Spacetime estimates and scattering theory for quasilinear Schrodinger equations in arbitrary space dimension

In this paper, we consider Cauchy problem of a quasilinear Schrodinger equation which has general form containing potential term, power type nonlinearity and Hartree type nonlinearity. The space dimension is arbitrary, that is, it is larger than or equals to one. First, we establish the local wellposedness of the solution and discuss the condition on the global existence of the solution. Next, we establish some conservation laws such as mass conservation law, energy conservation law, pseudoconformal conservation law of the solution. Based on these conservation laws, we give Morawetz type estimates, spacetime bounds for the global solution. Last, we take two ideas to establish scattering theory for the global solution in different functional spaces. The first idea is that we take different admissible pairs in Strichartz estimates for different terms on the right side of Duhamel’s formula in order to keep each term independent, another one is that we factitiously let a continuous function be the sum of two piecewise functions and choose different admissible pairs in Strichartz estimates for the terms containing these functions.

Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials

In this paper, we study the following quasilinear Schrödinger equation: $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ − div ( a ( x , ∇ u ) ) + V ( x ) | x | − α p ∗ | u | p − 2 u = K ( x ) | x | − α p ∗ f ( x , u ) in R N , where $N\geq 3$ N ≥ 3 , $1< p< N$ 1 < p < N , $-\infty <\alpha <\frac{N-p}{p}$ − ∞ < α < N − p p , $\alpha \leq e\leq \alpha +1$ α ≤ e ≤ α + 1 , $d=1+\alpha -e$ d = 1 + α − e , $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$ p ∗ : = p ∗ ( α , e ) = N p N − d p (critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.

Retraction Note: Existence of global solutions to a quasilinear Schrödinger equation with general nonlinear optimal control conditions

This article has been retracted. Please see the Retraction Notice for more detail: 10.1186/s13661-021-01502-z