An approach to elliptic equations with nonlinear gradient terms via a modulation framework

Abstract

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] is the nonlinearity, [Formula: see text] the potential and [Formula: see text] is a forcing term. Some examples of nonlinearities dealt with are [Formula: see text], [Formula: see text] and [Formula: see text], covering large values of [Formula: see text], and particularly variational supercritical powers for [Formula: see text] and super-[Formula: see text] ones for [Formula: see text] (superquadratic if [Formula: see text]). Moreover, we are able to consider some exponential growths, [Formula: see text] belonging to certain classes of power series, or [Formula: see text] satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

Keywords

• Nonlinear elliptic equations
• existence
• symmetry
• composition operator
• modulation-Lorentz spaces