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Degenerate elliptic equations with measure data and nonlinear potentials. (English) Zbl 0797.35052
The problem \(-\text{div} A (x,\nabla u) = \mu\), where \(\mu\) is a nonnegative Radon measure and \(A(x,\nabla u) \cdot \nabla u \approx | \nabla u |^ P\), is studied in the class of \(A\)-superharmonic functions.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35D05 Existence of generalized solutions of PDE (MSC2000)
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References:
[1] L. Boccardo - T. Gallouët , Non-linear elliptic and parabolic equations involving measure data , J. Funct. Anal. 87 ( 1989 ), 149 - 169 . MR 1025884 | Zbl 0707.35060 · Zbl 0707.35060
[2] R. Gariepy - W.P. Ziemer , A regularity condition at the boundary for solutions of quasilinear elliptic equations , Arch. Rational Mech. Anal. 67 ( 1977 ), 25 - 39 . MR 492836 | Zbl 0389.35023 · Zbl 0389.35023
[3] L.I. Hedberg - Th H. Wolff , Thin sets in nonlinear potential theory , Ann. Inst. Fourier , Grenoble 33 , 4 ( 1983 ), 161 - 187 . Numdam | MR 727526 | Zbl 0508.31008 · Zbl 0508.31008
[4] J. Heinonen - T. Kilpeläinen , A-superharmonic functions and supersolutions of degenerate elliptic equations , Ark. Mat. 26 ( 1988 ), 87 - 105 . MR 948282 | Zbl 0652.31006 · Zbl 0652.31006
[5] J. Heinonen - T. Kilpeläinen , Polar sets for supersolutions of degenerate elliptic equations , Math. Scand. 63 ( 1988 ), 136 - 150 . MR 994974 | Zbl 0706.31015 · Zbl 0706.31015
[6] J. Heinonen - T. Kilpeläinen , On the Wiener criterion and quasilinear obstacle problems , Trans. Amer. Math. Soc. 310 ( 1988 ), 239 - 255 . MR 965751 | Zbl 0711.35052 · Zbl 0711.35052
[7] J. Heinonen - T. Kilpeläinen - J. Malý , Connectedness in fine topologies , Ann. Acad. Sci. Fenn. Ser. A I Math. 15 ( 1990 ); 107 - 123 . MR 1050785 | Zbl 0715.31005 · Zbl 0715.31005
[8] J. Heinonen - T. Kilpeläinen - O. Martio , Fine topology and quasilinear elliptic equations , Ann. Inst. Fourier , Grenoble 39 , 2 ( 1989 ), 293 - 318 . Numdam | MR 1017281 | Zbl 0659.35038 · Zbl 0659.35038
[9] J. Heinonen - T. Kilpeläinen - O. Martio , Nonlinear potential theory of degenerate elliptic equations , Oxford University Press (in press). MR 1207810 · Zbl 0776.31007
[10] T. Kilpeläinen , Potential theory for supersolutions degenerate elliptic equations , Indiana Univ. Math. J. 38 ( 1989 ), 253 - 275 . MR 997383 | Zbl 0688.31005 · Zbl 0688.31005
[11] P. Lindqvist , On the definition and properties of p-superharmonic functions , J. Reine Angew. Math. 365 ( 1986 ), 67 - 79 . MR 826152 | Zbl 0572.31004 · Zbl 0572.31004
[12] P. Lindqvist - O. Martio , Two theorems of N. Wiener for solutions of quasilinear elliptic equations , Acta Math. 155 ( 1985 ), 153 - 171 . MR 806413 | Zbl 0607.35042 · Zbl 0607.35042
[13] V.G. Maz’ya , On the continuity at a boundary point of solutions of quasi-linear elliptic equations , Vestnik Leningrad. Univ. Mat. Mekh. Astronom . 25 ( 1970 ), 42 - 55 (Russian), Vestnik Leningrad Univ. Math. 3 , ( 1976 ), 225 - 242 . (English translation). MR 274948 | Zbl 0252.35024 · Zbl 0252.35024
[14] J.M. Rakotoson , Equivalence between the growth of f |\nabla u| pdy and T in the equation P[u]=T , J. Differential Equation 86 ( 1990 ), 102 - 122 . Zbl 0707.35033 · Zbl 0707.35033
[15] J.M. Rakotoson - W.P. Ziemer , Local behavior of solutions of quasilinear elliptic equations with general structure , Trans. Amer. Math. Soc. 319 ( 1990 ), 747 - 764 . MR 998128 | Zbl 0708.35023 · Zbl 0708.35023
[16] J. Serrin , Pathological solutions of elliptic differential equations , Ann. Scuola Norm. Sup. Pisa Cl. Sci . ( 1964 ), 385 - 387 . Numdam | MR 170094 | Zbl 0142.37601 · Zbl 0142.37601
[17] N.S. Trudinger , On Harnack type inequalities and their application to quasilinear elliptic equations , Comm. Pure Appl. Math. 20 ( 1967 ), 721 - 747 . MR 226198 | Zbl 0153.42703 · Zbl 0153.42703
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