We consider a nonlinear elliptic Dirichlet problem driven by the $(p,q)$-Laplacian and a reaction consisting of a parametric singular term plus a Carathéodory perturbation $f(z,x)$ which is $(p-1)$-linear as $x\to +\infty$. First we prove a bifurcation-type theorem describing in an exact way the changes in the set of positive solutions as the parameter $\lambda >0$ moves. Subsequently, we focus on the solution multifunction and prove its continuity properties. Finally we prove the existence of a smallest (minimal) solution $u_{\lambda }^{\ast }$ and investigate the monotonicity and continuity properties of the map $\lambda \to u_{\lambda }^{\ast }$.

中文翻译：

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参数奇异Dirichlet的正解 $（p，q）$-等式

我们考虑一个非线性椭圆Dirichlet问题，它由 $（p，q）$-拉普拉斯算子和由参数奇异项加上Carathéodory扰动组成的反应 $F（ž，X）$ 这是 $（p-\mathrm{1个}）$-线性为 $X\to +\infty$。首先，我们证明分叉型定理，以精确的方式描述正解集中的变化作为参数$\lambda >0$动作。随后，我们着重于解决方案的多功能性并证明其连续性。最后，我们证明了最小（最小）解的存在${ü}_{\lambda }^{\ast }$ 并研究地图的单调性和连续性 $\lambda \to {ü}_{\lambda }^{\ast }$。