论文标题
根据$ p $ - laplacian及其sublerear的抗临时原则
On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations
论文作者
论文摘要
我们研究了方程问题的弱解决方案的定性特性$-Δ_Pu =λm（x）| u |^{p-2} u + ha（x）| u | u |^{q-2} u + f（x）U + f（x）$ in Bounded域中$ω\ subset $Ω\ subset \ subset \ subset \ subseet \ subbb {r} r} r}^n $，在某些规律性和重量$ m，a $和源函数$ f $的定性假设下，我们确定了参数范围$λ$和$η$的范围，而解决方案满足弱和强形式的最大和抗触发原则。我们的某些结果，尤其是在低规律性假设下的反临时原理的有效性上，对于不受干扰的问题，$η= 0 $是新的，其中即使在线性案例$ p = 2 $中，结果也提供了新信息。特别是，我们表明，对于任何$ p> 1 $的解决方案，不受干扰的问题的解决方案满足了$ p $ -laplacian的第一个特征值的正确社区中的反临时原理，提供了$ m，f \ in l^γ（ω）$的$ m，f \ in $γ> n $。为了完整，我们还研究了解决方案的存在。
We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-Δ_p u = λm(x)|u|^{p-2}u + ηa(x)|u|^{q-2}u + f(x)$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $q<p$. Under certain regularity and qualitative assumptions on the weights $m, a$ and the source function $f$, we identify ranges of parameters $λ$ and $η$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $η=0$, and among them there are results providing new information even in the linear case $p=2$. In particular, we show that for any $p>1$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the $p$-Laplacian provided $m,f \in L^γ(Ω)$ with $γ>N$. For completeness, we also investigate the existence of solutions.
