Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory (Q2706209)

The semilinear Dirichlet problem: NEWLINE\[NEWLINE -\bigtriangleup v= \mu (1+\varepsilon v)^m e^{v/(1+\varepsilon v)} \quad \text{in } B,\qquad v=0\quad \text{on }\partial BNEWLINE\]NEWLINE arising in the combustion theory is considered, with \(B\) the open unit ball in \(\mathbb{R}^n\), \(\varepsilon >0\), \(m\geq 0\), \(\mu >0.\) The author obtains the exact number of positive solutions for \(n=2\) and \(0\leq m <1.\) It is shown that the main results are not true if \(n> 2\) or \(m\geq 1.\)