it ?

Below picture is from the 208 of this paper.

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