Multiple positive solutions to a class of modified nonlinear Schrödinger 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−2u+λk(x)u−u″+u−2((|u|2)″)u=a(x)|u|p−2u+λk(x)u where u∈H1(R)u∈H1(R). We will prove that the modified term 2((|u|2)″)u2((|u|2)″)u is helpful to find multiple positive solutions in the case of the nonlinearity with sign changing coefficient a(x)a(x). Surprisingly, we do not need a sign condition ∫RNa(x)e1pdx<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]).