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- W4297668614 abstract "This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the $m$-Laplacian operator begin{equation*} -Delta_{m}u=u^q|nabla u|^p mathrm{in} mathbb{R}^N, end{equation*} where $Ngeq1$, $m>1$ and $p,qgeq0$. The technique of Bernstein gradient estimates is ultilized to study the case $p<m$. Moreover, a Liouville-type theorem for supersolutions under subcritial range of exponents begin{equation*} q(N-m)+p(N-1)<N(m-1) end{equation*} is also established. Then, we use a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem of the type $-Delta_m u = f(x,u,nabla u)$, with $f$ satisfying certain structure conditions. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem in the half-space. As another application, some new Harnack inequalities are proved." @default.
- W4297668614 created "2022-09-30" @default.
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- W4297668614 date "2020-08-17" @default.
- W4297668614 modified "2023-09-25" @default.
- W4297668614 title "Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms" @default.
- W4297668614 doi "https://doi.org/10.48550/arxiv.2008.07211" @default.
- W4297668614 hasPublicationYear "2020" @default.
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