JournalsaihpcVol. 11, No. 6pp. 693–703

A priori regularity for weak solutions of some nonlinear elliptic equations

  • Frank Pacard

    ENPC-CERGRENE, La Courtine, 93167 Noisy-le-Grand Cedex, France
A priori regularity for weak solutions of some nonlinear elliptic equations cover
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Abstract

Let f(u) be some positive regular function bounded from above by unn2u^{\frac{n}{n−2}}, in ℝ+. We derive some necessary and sufficient conditions, in order for all positive solutions to Δu=f(u)Lloc1(Rn)−\mathrm{\Delta }u = f\left(u\right) \in L_{\mathrm{loc}}^{1}\left(ℝ^{n}\right) to be regular.

Résumé

Soit f(u) une fonction positive, suffisamment régulière, bornée par unn2u^{\frac{n}{n−2}} sur ℝ+. On démontre qu’il existe un critère permettant de déterminer si toutes les solutions faibles positives de Δu=f(u)Lloc1(Rn)−\mathrm{\Delta }u = f\left(u\right) \in L_{\mathrm{loc}}^{1}\left(ℝ^{n}\right) sont régulières.

Cite this article

Frank Pacard, A priori regularity for weak solutions of some nonlinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 6, pp. 693–703

DOI 10.1016/S0294-1449(16)30174-3