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Nonexistence results and estimates for some nonlinear elliptic problems

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Abstract

Here we study the local or global behaviour of the solutions of elliptic inequalities involving quasilinear operators of the type\(L_{\mathcal{A}^u } = - div\left[ {\mathcal{A}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^\sigma u^Q \) or\(\begin{gathered} L_{\mathcal{A}^u } = - div\left[ {\mathcal{A}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^\sigma u^S v^R \hfill \\ L_{\mathcal{B}^u } = - div\left[ {\mathcal{B}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^b u^Q u^T \hfill \\ \end{gathered} \). We give integral estimates and nonexistence results. They depend on properties of the supersolutions of the equationsL A u=0,L B v=0, which suppose weak coercivity conditions. Under stronger conditions, we give pointwise estimates in case of equalities, using Harnack properties.

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Authors and Affiliations

  1. CNRS UPRES-A 6083 Faculté des Sciences, Laboratoire de Mathématiques et Physique Théorique, Parc Grandmont, 37200, Tours, France

    Marie-Francoise Bidaut-Véron

  2. Steklov Mathematical Institute, Gubkina STR., 8, 117966, Moscow, Russia

    Stanislav Pohozaev

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  1. Marie-Francoise Bidaut-Véron
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  2. Stanislav Pohozaev
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Bidaut-Véron, MF., Pohozaev, S. Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math. 84, 1–49 (2001). https://doi.org/10.1007/BF02788105

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  • DOI: https://doi.org/10.1007/BF02788105

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