I'm reading an article and I saw the following affirmation without proof:

Let $u \in H^1(\mathbb{R}^2)$ and $\alpha>0$, then

$$\int_{\mathbb{R}^2}(e^{\alpha u^2}-1)dx<+\infty.$$

Is this claim really true? If yes, if possible, can anyone recommend me an reference for study? thank you so much in advance!