Let $u(x),x\in R_+$ be a non-negative decreasing smooth function with compact support $[0,L]$, I want to know the following inequality is true? $a\in (0,1)$ $$\int_0^\infty \frac{1}{1+x}u^{1+a}dx \le \epsilon \int_0^\infty |u_x|^2dx+C_\epsilon \int_0^\infty u^{2a}dx,$$ where we need $C_\epsilon$ is independent of $L$!!!

P.S. when the left hand is $\int_0^\infty \frac{1}{(1+x)^\beta}u^{1+a}dx$ with $\beta>1$, the inequality is true, I want to consider the case $\beta=1$.

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