Multiple positive solutions to a class of modified nonlinear Schrodinger equations

This paper is concerned with the existence and multiplicity of positive solutions of the following equation -u '' + u - 2((vertical bar u vertical bar(2)''))u = a(x)vertical bar u vertical bar(p-2)u + lambda k(x)u where u is an element of H-1(R). We will prove that the modified term 2((vertical bar u vertical bar(2))'')u is helpful to find multiple positive solutions in the case of the nonlinearity with sign changing coefficient a(x). Surprisingly, we do not need a sign condition integral(N)(R) a(x)e(1)(p) dx < 0, which has been proven to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with sign changing nonlinearity (Alama and Tarantello (1993) [1]). (C) 2014 Elsevier Inc. All rights reserved.