On the structure of the positive solutions of the logistic equation with nonlinear diffusion (Q1604239)

This paper is devoted to the structure of the positive solutions of the degenerate logistic equation, that is \[ \begin{cases} d{\mathcal J}w^m=\sigma w-b(x)w^z\quad &\text{in }\Omega\\ w=0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^N\), \(N\leq 1\), with a smooth boundary \(\partial\Omega\); \({\mathcal J}\) is a general second-order uniformly elliptic operator; \(b\) is a positive function, \(m\geq 1\), \(r>1\), \(d\) is a positive constant, and \(\sigma\) is a real parameter. An appropriate change of variable transforms (1) into \[ \begin{cases} {\mathcal J}u=\lambda u^q-b(x)u^p\quad &\text{in }\Omega\\ u=0\quad &\text{on }\partial\Omega,\end{cases}\tag{2} \] with \(\lambda\in \mathbb{R}\), \(0<q<p\), and \(q\leq 1\). Under suitable asumptions, on the data of (2) the authors prove that, there exists a unique positive solution of (2) if, and only if, \(\lambda>0\).