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arXiv 1508.04723math.AP
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半线性椭圆型方程半稳定解的新的先验估计

New a priori estimates for semistable solutions of semilinear elliptic equations

Asadollah Aghajani

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15 pages
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摘要：

考虑半线性椭圆型方程$-L u = f(u)$在一般光滑有界区域$Ω\subset R^{n}$中的Dirichlet边界条件为零，其中$L$是一致椭圆算子，$f$是$[0,\infty)$中的正的、非减的凸函数，使得$\frac{f(t)}{t}\rightarrow\infty$为$t\rightarrow\infty$ .我们证明，如果$u$是一个正半稳定的解决方案，然后对每一个$0\leqβ<1$，我们有$$f(u)\int_{0}^{u}f(t)f"(t)~e^{2β\int_{0}^{t}\sqrt{\frac{f"(s)}{f(s)}}ds}~dt\in L^{1}(Ω),$$由一个常数独立的$u$。正如我们将看到的，文献中关于先验界的大量结果都是这个估计的直接结果。特别是，在其他结果中，我们建立了先验$L^{\infty}$约束尺寸$n\leq 9$，额外的假设下，$\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}} < \frac{2}{9-2\sqrt{14}}\cong 1.318$。此外，我们建立了一个先验$L^{\infty}$界时，$n\leq 5$在很弱的假设下，对于一些$ε>0$，$\liminf_{t\rightarrow\infty} \frac{(tf(t))^{2-ε}}{f'(t)} > 0$或$\liminf_{t\rightarrow\infty} \frac{t^{2}f(t)f"(t)}{f'(t)^{\frac{3}{2}+ε}} > 0$。

英文摘要：

We consider the semilinear elliptic equation $-L u = f(u)$ in a general smooth bounded domain $Ω\subset R^{n}$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. We prove that if $u$ is a positive semistable solution then for every $0\leqβ<1$ we have $$f(u)\int_{0}^{u}f(t)f"(t)~e^{2β\int_{0}^{t}\sqrt{\frac{f"(s)}{f(s)}}ds}~dt\in L^{1}(Ω),$$ by a constant independent of $u$. As we shall see, a large number of results in the literature concerning a priori bounds are immediate consequences of this estimate. In particular, among other results, we establish a priori $L^{\infty}$ bound in dimensions $n\leq 9$, under the extra assumption that $\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}} < \frac{2}{9-2\sqrt{14}}\cong 1.318$. Also, we establish a priori $L^{\infty}$ bound when $n\leq 5$ under the very weak assumption that, for some $ε>0$, $\liminf_{t\rightarrow\infty} \frac{(tf(t))^{2-ε}}{f'(t)} > 0$ or $\liminf_{t\rightarrow\infty} \frac{t^{2}f(t)f"(t)}{f'(t)^{\frac{3}{2}+ε}} > 0$.

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