Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity (Q5936436)

Positive solutions to NEWLINE\[NEWLINEu''+\lambda f(u)=0, \quad x\in (-1,1), \qquad u(-1)=u(1)=0 ,NEWLINE\]NEWLINE where \(\lambda\) is a positive parameter and \(f(0)<0\), are considered. Recall that the nonlinearity \(f\) is called semipositon if \(f(0)<0\) and that semipositone problems were introduced by \textit{A. Castro} and \textit{R. Shivaji} [Proc. R. Soc. Edinb., Sect. A 108, No. 3/4, 291-302 (1988; Zbl 0659.34018)]. By symmetry, the previous problem can be reduced to NEWLINE\[NEWLINEu''+\lambda f(u)=0,\quad x\in (0,1),\quad u'(0)=u(1)=0,\quad u'(x)<0,\;x\in (0,1).NEWLINE\]NEWLINE On the other hand, all solutions to the last problem can be parametrized by their initial values \(\varrho=u(0)\). In turn, we get that the solution set to the last problem can be represented by \(\lambda=\lambda(\varrho)\). NEWLINENEWLINENEWLINEThe subject of this study is the precise behaviour of the bifurcation diagram (dependence of \(\lambda\) on \(\varrho\)) of the last problem for a class of semipositone nonlinearities. NEWLINENEWLINENEWLINEThe approach here uses a bifurcation result by \textit{M. G. Crandall} and \textit{P. H. Rabinowitz} [Arch. rat. Mech. Analysis 52, 161-180 (1973; Zbl 0275.47044)] and some integral comparison arguments from \textit{P. Korman, Y. Li} and \textit{T. Ouyang} [Proc. R. Soc. Edinb., Sect. A 126, No. 3, 599-616 (1996; Zbl 0855.34022)]. This approach enables one to obtain generalizations to radially symmetric solutions to semipositone problems for semilinear partial differential equations.