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Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity

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  • In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation

    $\beta \Delta^{2}u-\tau \Delta u=\frac{\lambda}{(1-u)^{p}} \mbox{in} B$,
    $0 < u \leq 1 \mbox{in} B$,
    $u=\Delta u=0 \mbox{on} \partial B$,

    where $B$ is the unit ball in $R^{n}$, $\lambda>0$ is a parameter, $\tau>0, \beta>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.

    Mathematics Subject Classification: Primary 35B45; Secondary 35J40.


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