THE MULTIPLICITY OF SOLUTIONS AND GEOMETRY OF A NONLINEAR ELLIPTIC EQUATION

Let Omega be a bounded domain in R(n) with smooth boundary partial der ivative Omega and let L denote a second order linear elliptic differen tial operator and a mapping from L(2)(Omega) into itself with compact inverse, with eigenvalues -lambda(i), each repeated according to its m ultiplicity, 0 < lambda(1) < lambda(2) < lambda(3) less than or equal to...less than or equal to lambda(i) less than or equal to... --> infi nity. We consider a semilinear elliptic Dirichlet problem Lu + bu(+) - au(-) = f(x) in Omega, u = 0 on partial derivative Omega. We assume t hat a < lambda(1), lambda(2) < b < lambda(3) and f is generated by phi (1) and phi(2). We show a relation between the multiplicity of solutio ns and source terms in the equation.