Multiple Positive Solutions to Nonlinear Schrödinger Equations with Competing Potential Functions

In this paper we obtain multiple positive solutions of the nonlinear elliptic equation −h2 Δu+V(x) u=K(x) up−2 u+Q(x) uq−2 u, x RN, where V, K, Q are competing potential functions. We relate the number of solutions with the topology of the global minima set of a suitable ground energy function and improve a recent existence result of X. Wang and B. Zeng (1997, SIAM J. Math. Anal.28, 633–655).