Semilinear elliptic boundary value problems in chemical reactor theory
This paper is devoted to the study of the semilinear elliptic boundary value problem \(Au(x)= f(\lambda, u)\) in a bounded domain \(D\subset \mathbb{R}^N\) \((N\geq 2)\) with Dirichlet or Neumann boundary condition on the smooth boundary \(\partial D\); \(A\) is a second-order strongly elliptic differential operator with real smooth coefficients defined in \(\overline D\). Such equations arise in chemical reactor theory. The function \(f(\lambda, u)= \lambda\exp[u/(1+ \varepsilon u)]\) \((\varepsilon, \lambda>0)\) describes the temperature dependence of reaction rate for exotermic reactions obeying the simple Arrhenius rate law in circumstances in which heat flow is purely conductive. The same authors have already shown in [Nonlinear Anal., Theory Methods Appl. 29, No. 7, 761-771 (1997; Zbl 0878.35048)] that for each positive \(\lambda\), the problem under consideration has at least one positive solution and it is unique if \(\varepsilon\geq 1/4\). In the present paper, the authors study the case \(0<\varepsilon< 1/4\) and prove the existence of at least three distinct positive solutions provided that certain relations between \(\lambda\) and \(\varepsilon\) are satisfied. Moreover, two existence and uniqueness results are stated for \(\lambda\) sufficiently small and sufficiently large.
scientific article; zbMATH DE number 1212879
scientific article