AN EXACT MULTIPLICITY RESULT FOR A CLASS OF SEMILINEAR EQUATIONS

For a class of Dirichlet problems in two dimensions, generalizing the model case Delta u + lambda u(u-b)(c-u) = 0 in \x\ < R, u = 0 on \x\ = R we show existence of a critical lambda(0) > 0, so that there are ex actly 0, 1 or 2 nontrivial solutions (in fact, positive), depending on whether lambda < lambda(0), lambda = lambda(0) or lambda > lambda(0). We show that all solutions lie on a single smooth solution curve, and study some properties of this curve. We use bifurcation approach. The crucial thing is to show that any nontrivial solution of the correspo nding linearized problem is of one sign.