A Study On The Number Of Peak Solutions To Schr(?)dinger Equations With Magnetic Field

We consider the following nonlinear Schr(?)dinger equation with electromagnetic fields(ih(?)+A(x))2 u+V(x)u=|u|p-1u,x ∈ RN,u:RN→C,where i is imaginary unit,h>0 is the Plank constant,A(x)=(A1(x),…,AN(x))is a vector valued function,Aj(x),j=1,2,…,N is real functions,V(x)is a positive bounded real function,1<p<(N+2)/(N-2)when N≥ 3,1<p<+∞ when N=1,2.We show that if A(x)and V(x)satisfy certain conditions,then there exists an h0>0 such that for 0<h<h0,for each integer K bounded by 1≤K≤αN,p/hmin{(p-1)/2p,1/3}|ln h|(N-1)/2+2 where αN,P is a constant depending on N and p only,there exists a solution with K interior peaks.Especially when h→0,the K peaks tend to the same position.