Singularités éliminables pour des équations semi-linéaires
Annales de l'Institut Fourier, Volume 34 (1984) no. 1, p. 185-206
Let L be an m th -order differential operator on an open subset Ω of R N , let K be a compact in Ω, γ>1 and γ =γ/(γ-1). We show that every solution of “Lu+u γ =0 on ΩK,u0” satisfies “Lu+u γ =0 on Ω” when the W m,γ -capacity of K is zero. This condition turns out to be necessary when L is a second-order elliptic operator. In the latter case, we also prove that, given μ a finite Radon measure on Ω, the equation ``Lu+u|u| γ-1 =μ,u| Ω =0 has a solution if and only if μ does not charge the sets of W 2,γ -capacity zero.
Étant donné L un opérateur différentiel d’ordre m sur un ouvert Ω de R N , K un compact de Ω, γ>1 et γ =γ/(γ-1), nous montrons que toute solution de “Lu+u γ =0 sur ΩK,u0” est solution de “Lu+u γ =0 sur Ω” dès que la W m,γ -capacité de K est nulle. Cette condition s’avère nécessaire quand L est un opérateur elliptique d’ordre 2. Dans ce cas, nous montrons aussi que ``Lu+u|u| γ-1 =μ,u| Ω =0 μ est une mesure de Radon bornée sur Ω, a une solution si et seulement si μ ne charge pas les ensembles de W 2,γ -capacité nulle.
@article{AIF_1984__34_1_185_0,
     author = {Baras, Pierre and Pierre, Michel},
     title = {Singularit\'es \'eliminables pour des \'equations semi-lin\'eaires},
     journal = {Annales de l'Institut Fourier},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {34},
     number = {1},
     year = {1984},
     pages = {185-206},
     doi = {10.5802/aif.956},
     mrnumber = {86j:35063},
     zbl = {0519.35002},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_1984__34_1_185_0}
}
Baras, Pierre; Pierre, Michel. Singularités éliminables pour des équations semi-linéaires. Annales de l'Institut Fourier, Volume 34 (1984) no. 1, pp. 185-206. doi : 10.5802/aif.956. https://aif.centre-mersenne.org/item/AIF_1984__34_1_185_0/

[1] D.R. Adams and J.C. Polking, The equivalence of two definitions of capacity, Proc. of A.M.S., 37 (1973), 529-534. | MR 48 #6451 | Zbl 0251.31005

[2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math, 12 (1959), 623-727. | MR 23 #A2610 | Zbl 0093.10401

[3] P. Baras et M. Pierre, Singularités éliminables d'équations elliptiques semi-linéaires, C.R.A.S., Paris, Série A, (1982). | MR 85a:35005 | Zbl 0517.35033

[4] Ph. Benilan et H. Brezis, Papier à paraître sur l'équation de Thomas-Fermi. Voir aussi H. BREZIS, Some Variational problems of the Thomas-Fermi type, in Variational Inequalities, Cottle, Gianessi, Lions éd., Reidel (1980). | Zbl 0643.35108

[5] H. Brezis et P.L. Lions, A note on isolated singularities for linear elliptic equations, Mathematical Analysis and Applications, Part. A. Volume dedicated to L. Schwartz, L. Nachbin éd., Acad. Press (1981), 263-266. | MR 83e:35039 | Zbl 0468.35036

[6] H. Brezis and W.A. Strauss, Semi-linear second-order elliptic equations in L1, J. of Math. Soc. Japan, 25 (1973), 565-590. | MR 49 #826 | Zbl 0278.35041

[7] H. Brezis and L. Veron, Removable singularities for some Nonlinear elliptic equations, Arch. for Rat. Mech. and Ana., 75 (1980), 1-6. | MR 83i:35071 | Zbl 0459.35032

[8] D. Feyel et A. De La Pradelle, Topologies fines et compactifications associées à certains espaces de Dirichlet, Ann. Inst. Fourier, Grenoble, 27-4 (1977), 121-146. | Numdam | MR 58 #22625 | Zbl 0357.31009

[9] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, 224 (1977). | MR 57 #13109 | Zbl 0361.35003

[10] M. Grun-Rehomme, Caractérisation du sous-différentiel d'intégrandes convexes dans les espaces de Sobolev, J. Math. Pures et Appl., 56 (1977), 149-156. | Zbl 0314.35001

[11] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. | MR 48 #12155 | Zbl 0246.35025

[12] A.E. Koselev, A priori estimates in Lp and generalized solutions of elliptic equations and systems, A.M.S. Transl. series, 2, 2D (1962), 105-171. | Zbl 0122.33702

[13] P.L. Lions, Isolated singularities in semilinear problems, J. of Diff. Equa., 38 (1980), 441-450. | MR 82g:35040 | Zbl 0458.35033

[14] N.G. Meyers, A theory of capacities for potentials of functions in Lebesgue Classes, Math. Scand., 26 (1970), 255-292. | MR 43 #3474 | Zbl 0242.31006

[15] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. | Numdam | MR 22 #823 | Zbl 0088.07601

[16] L. Veron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. Th., Meth. Appl., Vol. 5, n° 3 (1981), 225-242. | MR 82f:35076 | Zbl 0457.35031

[17] L. Veron, Singularités éliminables d'équations elliptiques non linéaires, J. of Diff. Equa., 41 (1981), 87-95. | MR 82k:35042 | Zbl 0431.35005