# zbMATH — the first resource for mathematics

Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions. (English) Zbl 0940.35078
The authors prove a maximum (comparison) principle for second-order nonlinear divergence form elliptic equations with quadratic growth conditions. Recent results of F. Murat and the first author are extended to a more general situation allowing $$L^N$$-dependence in $$x$$ instead of $$L^\infty.$$

##### MSC:
 35J60 Nonlinear elliptic equations 35B50 Maximum principles in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text:
##### References:
 [1] G. Barles - F. Murat , The Maximum Principle for quasilinear elliptic equations with quadratic growth conditions , Arch. Rational Mech. Anal. 133 ( 1995 ), 77 - 101 . MR 1367357 | Zbl 0859.35031 · Zbl 0859.35031 · doi:10.1007/BF00375351 [2] A. Bensoussan - L. Boccardo - F. Murat , On a nonlinear partial differential equation having natural growth terms and unbounded solution , Ann. Inst. H. Poincaré, Anal. Non Linéaire 5 ( 1988 ), 347 - 364 . Numdam | MR 963104 | Zbl 0696.35042 · Zbl 0696.35042 · numdam:AIHPC_1988__5_4_347_0 · eudml:78157 [3] L. Boccardo - F. Murat - J.P. Puel , Existence de solutions non bornées pour certaines équations quasi-linéaires , Portugal. Math. 41 ( 1982 ), 507 - 534 . Article | MR 766873 | Zbl 0524.35041 · Zbl 0524.35041 · eudml:115517 [4] L. Boccardo - F. Murat - J.-P. Puel , Existence de solutions faibles pour des équations elleptiques quasi-linéaires à croissance quadratique , In: ”Nonlinear Partial Differential and their Applications”, Collège de France Seminar , volume IV , H. Brezis and J.-L. Lions editors, Research Notes in Mathematics 84 , Pitman , London , 1983 , pp. 19 - 73 . MR 716511 | Zbl 0588.35041 · Zbl 0588.35041 [5] V. Ferone - F. Murat , Nonlinear problems having natural growth in the gradient: an existence result when the source term is small , Nonlinear Anal ., to appear. MR 1780731 | Zbl 01529452 · Zbl 1158.35358 [6] V. Ferone - F. Murat , Quasilinear problems having natural growth in the gradient: an existence result when the source term is small , In: ” Equations aux dérivées partielles et applications, article dédié à Jacques-Louis Lions ”, Gauthier-Villars , Paris , 1998 , pp. 497 - 515 . MR 1648236 | Zbl 0917.35039 · Zbl 0917.35039 [7] V. Ferone - M.R. Posteraro , On a class of quasilinear elliptic equations with quadratic growth in the gradient , Nonlinear Anal. 20 ( 1993 ). MR 1214736 | Zbl 0801.35028 · Zbl 0801.35028 · doi:10.1016/0362-546X(93)90028-Q [8] D. Gilbarg - N.S. Trudinger , ” Elliptic Partial Differential Equations of Second Order ”, Springer-Verlag , Berlin , 1977 . MR 473443 | Zbl 0361.35003 · Zbl 0361.35003 [9] J.L. Kazdan - R.J. Kramer , Invariant criteria for existence of solutions of second order quasi-linear elliptic equations , Comm. Pure Appl. Math. 31 ( 1978 ), 619 - 645 . MR 477446 | Zbl 0368.35031 · Zbl 0368.35031 · doi:10.1002/cpa.3160310505 [10] O.A. Lady - N.N. Ural’ceva , ” Équations aux derivées partielles de type elliptique , Dunod , Paris , 1968 . MR 239273 | Zbl 0164.13001 · Zbl 0164.13001 [11] J.M. Lasry - P.L. Lions , A remark on regularization in Hilbert spaces , Isr. J. Math . 55 ( 1986 ), 257 - 266 . MR 876394 | Zbl 0631.49018 · Zbl 0631.49018 · doi:10.1007/BF02765025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.