I'm trying to prove or disprove the inequality $$\int_{\mathbb{R}}f(x)^{p-2}f'(x)^4dx\leq \frac{3}{p(p-1)}\int_{\mathbb{R}}f(x)^pf''(x)^2dx$$ for any real-valued function $f$ (provided the RHS is finite) where $p\geq 6$ is an even integer.
I tried to disprove the inequality above by plugging in some nice functions (such as $e^{-x^2}, \frac{1}{1+x^2}$ or multiplying them by trig functions), but it failed. Proving the inequality seems to be very difficult since using Hölder, Galiardo-Nirenberg and so on splits the integral into two or more pieces. Is there any similar inequalities with a single integral on the right hand side like this, proved to be true? Otherwise, is there any counterexample to disprove such inequality? (Since this inequality is scaling-invariant, so scaling argument does not work.)
No, that inequality does not hold in general. A counterexample is $f(x) = e^{-x^4}$ with $p = 6$. The following calculations were done with the Maxima compute algebra system: $$ \int_{-\infty}^\infty f(x)^{p-2} f'(x)^4 \, dx = \frac{90}{8^{13/4}} \Gamma\left( \frac 14 \right) \\ > \frac{381}{5 \cdot 8^{13/4}} \Gamma\left( \frac 14 \right) = \frac{3}{p(p-1)}\int_{-\infty}^\infty f(x)^p f''(x)^2 \, dx \, . $$
Remark: Generally, if $f$ and all its derivatives vanish at infinity, repeated integration by parts gives $$ \int_{-\infty}^\infty f(x)^{p-2} f'(x)^4 \, dx = \int_{-\infty}^\infty f(x)^{p-2} f'(x) \cdot f'(x)^3 \, dx \\ = - \frac{3}{p-1}\int_{-\infty}^\infty f(x)^{p-1} \cdot f'(x)^2 f''(x) \, dx \\ = - \frac{3}{p-1}\int_{-\infty}^\infty f(x)^{p-1} f'(x) \cdot f'(x) f''(x) \, dx \\ = \frac{3}{p(p-1)} \int_{-\infty}^\infty f(x)^{p} \left(f''(x)^2 + f'(x) f'''(x)\right) \, dx \\ = \frac{3}{p(p-1)} \int_{-\infty}^\infty f(x)^{p} f''(x)^2 \, dx - \frac{3}{(p+1)p(p-1)} \int_{-\infty}^\infty f(x)^{p+1} f^{(4)}(x) \, dx $$ so that your conjectured inequality holds if and only if $$ \int_{-\infty}^\infty f(x)^{p+1} f^{(4)}(x) \, dx \ge 0 \, . $$