My friends and I have a question about the Sobolev space $H_0^1$ as follows:
Problem. Let $\Omega\subset \mathbb R^N, N=2,3$ be a regular bounded open set. Define $L(f):=\|f\|_{L^2}^2-\|f\|^4_{L^4}$ on $H_0^1(\Omega)$. Then does there exist $r>0$ with $\|f\|_{H_0^1}\leq r$ such that $L(f)>0.$
I think this result holds for all $N\geq 1.$ However, my friends doubt it, he tries to solve this problem by Gagliardo–Nirenberg interpolation inequality in $N=2$ case. Indeed, from Gagliardo–Nirenberg interpolation inequality (see "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis, Page 314, link to http://www.math.utoronto.ca/almut/Brezis.pdf), there holds $$\|f\|_{L^4}\leq C\|f\|_{L^2}^{\frac{1}{2}}\|f\|_{H^1}^{\frac{1}{2}}\text{ for any }u\in H^1(\Omega),\Omega\subset \mathbb R^2.$$ Hence $L(f)\geq\|f\|_{L^2}^2(1-C^2\|f\|_{H^1}^2)>0$ if $\|f\|_{H^1}$ is sufficiently small.
I think the Gagliardo–Nirenberg inequality is too strong to solve this problem since it claims the result for any $f\in H^1(\Omega)$ while the result we want to prove holds for any $f\in H_0^1(\Omega)$. Moreover, Gagliardo–Nirenberg inequality is too rough since the inequality may be wrong for $N>2$.
My proof is very similar to the proof of Poincare's inequality. Take N=1 for example (N>1 is similar). Without loss of generality, let $u(t_0)=0$ in $\mathbb R^1$. Then there holds the equality $|f(t)|=|\int f'(t)|.$ From the Cauchy Schwartz's inequality, one has $$\int |f|^4=\int|f|^2|\int f'(t)|^2\leq |B|\int |f|^2\int |f'(t)|^2,$$ where $|B|=\int 1$. Hence by Poincare's inequality, $$L(f)\geq \|f\|^2_{L^2}-|B|\|f\|^2_{L^2}\|\nabla f\|^2_{L^2}\geq \|f\|^2_{L^2}-C\|f\|_{L^2}^2\|f\|^2_{H_0^1},$$ where $C$ depends on $|B|$. From the above inequality, it is clear that if $\|f\|_{H_0^1}$ is sufficiently small, $L(f)>0$.
I can't find mistakes in it. But in my proof the result should not depend on the dimension of $\Omega$. Can someone help us to solve this problem? Please!