MULTIPLE COMPLEX-VALUED SOLUTIONS FOR THE NONLINEAR SCHRODINGER EQUATIONS INVOLVING MAGNETIC POTENTIALS

作者：Gan, Lu; Liu, Weiming^*

来源：Communications on Pure and Applied Analysis, 2017, 16(6): 1957-1975.

DOI：10.3934/cpaa.2017096

• Lyapunov-Schmidt reduction
• magnetic potentials
• nonlinear Schrodinger equations
• multiple complex-valued solutions

This paper is concerned with the following nonlinear Schrodinger equations with magnetic potentials @@@ (del/i - alpha A(vertical bar x vertical bar))(2) u + (1 + alpha V(vertical bar x vertical bar))u = vertical bar u vertical bar(p-2)u, u is an element of H-1 (R-N, C), (0.1) @@@ where 2 < p < 2N/N-2 if N >= 3 and 2 < p < + infinity if N = 2. alpha can be regarded as a parameter. A(vertical bar x vertical bar) = (A(1)(vertical bar x vertical bar); A(2) (vertical bar x vertical bar), . . . , A(N)(vertical bar x vertical bar)) is a magnetic fi eld satisfying that A (j) (vertical bar x vertical bar) > 0(j = 1, . . . , N) is a real C 1 bounded function on R N and V(vertical bar x vertical bar) > 0 is a real continuous electric potential. Under some decaying conditions of both electric and magnetic potentials which are given in section 1, we prove that the equation has multiple complex-valued solutions by applying the fi nite reduction method.

• 出版日期2017-11
• 单位湖北师范大学