×

Singular solutions of the \(p\)-Laplace equation. (English) Zbl 0592.35031

This paper deals with the study of isolated singularities of solutions of the \(p\)-Laplace equation \[ \mathop{div}(| \nabla u|^{p-2} \nabla u)=0 \tag{1} \] and extends some previous well known results of J. Serrin [Acta. Math. 111, 247–302 (1964; Zbl 0128.09101), 113, 219–240 (1965; Zbl 0173.39202)].
Correction see Zbl 0653.35018.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J30 Higher-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Dibenedetto, E.:C 1+? local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal., Theory Methods Appl.7, 827-850 (1983) · Zbl 0539.35027
[2] Bojarski, B., Iwaniec, T.:p-Harmonic equation and quasilinear mappings. Sonderforschungsbereich 72, Univ. Bonn (1983) · Zbl 0548.30016
[3] Chrusciel, P.: Sur l’existence de solutions singulières d’une équation “condition de coordonnées{” utilisée par J. Kijowski dans l’analyse symplectique de la relativité générale. C. R. Acad. Sci. Paris299I, 891-894 (1984)}
[4] Chrusciel, P.: Conformally minimal foliations of three dimensional Riemannian manifolds and the energy of the gravitational field. Ph.D. Thesis, Inst. Th. Phys. Polish Acad. Sci. (1986)
[5] Dobrowolski, M.: Nichtlineare Eckenprobleme und finite Elementenmethode. Z. Angew. Math. Mech.64, 270-271 (1984) · Zbl 0578.35026
[6] Evans, L.C.: A new proof of localC 1,? regularity for solutions of certain degenerate elliptic P.D.E.. J. Differ. Equations50, 315-338 (1982)
[7] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1982 · Zbl 1042.35002
[8] Kichenassamy, S., Veron, L.: Singularités isolées de l’équation 614-1. C.R. Acad. Sci. Paris301, 149-151 (1985)
[9] Krol, I.N.: On the behaviour of the solutions of a quasilinear equation near null salient points of the boundary. Proc. Steklov Inst. Math.125, 130-136 (1973) · Zbl 0306.35047
[10] Krol, I.N., Mazja, V.G.: The lack of continuity and Hölder continuity of the solutions of a certain quasilinear equation Sem. Math. V. A. Steklov Math. Inst. Leningrad14, 44-45 (1969)
[11] Ladyzhenskaya, O.A., Ural’Ceva, N.N.: Linear and quasilinear elliptic equations. London, New York: Academic Press 1968 · Zbl 0164.13002
[12] Lewis, J.L.: Smoothness of certain degenerate elliptic equations. Proc. AMS80, 259-265 (1980) · Zbl 0455.35064
[13] Lewis, J.L.: Regularity of the derivatives of solutions to certain degenerate elliptic equations. Indiana Univ. Math. J.32, 849-858 (1983) · Zbl 0554.35048
[14] Lions, J.L.: Quelques méthodes de résolutions des problèmes aux limites non linéaires. Paris: Dunod-Gauthier-Villars 1969
[15] Reshetniak, Y.G.: Mappings with bounded deformation as extremals of Dirichlet type integrals. Sibirsk. Mat. Zh.9, 652-666 (1966)
[16] Serrin, J.: Local behaviour of solutions of quasilinear equations. Acta Math.111, 247-302 (1964) · Zbl 0128.09101
[17] Serrin, J.: Isoled singularities of solutions of quasilinear equations. Acta Math.113, 219-240 (1965) · Zbl 0173.39202
[18] Serrin, J.: Singularities of solutions of nonlinear equations. Proc. Symp. Appl. Math.17, 68-88 (1965) · Zbl 0149.30701
[19] Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equations51, 126-150 (1984) · Zbl 0522.35018
[20] Tolksdorf, P.: On the Dirichlet problem for quasilinear equations in domain with conical boundary points. Commun. Partial Differ. Equations8, 773-817 (1983) · Zbl 0515.35024
[21] Uhlenbeck, K.: Regularity for a class of nonlinear elliptic systems. Acta Math.138, 219-240 (1977) · Zbl 0372.35030
[22] Ural’Ceva, N.N.: Degenerate quasilinear systems. Sem. Math. V.A. Steklov, Math. Inst. Leningrad7, 83-99 (1968)
[23] Véron, L.: Anisotropic singularities of degenerate eliptic equations (in preparation)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.