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!

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!

Edit: The article I'm reading is "A nonhomogeneous elliptic problem involving critical growth in dimension two" by João Marcos do Ó, Everaldo Medeiros and Uberlandio Severo. The claim is written in Lemma 2.1 of the article and it is said that the proof can be found in [9], [13] or [21] at the references of the article, However I only found at this references the proof of the part "Moreover..."