Positive solutions for nonlinear singular elliptic equations of \(p\)-Laplacian type with dependence on the gradient
Given a bounded domain \(\Omega\subset \mathbb{R}^N, N \geq 3\), with a \(C^2\)-boundary \(\partial\Omega\), the authors study the following singular elliptic equation with Dirichlet boundary condition \[ \begin{cases} -\Delta_{p} u(x) = f(x,u(x),\nabla u(x))+g(x,u(x)) &\text{in }\Omega,\\ u > 0 &\text{in }\Omega,\\ u=0 &\text{on }\partial \Omega, \end{cases} \] where \(\Delta_p u\) stands for the \(p\)-Laplacian, the convextion term \(f:\Omega\times\mathbb{R}\times \mathbb{R}^N\to\mathbb{R}\) satisfies suitable growth and the semilinear function \(g: \Omega \times (0,\infty)\to\mathbb{R}\) is singular at \(s=0\). Based on the Leray-Schauder alternative principle, the method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis, the authors prove an existence theorem for positive solutions and show the compactness of the solution set.
scientific article; zbMATH DE number 6999929
scientific article