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∫_(R)N a(x)e_(1)^(p)dx<0(also named“a necessary condition”see[2,3])does not hold.