论文标题
$ \ mathbb {r}^n $中半磷灰元素问题的非平滑变量方法
A nonsmooth variational approach to semipositone quasilinear problems in $\mathbb{R}^N$
论文作者
论文摘要
本文涉及存在以下半尖酮准线性问题的解决方案 \ begin {equation*} \左边 \{ \ begin {array} {rclcl} -Δ_Pu = h（x）（f（u）-a），\＆ u> 0＆\ mbox {in}＆\ mathbb {r}^n， \ end {array} \正确的。 \ end {equation*}其中$ 1 <p <n $，$ a> 0 $，$ f：[0，+\ infty）\ to [0，+\ infty）$是一个具有亚临界增长的功能，而$ f（0）= 0 $，而$ h：h：\ h：\ mathbb {r}^n \ to（r}^n \ forking（r}^n \ to（0）我们通过非平滑关键点理论和比较原则证明，解决方案的存在于$ a $小的情况下。我们还提供了Hopf的引理的版本和整个$ \ Mathbb {r}^n $中$ p $ -laplacian的liouville-type结果。
This paper concerns the existence of a solution for the following class of semipositone quasilinear problems \begin{equation*} \left \{ \begin{array}{rclcl} -Δ_p u = h(x)(f(u)-a),\ & u > 0 & \mbox{in} & \mathbb{R}^N, \end{array} \right. \end{equation*} where $1<p<N$, $a>0$, $ f:[0,+\infty) \to [0,+\infty)$ is a function with subcritical growth and $f(0)=0$, while $h:\mathbb{R}^N \to (0,+\infty)$ is a continuous function that satisfies some technical conditions. We prove via nonsmooth critical points theory and comparison principle, that a solution exists for $a$ small enough. We also provide a version of Hopf's Lemma and a Liouville-type result for the $p$-Laplacian in the whole $\mathbb{R}^N$.
