Abstract 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 ) (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 , ∇ u ) -\hspace{-0.25em}{\text{Δ}}_{p}u-{\text{Δ}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain Ω ⊂ ℝ N \text{Ω}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case f μ ( x , u , ∇ u ) = λ 1 ( p ) | u | p − 2 u + μ | u | q − 2 u {f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u , where λ 1 ( p ) {\lambda }_{1}(p) is the first eigenvalue of the p-Laplacian.

中文翻译：

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广义 Picone 不等式及其在 (p,q)-Laplace 方程中的应用

摘要 我们获得了 Picone 不等式的推广，它与经典的 Picone 不等式相结合，似乎对 ( p , q ) (p,q) -Laplace 类型算子的问题很有用。在它的帮助下，以及在其他几个已知的广义 Picone 不等式的帮助下，我们提供了一些关于零狄利克雷问题的正解的存在和不存在的重要事实 - Δ pu - Δ qu = f μ ( x , u , ∇ u ) -\hspace{-0.25em}{\text{Δ}}_{p}u-{\text{Δ}}_{q}u={f}_{\mu }(x ,u,\nabla u) 在有界域 Ω ⊂ ℝ N \text{Ω}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} 中，在非线性的某些假设下，特别注意共振情况 f μ ( x , u , ∇ u ) = λ 1 ( p ) | 你| p − 2 u + μ | 你| q − 2 u {f}_{\mu }(x,u,