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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
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