Global branching for discontinuous problems. (English) Zbl 0732.35101

The elliptic problem with a discontinuous nonlinearity: \(-\Delta u=f(u- a)\) in \(\Omega\) and \(u=0\) on \(\partial \Omega\), is considered, where \(\Omega \subset {\mathbb{R}}^ N\) is a bounded domain with suitable symmetry, e.g. the ball B(R). The nonlinear term f(s) satisfies some growth condition, but has a discontinuity at \(s=0\). Taking \(a\in {\mathbb{R}}\) as a bifurcation parameter, the authors prove the existence of a global branch of positive solutions, starting from \((a,u)=(0,0)\) and bending at a point with \(a=a_ 0\), and consequently the existence of two distinct positive solutions for every \(a\in (0,a_ 0)\). By a limiting procedure based upon an a priori estimate of the free boundary \(\{u=a\}\), they also obtain the similar result for the case where the domain \(\Omega\) is \({\mathbb{R}}^ N\) (N\(\geq 3)\) itself and u(x) converges to 0 as \(| x| \to 0\).


35R05 PDEs with low regular coefficients and/or low regular data
35B32 Bifurcations in context of PDEs
35R35 Free boundary problems for PDEs
35J60 Nonlinear elliptic equations
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