Existence of positive solutions for semilinear elliptic equations in general domains (Q1114856)

The authors consider the existence of positive solutions to the Dirichlet problem \[ (1)\quad \Delta u(x)+f(u(x))=0,\quad x\in \Omega;\quad u(x)=0,\quad x\in \partial \Omega, \] where \(\Omega\) is a bounded domain in \(R^ n\) with smooth boundary and f is a continuous function on R. For it, they introduce the notion of ``eccentricity'', e(\(\Omega)\), of a domain \(\Omega\) with the property that \(1\leq e(\Omega)<+\infty\) and \(e(\Omega)=1\) if and only if \(\Omega\) is a ball, and the notion of the ``nonlinearity'' of f, N(f) (for instance, if f is a linear function, \(N(f)=1\); if \(f(u)=u^ k\), \(k<1\), \(N(f)=\infty\) and if \(f(u)=u^ k\), \(k>1\), \(N(f)<1)\). They prove, using a variation of the method of upper and lower solutions, that if \(N(f)>e(\Omega)\) then there exist positive solutions to (1) on all domains \(\lambda\) \(\Omega\) if \(\lambda\) is sufficiently large (equivalently, positive solutions to the Dirichlet problem for \(\Delta u+\mu f(u)=0\) exist on \(\Omega\) for some range of \(\mu)\). Then, they give some applications of this result. Also, they consider in more detail the case where \(\Omega\) is an n-ball and the Neumann problem on n-balls. The used method may be applied to more general elliptic operators.