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Abstract <p> We study the existence of global positive solutions of the differential inequalities <Equation ID="Equi"> <EquationSource Format="TEX">\(-\operatorname{div}A(x,u,\nabla u) \ge f(u)\qquad \text{in}\quad \mathbb R^n,\)</EquationSource> </Equation> where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 2\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(A\)</EquationSource> </InlineEquation> is a Carathéodory function such that <Equation ID="Equii"> <EquationSource Format="TEX">\(\begin{gathered} \, \bigl(A(x,s,\zeta)-A (x,s,\xi)\bigr)(\zeta-\xi)\ge 0, \\ C_1|\xi|^p \le\xi A(x,s,\xi),\qquad |A(x,s,\xi)| \le C_2|\xi|^{p-1},\qquad C_1,C_2&gt;0,\quad p&gt;1, \end{gathered}\)</EquationSource> </Equation> for almost all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(x\in\mathbb R^n\)</EquationSource> </InlineEquation> and all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(s\in\mathbb R\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\zeta,\xi\in\mathbb R^n\)</EquationSource> </InlineEquation>. </p>

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On Existence of Global Solutions of Second-Order Quasilinear Elliptic Inequalities

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • A. A. Kon’kov,
  • M. D. Surnachev,
  • A. E. Shishkov

摘要

Abstract

We study the existence of global positive solutions of the differential inequalities \(-\operatorname{div}A(x,u,\nabla u) \ge f(u)\qquad \text{in}\quad \mathbb R^n,\) where \(n\ge 2\) and \(A\) is a Carathéodory function such that \(\begin{gathered} \, \bigl(A(x,s,\zeta)-A (x,s,\xi)\bigr)(\zeta-\xi)\ge 0, \\ C_1|\xi|^p \le\xi A(x,s,\xi),\qquad |A(x,s,\xi)| \le C_2|\xi|^{p-1},\qquad C_1,C_2>0,\quad p>1, \end{gathered}\) for almost all \(x\in\mathbb R^n\) and all \(s\in\mathbb R\) and \(\zeta,\xi\in\mathbb R^n\) .