Semilinear elliptic problems of the form \[ -\Delta u = f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial\Omega, \] where \(\Omega\) is a smooth bounded domain in \({\mathbb R}^N\), have received a great deal of attention. Of particular interest here is the existence of nonnegative solutions in the case where \(f\) is not Lipschitz continuous at \(0\). In this situation there may be nonnegative solutions that are positive only in a proper open subset of \(\Omega\), so a free boundary \(\Gamma=\partial\{x\in\Omega:u(x)=0\}\) occurs. This can also occur if \(\Delta\) is replaced by other elliptic operators such as the \(p\)-Laplacian.
Motivated by this phenomenon, the authors consider the one-dimensional problem \[ -(|u'|^{p-2}u')' = f(u) \text{ in }(-L,L), \qquad u(\pm L)=0, (*) \] where \(p>1\) and \(f\) satisfies a number of conditions. The main ones are that \(f\) is defined at least on \([0,\infty)\) with \(f\in C^0(0,\infty)\cap L^1_{loc}(0,\infty)\) with \(\limsup_{s\searrow 0} f(s) \leq 0\), and in addition, there is a number \(r>0\) such that \(f(r)=0\) and \(f(s)<0\) if \(s\in(0,r)\) and \(f(s)>0\) if \(s\in(r,\infty)\). They prove several existence and multiplicity results for solutions of \((*)\) which extend the work of the first two authors [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 7, 587–592 (1999; Zbl 0937.35064)] to the case that \(f\) is singular at the origin, but now incorporating a phenomenon on the number of positive solutions arising for suitable singular functions \(f\) as in the works of Y. S. Choi, A. C. Lazer and P. J. McKenna [Nonlinear Anal., Theory Methods Appl. 32, No. 3, 305–314 (1998; Zbl 0940.35089)] and T. Horváth and P. L. Simon [“On the exact number of solutions of a singular boundary value problem”, preprint].
The authors also refine their results for the special case \[ -(|u'|^{p-2}u')' + \alpha u^m = \lambda u^q \text{ in }(-1,1), \qquad u(\pm 1)=0, \] where \(p>1\) and \(-1<m<q<p-1\).