Existence of global branches of positive solutions for semilinear elliptic degenerate problems (Q1583578)

The paper is devoted to the semilinear elliptic degenerate problem \[ \begin{cases} -|x|^2 \Delta u= \lambda u +g(u) & \text{in} B_1,\\ u\geq 0 & \text{in} B_1,\\ u=0 & \text{on} \partial B_1, \end{cases} \] where \(B_1=\{x\in \mathbb{R}^n : |x|<1\}\) and \(g\) is nonlinear. The authors prove the existence of global and connected branches of the solution in \(\mathbb{R}\times L_\infty (B_1)\) or \(\mathbb{R}\times H^1_0(B_1)\) for the following types of \(g(u)\): (1) \(u^p-u^q\), \(1<p<q\); (2) \(-u^p\), \(p>1\); (3) \(u^a\), \(0<a<1.\)