论文标题
存在参数分数$ p $ -laplacian问题的积极解决方案与半蛋白酶非线性
Existence of positive solutions for a parameter fractional $p$-Laplacian problem with semipositone nonlinearity
论文作者
论文摘要
在本文中，我们证明了至少存在一种非局部半明苏酮问题的阳性解决方案 \ [ \ displaystyle \ left \ {\ begin {array} {rcll} （-Δ）_p^s（u）＆=＆λf（u）\ qquad＆\ text {in} \ \ω \\ u＆=＆0＆\ text {in} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \}^n-Ω， \ end {array} \ right。 \] 每当$λ> 0 $都是一个足够小的参数时。这里$ c^{1,1} $ bouncy，$ c^{1,1} $ boundard，$ 2 \ leqslant p <n $，$ s \ in（0,1）$和$ f $ superlineal and subcritical和subcritical。我们证明，如果选择$λ> 0 $足够小，则相关的能量功能与问题的功能具有山通结构，因此，它具有关键点$u_λ$，这是一个薄弱的解决方案。之后，我们设法通过使用新的规律性结果到边界和HOPF的引理来证明该解决方案是积极的。
In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-Δ)_p^s(u) &=& λf(u) \qquad & \text{in} \ \ Ω \\u &=& 0 & \text{in} \ \ \mathbb{R}^N -Ω, \end{array}\right. \] whenever $λ>0$ is a sufficiently small parameter. Here $Ω\subseteq \mathbb{R}^N$ a bounded domain with $C^{1,1}$ boundary, $2\leqslant p <N$, $s\in (0,1)$ and $f$ superlineal and subcritical. We prove that if $λ>0$ is chosen sufficiently small the associated Energy Functional to the problem has a mountain pass structure and, therefore, it has a critical point $u_λ$, which is a weak solution. After that we manage to prove that this solution is positive by using new regularity results up to the boundary and a Hopf's Lemma.
