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Some Dirichlet problems with bad coercivity

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  • Here I summarize and I translate in English my lectures, devoted to some Dirichlet problems with a common feature: bad coercivity.
    Very simple examples are:

    $-\Delta u = f(x)\in L^1(\Omega)\quad$ in $\Omega$

    $u = 0\quad$ on $\partial \Omega$

    since the term $\int_\Omega f(x)v(x)$ does not make sense, if $f\in L^1(\Omega), v\in W^{1,2}_0(\Omega)$;

    -div$(\frac{\nabla u}{(1+|u|)^\theta})=f(x)\in L^2(\Omega)\quad$ in $\Omega$

    $u = 0\quad$ on $\partial \Omega$

    Since the term $\int_\Omega \frac{|\nabla v|^2}{(1+|v|)^\theta}$ goes to zero, if $v$ is large.

    Mathematics Subject Classification: Primary: 35J60, 35J70; Secondary: 35J65, 49J10.


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