论文标题
普遍的Picone不平等现象及其应用于$（P，Q）$ - 拉普拉斯方程
Generalized Picone inequalities and their applications to $(p,q)$-Laplace equations
论文作者
论文摘要
我们获得了Picone不平等的概括，该概括与经典的Picone不等式相结合，似乎对于$（P，Q）$ -Laplace类型运算符的问题很有用。借助它的帮助以及在其他几个已知的广义Picone不平等的帮助下，我们提供了一些关于方程式$-Δ_Pu-Δ_Qu-Δ_qu =f_μ（x，u，u，u，\ nabla u）的零dirichlet问题的积极解决方案的存在和不存在的事实，该问题是在一个边界$ puneped domain $ phumain $ phumain $ phup的基础上。非线性和特别注意共振案例$f_μ（x，u，\ nabla u）=λ_1（p）| u | u |^{p-2} u + + + +μ| u | u | u |^{q-2} u $，其中$λ_1（p）$是$ p $ p $ p $ -laplacian的第一个eigenvalue。
We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = f_μ(x,u,\nabla u)$ in a bounded domain $Ω\subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_μ(x,u,\nabla u) = λ_1(p) |u|^{p-2} u + μ|u|^{q-2} u$, where $λ_1(p)$ is the first eigenvalue of the $p$-Laplacian.
