## Global positive solution branches of positone problems with nonlinear boundary conditions.(English)Zbl 0983.35051

The paper is concerned with the existence of smooth branches of positive solutions $$(\lambda, u_\lambda)$$ of the semilinear elliptic boundary value problem with nonlinear boundary condition $(-\Delta+ c(x))u= \lambda f(u)\quad\text{in }D,\quad {\partial u\over\partial\nu}+ b(x) g(u)= 0\quad\text{on }\partial D,$ where $$D$$ is a bounded domain in $$\mathbb{R}^N$$ $$(N\geq 2)$$ with smooth boundary $$\partial D$$, $$c$$, $$f$$, $$g$$ are smooth functions satisfying $$c(x)> 0$$ in $$D$$, $$b(x)\geq 0$$ on $$\partial D$$; $$g\geq 0$$, and $$\lambda$$ is a positive parameter. When $$f(t)> 0$$ $$(t\geq 0)$$, under several conditions on $$f$$ and $$g$$, the author showed that some $$\mu\in (0,\infty]$$ is determined by the first eigenvalue of some eigenvalue problem and by the conditions on $$f$$ and $$g$$ such that there exists a unique $$C^1$$-branch $$(\lambda, u_\lambda)$$ of positive solutions parameterized by $$\lambda\in (0,\mu)$$ and there exist no positive solutions if $$\lambda\geq \mu$$. Furthermore, the $$C^1$$-branch $$(\lambda, u_\lambda)$$ satisfies $\|u_\lambda\|_{C(\overline D)}\to 0\quad\text{as }\lambda\downarrow 0,\quad\text{and }\|u_\lambda\|_{C(\overline D)}\to \infty\quad\text{as }\lambda\uparrow\mu.$ When $$f(t)$$ has a zero $$t_0> 0$$ such that $f(t)> 0\quad\text{if }0\leq t< t_0,\quad\text{and }f(t)< 0\quad\text{if }t> t_0,$ under several conditions on $$f$$ and $$g$$, he showed that there exists a unique $$C^1$$-branch $$(\lambda, u_\lambda)$$ of positive solutions parameterized by $$\lambda\in (0,\infty)$$ satisfying $\|u_\lambda\|_{C(\overline D)}\to 0\quad\text{as }\lambda\downarrow 0,\quad\text{and }\|u_\lambda\|_{C(\overline D)}\to t_0\quad\text{as }\lambda\uparrow \infty.$

### MSC:

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