Resonance phenomenon for a Gelfand-type problem (Q395410)

The authors investigate the structure of the solution set of the boundary value problem NEWLINE\[NEWLINE -\Delta u = \lambda (e^u-1), \;u>0 \;\text{in} \;B, \quad u=0 \;\text{on} \;\partial B \tag{1} NEWLINE\]NEWLINE where \(B\) is the unit ball in \(\mathbb R^N\), \(N\geq 3\), and \(\lambda>0\) is a real parameter. Smooth solutions of (1) are known to be radially symmetric and decreasing. Moreover, classical solutions of (1) can exist only for some values of \(\lambda\).NEWLINENEWLINEIn this paper the following results are proved:NEWLINENEWLINE-- For \(N \geq 3\) there exists a unique \(\lambda^*\) such that for \(\lambda=\lambda^*\), problem (1) admits a unique radial singular solution.NEWLINENEWLINE-- For \(3 \leq N \leq 9\) and \(\lambda=\lambda^*\), problem (1) admits infinitely many regular radial solutions. For \(\lambda\) close to \(\lambda^*\) there is a ``large'' number of regular radial solutions.NEWLINENEWLINE-- For \(N \geq 10\) the number of regular solutions of (1) is bounded.NEWLINENEWLINEThe proof of the multiplicity result is based on geometric theory of three-dimensional dynamical systems, while the bound on the number of solutions is related to the Morse index of the solutions.