On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large.

*(English)*Zbl 0572.35040We 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.

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.