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Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in H_{0}^{1}(\Omega)$. By Sobolev Embedding Theorem for 1 dimensional space, we can obtain $$S ||u||_{p}^{2} \leq ||u||_{H^{1}_{0}(\Omega)}^{2}$$ Here, $S := \inf\limits_{u\in H_{0}^{1}(\Omega)\backslash\{0\}}\frac{||u||_{H^{1}_{0}(\Omega)}^{2}}{||u||_{p}^{2}}$

So, my question is how to show that $S^{\frac{p}{p-2}}\geq \inf\limits_{u\in H_{0}^{1}(\Omega)\backslash\{0\}}\bigg(\frac{||u||_{H^{1}_{0}(\Omega)}^{2}}{||u||_{p}^{2}} \bigg)^{\frac{p}{p-2}}$

I apologize for my elementary question but I am not sure how to proceed. Any hint is much appreciated! Thank you very much!

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  • $\begingroup$ If $p>2$ this follows from monotonicity if $x^{\frac 2 {p-2}}$. $\endgroup$ – Kavi Rama Murthy Feb 18 at 5:27
  • $\begingroup$ Thank you very much, I will try! $\endgroup$ – Evan William Chandra Feb 18 at 6:02
  • $\begingroup$ From your hint, I obtain "less or equal" but what I need is "greater or equal" $\endgroup$ – Evan William Chandra Feb 18 at 6:12
  • $\begingroup$ You need both continuity and monotonicity. I have posted an answer. $\endgroup$ – Kavi Rama Murthy Feb 18 at 6:30
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Suppose $S$ is infimum of a set $A$ of real numbers and suppose $f$ is continuous and increasing. Then, for any $a \in A$ we have $f(a) \geq f(S)$ because $ a \geq S$ and $f$ is increasing. Hence $\inf f(A) \geq f(S)$. Also there exists a sequencce $\{a_n\} \in S$ converging to $S$. Since $f(a_n) \to f(S)$ and $\inf f(A) \leq f(a_n)$ for all $n$ we get $\inf f(A) \leq f(S)$. Thus $\inf f(A) =f(S)$.

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