NONLOCAL SCHRODINGER-KIRCHHOFF EQUATIONS WITH EXTERNAL MAGNETIC FIELD

The paper deals with the existence and multiplicity of solutions of the fractional Schrodinger Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case (a + b[u](s,A)(2u-2)) (-Delta)(A)(s)u + V(x)u = f(x,vertical bar u vertical bar)u in R-N , where s is an element of(0,1), N> 2s, a is an element of R-0(+), b is an element of R-0(+) [1, N/(N- 2s)), A: R-N -> R-N magnetic potential, V: R-N -> R-N is an electric potential, (-Delta)(A)(s) is the fractional magnetic operator. In the super and sub linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.