Local Well-Posedness of a Critical Inhomogeneous Bi-harmonic Schrodinger Equation

This note studies the inhomogeneous fourth-order Schrodinger equation iu + Delta(2)u = +/-|x|b |u|p(-1) u, b<0 ,p>1. Indeed, the local existence of solutions is established in some Sobolev critical spaces H-sc. In particular, one considers the mass-critical regime s(c)=0 and the energy-critical one s(c)=2. For more efficient way to handle the spatially decaying factor |x|(b) in the nonlinearity, we approach to the matter in a weighted Lebesgue space which seems to be more suitable to perform a finer analysis for this problem. In fact, this way enables to investigate the critical regime which seems to be still an open problem. The method used to prove the existence of energy subcritical local solutions consists on dividing the integrals on the unit ball of R-N and its complementary and use the fact that |x|(b) is an element of LN/-b (-) (epsilon)(|x|<1) and |x|(b) is an element of L (N/-b +) (epsilon)(|x|>1). This method is no more sufficient to investigate the critical regime. The proof combines a standard fixed point argument with some Strichartz estimates in weighted Lebesgue spaces. This follows the method developed recently (Kim et al. in J Differ Equ 280:179-202, 2021). In a paper in progress, the authors investigate the scattering of global solutions versus the finite time blow-up of nonglobal solutions.