Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Farman Mamedov, Jasarat Gasimov

Abstract

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.

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

Abstract

In this paper, we study a solvability result for the nonlinear problem assuming for the weight functions to belong the Muckenhoupt class and a balance condition of Chanillo-Wheeden's type, with degenerate gradient and its module the domain is bounded, and The range conditions and (or and additionally) and with sufficiently small are assumed also.
Paper Structure (3 sections, 1 theorem, 119 equations)

This paper contains 3 sections, 1 theorem, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Corollary 6

. Let $\Omega \subset R^{N}$ be a bounded domain, let $1<\gamma <N/(N-1),$ let the function $\omega$ of the variable x belongs to the class $A_{p}$ in $R^{n}$, let $\mu \in (0,\Lambda),$ where $\Lambda$ is small enough, and let the following assumptions holds: Then the following problem: for $\alpha >0$ has at least two different positive solutions from the space $\mathring { \mathcal{W}}_{p}^1

Theorems & Definitions (1)

  • Corollary 6