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Uniqueness and stability of nonnegative solutions for semipositone problems in a ball. (English) Zbl 0770.34019
If \(f:[0,\infty)\to\mathbb{R}\) is monotonically increasing, \(f(0)<0\), \(f(u)>0\) for some \(u>0\) and \(f\) is concave, then there exists \(\mu_ 1\) such that if \(\lambda>\mu_ 1\), then the problem \(\Delta u+\lambda f(u)=0\) in \(\Omega\), \(\Omega\) is the unit ball in \(\mathbb{R}^ n\), \(n\geq 2\), \(u=0\) on \(\partial\Omega\), has at most one nonnegative solution. Furthermore, there exists \(\mu_ 2\) such that all nonnegative solutions for \(\lambda>\mu_ 2\) are stable. A similar result is obtained if \(f\) is convex and some other conditions hold.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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