Let be a bounded domain in . The paper is concerned with the semilinear elliptic problem
where , as , is the positive eigenvalue of the Laplacian and . To there corresponds a functional
in and critical points of are solutions of . It is shown that for in certain regions of , if is large enough, then has at least one, two, three, respectively four solutions. Existence of one solution is shown by using a variant of the saddle point theorem of Rabinowitz. Two and three solutions are obtained by linking-type arguments where careful estimates are needed in order to show that certain linking levels are different. An additional argument gives a fourth critical point. It should also be noted that a rather general sufficient condition for to satisfy the Palais-Smale condition is given in this paper.