Let $p>1, \Omega\subset\mathbb{R}^N$ be an open bounded domain. Moreover, let $(u_n)_n$ be a sequence in $W_0^{1, p}(\Omega)$ such that $$a\ge \|u_n\|_{W_0^{1, p}} +\int_{\Omega} h(x),$$ where $a\in\mathbb{R}^+$ and $h$ is a continuous function such that $$h(x)\ge e^{\alpha x^2}- c_1$$ with $\alpha\in\mathbb{R}^+$ and $c_1$ is a positive constant.
My question is: in these assumptions, we can say that $\|u_n\|_{W_0^{1, p}}$ is bounded?
About me the answer is yes, since it is (I think): $$a +c_1 {\rm meas}(\Omega)\ge \|u_n\|_{W_0^{1, p}} +\int_{\Omega}e^{\alpha x^2} dx \ge \|u_n\|_{W_0^{1, p}}. $$
Could someone please tell me if my reasoning holds true or not?
Thank you in advance!
Following your solution, you can arguee by this way:
$$ \| u_n\|_{W^{1,p}_0} \leq a - \int_{\Omega} h(x)dx \leq a + \int_{\Omega}( c_1 -e^{\alpha x^2})dx$$
and arguing similarly with your answer, you get the thesys.