Multiple positive solutions to a class of modified nonlinear Schrِdinger equations

This paper is concerned with the existence and multiplicity of positive solutions of the following equation − u ″ + u − 2 ( ( | u | 2 ) ″ ) u = a ( x ) | u | p − 2 u + λ k ( x ) u where u ∈ H 1 ( R ) . We will prove that the modified term 2 ( ( | u | 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 ∫ R N a ( x ) e 1 p d x < 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]).