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

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely (-Delta.) a2 u = V (x)u + g(u,del u) + f in Rn, where 0 < a < n, g is the nonlinearity, V the potential and f is a forcing term. Some examples of nonlinearities dealt with are u|u|.-1, |.u|. and |u|.1|.u|.2, covering large values of.,.1,.2, and particularly variational supercritical powers for u and super-a ones for |.u| (superquadratic if alpha = 2). Moreover, we are able to consider some exponential growths, g belonging to certain classes of power series, or g 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, H<spacing diaeresis>older-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.