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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.

34B15 Nonlinear boundary value problems for ordinary differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI
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