Quasilinear equations and equations with large nonlinearities

Quasilinear Schrödinger Equations Ii: Small Data and Cubic Nonlinearities

In part I of this project we examined low regularity local well-posedness for generic quasilinear Schrödinger equations with small data. This improved, in the small data regime, the preceding results of Kenig, Ponce, and Vega as well as Kenig, Ponce, Rolvung, and Vega. In the setting of quadratic interactions, the (translation invariant) function spaces which were utilized incorporated an l sum...

Large Solutions for Quasilinear Elliptic Equations

Wave Equations with Concentrated Nonlinearities

In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field V on an open subset of C and a discrete set Y ⊂ R with n elements, we define a nonlinear operator ∆V,Y on L (R) which coincides with the free Laplacian when restricted to regular functions vanishing at Y , and which reduces to the usual Laplacian with point interactions placed ...

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain the asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as...

Existence of Radial Solutions for Quasilinear Elliptic Equations with Singular Nonlinearities

We prove the existence of radial solutions of the quasilinear elliptic equation div(A(|Du|)Du) + f(u) = 0 in R, n > 1, where f is either negative or positive for small u > 0, possibly singular at u = 0, and growths subcritically for large u. Our proofs use only elementary arguments based on a variational identity. No differentiability assumptions are made on f .