We consider a nonlinear Dirichlet problem, driven by the p -Laplacian with a reaction involving two parameters $$\lambda \in {\mathbb {R}}, \theta >0$$ λ ∈ R , θ > 0 . We view the problem as a perturbation of the classical eigenvalue problem for the Dirichlet problem. The perturbation consists of a parametric singular term and of a superlinear term. We prove a nonexistence and a multiplicity results in terms of the principal eigenvalue $${\hat{\lambda }}_1>0$$ λ ^ 1 > 0 of $$(-\Delta _p, W_0^{1,p}(\Omega ))$$ ( - Δ p , W 0 1 , p ( Ω ) ) . So, we show that if $$\lambda \ge {\hat{\lambda }}_1$$ λ ≥ λ ^ 1 and $$\theta >0$$ θ > 0 , then the problem has no positive solution, while if $$\lambda <{\hat{\lambda }}_1$$ λ < λ ^ 1 and $$\theta >0$$ θ > 0 is suitably small (depending on $$\lambda $$ λ ), there are two positive smooth solutions.

中文翻译：

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Dirichlet p-Laplacian 特征值问题的奇异和超线性摄动

我们考虑非线性狄利克雷问题，由 p -Laplacian 驱动，反应涉及两个参数 $$\lambda \in {\mathbb {R}}, \theta >0$$ λ ∈ R , θ > 0 。我们将该问题视为狄利克雷问题的经典特征值问题的扰动。扰动由参数奇异项和超线性项组成。我们证明了不存在性和多重性结果的主要特征值 $${\hat{\lambda }}_1>0$$ λ ^ 1 > 0 of $$(-\Delta _p, W_0^{1,p} (Ω ))$$ (-Δ p , W 0 1 , p ( Ω ) ) 。所以，我们证明如果 $$\lambda \ge {\hat{\lambda }}_1$$ λ ≥ λ ^ 1 并且 $$\theta >0$$ θ > 0 ，那么问题没有正解，而如果 $$\lambda <{\hat{\lambda }}_1$$ λ < λ ^ 1 和 $$\theta >0$$ θ > 0 适当小（取决于 $$\lambda $$ λ ），