Multiple positive solutions for semilinear Schrödinger equations with critical growth in ℝN

Abstract

In this paper, we study the existence, multiplicity, and concentration of positive solutions for the semilinear Schrödinger equation −ε2Δu+K(x)u=Q(x)up−2u+f(u),u∈H1(RN), where ε > 0 is a small parameter, N ≥ 3 and 2<p<2∗=2NN−2, K and Q are positive continuous functions, f is a continuous superlinear nonlinearity with critical growth. First of all, we prove that there are two families of semiclassical positive solutions for ε > 0 small, one is concentrating on the set of minimal points of K, another is concentrating on the sets of maximal points of Q. Second of all, we investigate the relation between the number of solutions and the topology of the set of the global minima (or maxima) of the potentials (K and Q) by the Ljusternik-Schnirelmann and theory minimax theorems. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solution.