SEMI-CLASSICAL STATES FOR FRACTIONAL SCHRODINGER EQUATIONS WITH MAGNETIC FIELDS AND FAST DECAYING POTENTIALS

摘要

This paper deals with the following fractional magnetic Schrodinger equations @@@ epsilon(2s)(-Delta)(A/epsilon)(s) u + V(x)u = vertical bar u vertical bar(p-2)u, x is an element of R-N, @@@ where epsilon > 0 is a parameter, s is an element of (0, 1), N >= 3, 2 + 2s/(N - 2s) < p < 2(s)* := 2N/(N - 2s), A is an element of C-0,C-alpha(R-N, R-N) with alpha is an element of (0, 1] is a magnetic field, V : R-N -> R is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of V as epsilon -> 0. There is no restriction on the decay rates of V. Especially, V can be compactly supported. The appearance of A and the nonlocal of (-Delta)(s) makes the proof more difficult than that in [7], which considered the case A equivalent to 0.

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