A trouble with the existence of an $C_{0}^{\infty}$- function $v,v>0$ such that $\int_{\Omega}hv^{\alpha}>0$

Question

Let $\Omega$ be a smooth bounded subset of $\mathbb{R}^{n}$ , an $L^{\sigma_{\alpha}}$ -function $h$ with $h^{+}\neq0$ , $\dfrac{1}{\sigma_{\alpha}}+\dfrac{\alpha}{p*}=1$ , does there exist an $C_{0}^{\infty}$- function $v,v>0$ such that $\int_{\Omega}hv^{\alpha}>0$ ?

Answer 1

Let $h \in L^r$. Let $r'$ be the conjugate exponent.

Set $w = (1 + sign(h))/2$. Thus $\int_\Omega h \cdot w = \|h^+\|_{L^1}> 0$.

Then take $u$ to be a suitable non-negative $C^\infty_0$ approximation of $w$, such that $\|u - w\|_{L^{r'}} = \varepsilon$. Therefore $$ \int h \cdot u = \int h \cdot w + \int h \cdot (u-w) \ge \int |h^+|- \varepsilon \|h\|_{L^r} $$ If $\varepsilon$ is sufficiently small, this is still positive. Now set $v = u^{1/\alpha}$ so that $v^\alpha = u$. Then $v \in C^\infty_0$.