Geodesic
Existence results for a class of nonlinear degenerate Navier problems
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 647-667.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we are interested in the existence of solutions for Navier problem associated with the degenerate nonlinear elliptic equations \begin{eqnarray*} {\Delta}{\big[}{\omega}_1(x) {\vert{\Delta}u\vert}^{p-2}{\Delta}u + {\omega}_2(x) {\vert{\Delta}u\vert}^{q-2}{\Delta}u {\big]} -\sum_{j=1}^n D_j{\bigl[}{\omega}_3(x) {\mathcal{A}}_j(x, u, {\nabla}u){\bigr]}\\ = f_0(x) - \sum_{j=1}^nD_jf_j(x), \ \ {\mathrm{in}} \ \ {\Omega} \end{eqnarray*} in the setting of the weighted Sobolev spaces.
Keywords: degenerate nonlinear elliptic equations, weighted Sobolev spaces. 
@article{SEMR_2021_18_1_a49,
     author = {A. C. Cavalheiro},
     title = {Existence results for a class of nonlinear degenerate {Navier} problems},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {647--667},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a49/}
}
TY  - JOUR
AU  - A. C. Cavalheiro
TI  - Existence results for a class of nonlinear degenerate Navier problems
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2021
SP  - 647
EP  - 667
VL  - 18
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a49/
LA  - en
ID  - SEMR_2021_18_1_a49
ER  - 
%0 Journal Article
%A A. C. Cavalheiro
%T Existence results for a class of nonlinear degenerate Navier problems
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2021
%P 647-667
%V 18
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a49/
%G en
%F SEMR_2021_18_1_a49
A. C. Cavalheiro. Existence results for a class of nonlinear degenerate Navier problems. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 647-667. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a49/

[1] L.-E. Anderson, T. Elfving, G.H. Golub, “Solution of biharmonic equations with application to radar imaging”, J. Comput. Appl. Math., 94:2 (1998), 153–180 | DOI | MR | Zbl

[2] D. Bresch, J. Lemoine, F. Guíllen-Gonzalez, “A note on a degenerate elliptic equations with applications for lakes and seas”, Electron. J. Differ. Equ., 2004, 42 | MR | Zbl

[3] A.C. Cavalheiro, “Existence and uniqueness of solutions for some degenerate nonlinear Dirichlet problems”, J. Appl. Anal., 19:1 (2013), 41–54 | DOI | MR | Zbl

[4] A.C. Cavalheiro, “Existence results for Dirichlet problems with degenerate p-Laplacian”, Opusc. Math., 33:3 (2013), 439–453 | DOI | MR | Zbl

[5] A.C. Cavalheiro, Topics on Degenerate Elliptic Equations, Lambert Academic Publishing, Germany, 2018

[6] M. Chipot, Elliptic Equations. An Introductory Course, Birkhäuser, Basel, 2009 | MR | Zbl

[7] M. Colombo, Flows of non-smooth vector fields and degenerate elliptic equations, With applications to the Vlasov-Poisson and semigeostrophic systems, Publications on the Scuola Normale Superiore Pisa, 22, Pisa, 2017 | MR | Zbl

[8] M. Colombo, A. Figalli, “Regularity results for very degenerate elliptic equations”, J. Math. Pures Appl., 101:1 (2014), 94–117 | DOI | MR | Zbl

[9] P. Drábek, A. Kufner, F. Nicolosi, Quasilinear elliptic equations with degenerations and singularities, Walter de Gruyter, Berlin, 1997 | MR | Zbl

[10] E. Fabes, C. Kenig, R. Serapioni, “The local regularity of solutions of degenerate elliptic equations”, Commun. Partial Differ. Equations, 7 (1982), 77–116 | DOI | MR | Zbl

[11] A. Kufner, O. John, S. Fuc̆ik, Function spaces, From the Czech, Noordhoff International Publ., Leyden, 1977 | MR | Zbl

[12] J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116, Amsterdam etc., 1985 | MR | Zbl

[13] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin etc, 1983 | MR | Zbl

[14] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Clarendon Press, Oxford, 1993 | MR | Zbl

[15] A.A. Kovalevsky, Y.S. Gorban, “Solvability of degenerate anisotropic elliptic second-order equations with $L^1$-data”, Electron.J. Differ. Equ., 2013 (2013), 167, 1–17 | DOI | MR | Zbl

[16] A. Kufner, Weighted Sobolev spaces, John Wiley Sons, Chichester etc, 1985 | MR | Zbl

[17] A. Kufner, B. Opic, “How to define reasonably weighted Sobolev spaces”, Commentat. Math. Univ. Carol., 25 (1984), 537–554 | MR | Zbl

[18] M.-C. Lai, H.-C. Liu, “Fast direct solver for the biharmonic equations on a disk and its application to incompressible flows”, Appl. Math. Comput., 164:3 (2005), 679–695 | MR | Zbl

[19] B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function”, Trans. Am. Math. Soc., 165 (1972), 207–226 | DOI | MR | Zbl

[20] J. Petean, “Degenerate solutions of a nonlinear elliptic equation on the sphere”, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 100 (2014), 23–29 | DOI | MR | Zbl

[21] M. Talbi, N. Tsouli, “On the spectrum of the weighted p-biharmonic operator with weight”, Mediterr. J. Math., 4:1 (2007), 73–86 | DOI | MR | Zbl

[22] A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, Orlando etc, 1986 | MR | Zbl

[23] B.O. Turesson, Nonlinear potential theory and weighted Sobolev spaces, Lecture Notes in Mathematics, 1736, Springer-Verlag, Berlin, 2000 | DOI | MR | Zbl

[24] E. Zeidler, Nonlinear functional analysis and its applications, v. I, Springer-Verlag, New York etc, 1990 | MR | Zbl

[25] E. Zeidler, Nonlinear functional analysis and its applications, v. II/B, Springer-Verlag, New York etc, 1990 | MR | Zbl