Global branching for discontinuous problems in the exterior domain of a ball. (English) Zbl 0881.35041

Let \(f:\mathbb{R}\to[0,+ \infty[\) be a nondecreasing function, \(f(s)=0\) for \(s\leq 0\), \(f\in C^{0,\alpha}(]0,+ \infty[,]0,+ \infty[)\), where \(0<\alpha<1\), there exist \(c\geq d>0\) such that \(d\leq f(s)\leq c\) for every \(s>0\); \(\Omega= \{x\in\mathbb{R}^N:|x|>1\}\), \(N\geq 3\), \(a\geq 0\); \[ \begin{split} \Sigma= \{(a,u)\in [0,+\infty[\times L^1_{\text{loc}}(\Omega):\nabla u\in L^2(\Omega),\;u(x)>0\text{ for }x\in\Omega,\;u(x)=0\text{ for }x\in\partial\Omega,\\ u\in C^2(\overline\Omega\backslash \Omega(a))\text{ with }\Omega(a)= \{x\in \Omega:u(x)= a\},\;-\Delta u(x)= f(u(x)- a)\\ \text{for }x\in \Omega\backslash\Omega(a),\;u(x)\to 0\text{ as }x\to\infty,\;u\text{ is radial}\}.\end{split} \] The authors prove the following Theorem: There is an unbounded, closed, connected component \(S\) of \(\overline\Sigma\) such that \((0,0)\in S\); for every closed bounded interval \(I\subset[0,+\infty[\) there exists \(M(I)>0\) such that \(a\leq|\nabla u|_{L^2(\Omega)}\leq M(I)\) for every \((u,a)\in \Sigma\), \(a\in\dot I\); for every \(a>0\) there is \(u\) such that \((a,u)\in S\).
Reviewer: G.Bottaro (Genova)


35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs