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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)
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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
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