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- W3103266578 abstract "We prove the nonexistence of smooth stable solution to the biharmonic problem $Delta^2 u= u^p$, $u>0$ in $R^N$ for $1 < p < infty$ and $N < 2(1 + x_0)$, where $x_0$ is the largest root of the following equation: $$x^4 - frac{32p(p+1)}{(p-1)^2}x^2 + frac{32p(p+1)(p+3)}{(p-1)^3}x -frac{64p(p+1)^2}{(p-1)^4} = 0.$$ In particular, as $x_0 > 5$ when $p > 1$, we obtain the nonexistence of smooth stable solution for any $N leq 12$ and $p > 1$. Moreover, we consider also the corresponding problem in the half space $R^N_+$, or the elliptic problem $Delta^2 u= l(u+1)^p$ on a bounded smooth domain $O$ with the Navier boundary conditions. We will prove the regularity of the extremal solution in lower dimensions. Our results improve the previous works." @default.
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- W3103266578 date "2014-08-02" @default.
- W3103266578 modified "2023-10-10" @default.
- W3103266578 title "On stable solutions of the biharmonic problem with polynomial growth" @default.
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- W3103266578 doi "https://doi.org/10.2140/pjm.2014.270.79" @default.
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