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- W4298136265 abstract "We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $fin C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $nleq 9$, but only established for $nleq 4$. In this paper we prove the $L^infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $varepsilon>0$, there exist $T=T(varepsilon)$ and $C=C(varepsilon)$ such that $f'(t)leq Cf(t)^{1+varepsilon}$ for all $t>T$. This bound follows from a $L^p$-estimate for $f'(u)$ for every $p<3$ and $ngeq 2$. Under a similar but more restrictive assumption on $f$, we also prove the $L^infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$." @default.
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- W4298136265 date "2014-07-01" @default.
- W4298136265 modified "2023-10-06" @default.
- W4298136265 title "A priori estimates for semistable solutions of semilinear elliptic equations" @default.
- W4298136265 doi "https://doi.org/10.48550/arxiv.1407.0243" @default.
- W4298136265 hasPublicationYear "2014" @default.
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