, Volume 84, Issue 1, pp 1-49

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 \(\begin{gathered} L_{\mathcal{A}^u } = - div\left[ {\mathcal{A}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^\sigma u^S v^R \\ L_{\mathcal{B}^u } = - div\left[ {\mathcal{B}\left( {x,u,\nabla u} \right)} \right] \geqslant \left| x \right|^b u^Q u^T \\ \end{gathered} \) . 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.