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Farman Mamedov

Publications and source records attributed to Farman Mamedov.

3 recordsLinked to original sources
arXiv2024

On Harnack inequality to the homogeneous nonlinear degenerate parabolic equations

In this paper, the Harnack inequality result are established for a new class of the homogeneous nonlinear degenerate parabolic equations \begin{align*} div A(t,x,u,\nabla_x u)-\partial_t \vert u\vert^{p-2}u=0 \end{align*} on a bounded domain $ D \subset R^{n+1}. $ Let $A(t,x,ξ,η)$ be measurable function on $R\times R^n\times R\times R^n\to R^n$ that satisfies the Caratheodory conditions for $ \, \text{arbitrary } \, (t,x)\in D$ and $(ξ,η)\in R^{1}\times R^n.$ The following growth conditions are also satisfied: \begin{equation*} A(t,x,ξ,η)η\geq c_{1}ω(t,x)\vertη\vert^{p} \end{equation*} \begin{equation*} \vert A(t,x,ξ,η)\vert\leq c_{2}ω(t,x)\vertη\vert^{p-1},\quad p>1. \end{equation*} The exclusive Muckenhoupt condition $ ω^α \in A_{1+α/r} . $

math.AP↗
arXiv2024

On existence of two positive solutions for the nonlinear subelliptic equations involving nonuniformly p-Laplacian

In this paper, we study a solvability result for the nonlinear problem $$ \mbox {div } \left ( \vert \nabla_ωu\vert^{p-2}\nabla_ωu \right )+v(x) u^{q-1}+μu^{γ-1}=0, \quad z\in Ω, \quad u \Big \vert_{\partial Ω}=0. $$ assuming for the weight functions $ v \in A_\infty, \, ω\in A_p $ to belong the Muckenhoupt class and a balance condition of Chanillo-Wheeden's type, with degenerate gradient $\nabla_ωu =\left ( ω^{1/p} \nabla_x, \, \nabla_y \right ) $ and its module $ \vert \nabla_ωu\vert= \left (ω(x)^{2/p} \vert \nabla_{x}u \vert ^2+\vert \nabla_{y}u\vert^2 \right )^{\frac{1}{2}}; $ the domain $ Ω\subset \mathbb{R}^N $ is bounded, $ N=n+m, x\in \mathbb{R}^n, \, y\in \mathbb{R}^m$ and $z=(x, y) \in \mathbb{R}^N.$ The range conditions $ q \in (p, pN/(N-p)) $ and $ γ\in \left (1, N/(N-1)\right ) $ (or $γ\in (1, p)$ and $v^{-γ/(q-γ)}\in L_{1,loc}(Ω)$ additionally) and $ μ\in (0, Λ) $ with sufficiently small $ Λ$ are assumed also.

math.AP↗