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I am looking for unbounded functions that grow rapidly fast near the origin, but are in the Sobolev space $H^1{(\Omega)}$, where $\Omega$ is a unit square centered at the origin.

I already know about functions like $\log\Big(\log\big(\frac{2}{\sqrt{x^2+y^2}}\big)\Big)$ which though are unbounded, have a very slow growth rate and as a result $$\lim_{x\to 0} x f(x,y)=0.$$

I am looking for functions $f(x,y) \in H^1(\Omega)$ such that $$\lim_{x\to 0} x f(x,y)= \infty.$$

Does anyone know about such functions? Do such functions even exist?

Thanks!

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    $\begingroup$ Such a function can not be in $L^2(\Omega)$, let alone in $H^1(\Omega)$. $\endgroup$
    – Aleksei Kulikov
    Commented Jul 5, 2018 at 4:19
  • $\begingroup$ @AlekseiKulikov Can you explain why? $\endgroup$
    – approximation
    Commented Jul 5, 2018 at 4:23
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    $\begingroup$ Well, because $\int_{-\varepsilon}^{\varepsilon}\int_{-\varepsilon}^{\varepsilon} \frac{1}{x^2}dxdy$ is divergent for any $\varepsilon > 0$. $\endgroup$
    – Aleksei Kulikov
    Commented Jul 5, 2018 at 4:30

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