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We prove the existence of a nontrivial solution for a nonlinear elliptic problem &2u=+u+a(x) g(u) with Dirichlet boundary condition on a bounded domain, where g is superlinear both at zero and at infinity, a(x) changes sign and +>0.1998 Academic Press

ABSTRACT. Assuming B R is a ball in R N , we analyze the positive solutions of the problemthat branch out from the constant solution u = 1 as p grows from 2 to +∞. The non-zero constant positive solution is the unique positive solution for p close to 2. We show that there exist arbitrarily many positive solutions as p → ∞ (in particular, for supercritical exponents) or as R → ∞ for any fixed value of p > 2, answering partially a conjecture in [12]. We give the explicit lower bounds for p and R so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.