论文标题
M-Convex非线性的半线性椭圆方程的极端解决方案的规律性
Regularity of extremal solutions of semilinear elliptic equations with m-convex nonlinearities
论文作者
论文摘要
我们考虑了有界平滑域中的gelfand问题$ω\ subset \ mathbb {r}^n $ at dirichlet边界条件。我们对极端解决方案$ u^*$的界限感兴趣。当尺寸$ n \ ge10 $时，众所周知，可以为非线性$ f（u）= e^u $和$ω= b_1 $构建单数极值解决方案。当$ 3 \ le n \ le 9 $时，Cabré，Figalli，Ros-Oton和Serra（2020）证明了以下令人惊讶的结果：如果非线性$ f $是正面，非销售的，则极端解决方案$ u^*$是有限的。 在本文中，我们成功地将其结果推广到一般$ m $ convex的非线性。此外，我们通过考虑$ M $ - 概念性来对先前研究的结果统一观点。我们为具有$ M $ -CONVEX非线性的稳定解决方案提供了封闭性结果。结果，我们提供了一个liouville型的结果，并通过使用爆炸的论点，证明了极端解决方案的界限。
We consider the Gelfand problem in a bounded smooth domain $Ω\subset \mathbb{R}^N$ with the Dirichlet boundary condition. We are interested in the boundedness of the extremal solution $u^*$. When the dimension $N\ge10$, it is known that a singular extremal solution can be constructed for the nonlinearity $f(u)=e^u$ and $Ω=B_1$. When $3\le N\le 9$, Cabré, Figalli, Ros-Oton, and Serra (2020) proved the following surprising result: the extremal solution $u^*$ is bounded if the nonlinearity $f$ is positive, nondecreasing, and convex. In this paper, we succeed in generalizing their result to general $m$-convex nonlinearities. Moreover, we give a unified viewpoint on the results of previous studies by considering $m$-convexity. We provide a closedness result for stable solutions with $m$-convex nonlinearities. As a consequence, we provide a Liouville-type result and by using a blow-up argument, we prove the boundedness of extremal solutions.
