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

Universidade Estadual de Campinas, IMECC-Departamento de Matemática, Rua Sérgio, Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil

E-mail Address: lcff@ime.unicamp.br

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and

Wender S. Lagoin

Universidade Estadual de Campinas, IMECC-Departamento de Matemática, Rua Sérgio, Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil

E-mail Address: wendertnb@gmail.com

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https://doi.org/10.1142/S1664360723500030Cited by:6 (Source: Crossref)

Abstract

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely ${(- Δ)}^{\frac{α}{2}} u = V (x) u + g (u, \nabla u) + f$ in $ℝ^{n}$ , where $0 < α < n$ , $g$ is the nonlinearity, $V$ the potential and $f$ is a forcing term. Some examples of nonlinearities dealt with are $u | u |^{ρ - 1}$ , $| \nabla u |^{ρ}$ and $| u |^{ρ_{1}} | \nabla u |^{ρ_{2}}$ , covering large values of $ρ, ρ_{1}, ρ_{2}$ , and particularly variational supercritical powers for $u$ and super- $α$ ones for $| \nabla u |$ (superquadratic if $α = 2$ ). 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, 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.

Communicated by Neil Trudinger

Keywords:

AMSC: 35J60, 35J61, 35J75, 42B35, 42B37

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Vol. 13, No. 03

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History

Received 1 September 2022

Revised 18 January 2023

Accepted 21 January 2023

Published: 24 February 2023

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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

Abstract

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