Multiple Positive Solutions to a Class of Modified Nonlinear Schrödinger Equation in a High Dimension

The existence and multiplicity of positive solutions for equation (1.1) with the new critical exponent 4 < p < 2·2* shall be investigated in a high dimension. The conclusions extend the relative results recently attained in [1] for the one-dimensional case. More precisely, as the coefficient a ( x ) in the nonlinearity is sign-changing, the modified term 2(Δ(∣ u ∣ 2 )) u is still helpful for obtaining multiple positive solutions in a high dimension, even if a sign condition like ∫_ℝ^Na(x)e_1^pdx < 0 (also named “a necessary condition” see [2, 3]) does not hold.