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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.

MSC:
 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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