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On the number of positive solutions of some semilinear Dirichlet problems in a ball. (English) Zbl 0940.35069

The number of solutions of the problem \[ -\Delta u = u^p + \lambda u^q \text{ in } B_1,\;u >0 \text{ in } B_1,\;u=0 \text{ on } \partial B_1 \] is discussed where \(B_1\) is the unit ball in \(\mathbb{R}^n\) \((n\geq 3)\), \(\lambda \) is a positive real number, \(0\leq q<p<\infty \). First, the case \(1<p\leq \frac {n+2}{n-2}\) is considered. It is shown that for \(q=0\), there are at most two solutions and it follows from previous results that there is \(\lambda ^*\) such that the problem has exactly two or one or none solution for \(\lambda \in (0,\lambda ^*)\) or \(\lambda = \lambda ^*\) or \(\lambda >\lambda ^*\), respectively. For \(0<q<1\) and \(\lambda \) sufficiently small, the existence of exactly two solutions is proved. Further, the case \(1\leq q < \frac {n+2}{n-2} <p\) is studied and the nonexistence of solutions for \(\lambda \) sufficiently small is given. Moreover, the problem \[ -\Delta u = \lambda [ (1+ \alpha u)^p + \mu (1+ \alpha u)] \text{ in } B_1,\;u >0 \text{ in } B_1,\;u=0 \text{ on } \partial B_1 \] is considered. It is proved that for any \(\lambda >0\), \(\alpha >0\), \(\mu \geq 0\), \(1<p\leq \frac {n+2}{n-2}\) there are at most two solutions. Actually, for fixed \(\alpha ,\;\mu \), there is \(\lambda ^*\) such that the problem has exactly two or one or none solution for \(\lambda \in (0,\lambda ^*)\) or \(\lambda = \lambda ^*\) or \(\lambda >\lambda ^*\), respectively.
Reviewer: M.Kučera (Praha)

MSC:

35J25 Boundary value problems for second-order elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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