Existence and dynamics of normalized solutions to nonlinear Schrodinger equations with mixed fractional Laplacians

In this paper, we are concerned with the existence and dynamics of solutions to the equation with mixed fractional Laplacians (-Delta)(s1) u + (-Delta)(s2) u +lambda u = |u|(p-2) u Under the constraint integral(RN) |u|(2) dx = c > 0, where N >= 1,0 <s(2) < s(1) < 1,2 + 4s(1)/N <= p < infinity if N <= 2s(1) , 2 + 4s(1)/N <= p < p < 2N/N-2s(1) if N > 2s(1), lambda is an element of R appearing as Lagrange multiplier is unknown. The fractional Laplacian. The fractional Laplacian (-Delta)(s) is character characterized as F((-Delta)(s) u)(xi) = |xi|(2s) F(u) (xi) for xi is an element of R-N , where F denote the Fourier transform. First we establish the existence of ground state solutions and the multiplicity of bound state solutions. Then we study dynamics of solutions to the Cauchy problem for the associated time-dependent equation. Moreover, we establish orbital instability of ground state solutions.