Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]

Question

Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?

$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$

Answer 1

This inequality cannot hold because of the homogeneity matter. Indeed, take any $u$ with $\int u^3\in(0,\infty)$ and $\int u^2+\int(u')^2<\infty$. Then replace $u$ by $cu$ for a real number $c$, and let $c\to\infty$. Then the left-hand side of your inequality will grow to $\infty$ as $c^3$, whereas the right-hand side of your inequality will grow to $\infty$ only as $c^2$. So, for large enough $c>0$, the left-hand side of your inequality will be greater than its right-hand side.