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On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large. (English) Zbl 0572.35040
We consider positive solutions of \(-\Delta u=\lambda f(u)\) in D, \(u=0\) on \(\partial D\). We assume that \(a>0\) (possibly \(+\infty)\), f: [0,a]\(\to R\) is \(C^ 1\) such that \(f(t)>0\) on (0,a) \(f(a)=0\) if \(a<\infty\) and \(f(t)\to C>0\) as \(t\to \infty\) if \(a=\infty\). We only consider solutions such that u(x)\(\leq a\) in D. We also assume that D is bounded in \(R^ n\) with smooth boundary. If \(f(0)>0\) or \(f'(0)>0\) and some technical conditions are satisfied, we prove that there is a unique positive solution for large \(\lambda\).
Now assume that \(f(0)=f'(0)=0\) and that there is a \(p>1\) such that \(y^{1-p}f'(y)\to bp\) as \(y\to 0\) where \(b>0\). If \(p\leq n(n-2)^{-1}\), we prove that there is a unique positive solution whose sup norm is not small and prove that the small positive solutions are largely detemied by the positive solutions of the equation \(-\Delta u=u^ p\) in D, \(u=0\) on \(\partial D.\)
We prove a similar result if \(p\in (n(n-2)^{-1}, (n+2)(n-2)^{-1})\) under a global condition on f and give examples showing that this condition can not be entirely removed. We also give an example where \(a=\infty\) and the set of positive solutions is not connected.

35J65 Nonlinear boundary value problems for linear elliptic equations
47J05 Equations involving nonlinear operators (general)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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