MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHR?DINGER 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■(also named "a necessary condition" see [2,3]) does not hold.