## 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)

### MSC:

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

### Keywords:

global solution branch