Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2016)

  • Volume: 70, Issue: 2
  • ISSN: 0365-1029

Abstract

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In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations Δ ( v ( x ) | Δ u | p - 2 Δ u ) - ∑ j = 1 n D j [ ω 1 ( x ) 𝒜 j ( x , u , ∇ u ) ] + b ( x , u , ∇ u ) ω 2 ( x ) = f 0 ( x ) - ∑ j = 1 n D j f j ( x ) , in Ω in the setting of the weighted Sobolev spaces.

How to cite

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Albo Carlos Cavalheiro. "Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 70.2 (2016): null. <http://eudml.org/doc/289841>.

@article{AlboCarlosCavalheiro2016,
abstract = {In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations \begin\{align\} \{\Delta \}(v(x)\, \{\vert \{\Delta \}u\vert \}^\{p-2\}\{\Delta \}u) &-\sum \_\{j=1\}^n D\_j\{\bigl [\}\{\omega \}\_1(x) \mathcal \{A\}\_j(x, u, \{\nabla \}u)\{\bigr ]\}+ b(x,u,\{\nabla \}u)\, \{\omega \}\_2(x)\\ & = f\_0(x) - \sum \_\{j=1\}^nD\_jf\_j(x), \ \ \{\rm in \} \ \ \{\Omega \} \end\{align\} in the setting of the weighted Sobolev spaces.},
author = {Albo Carlos Cavalheiro},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Degenerate nonlinear elliptic equations; weighted Sobolev spaces},
language = {eng},
number = {2},
pages = {null},
title = {Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations},
url = {http://eudml.org/doc/289841},
volume = {70},
year = {2016},
}

TY - JOUR
AU - Albo Carlos Cavalheiro
TI - Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2016
VL - 70
IS - 2
SP - null
AB - In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations \begin{align} {\Delta }(v(x)\, {\vert {\Delta }u\vert }^{p-2}{\Delta }u) &-\sum _{j=1}^n D_j{\bigl [}{\omega }_1(x) \mathcal {A}_j(x, u, {\nabla }u){\bigr ]}+ b(x,u,{\nabla }u)\, {\omega }_2(x)\\ & = f_0(x) - \sum _{j=1}^nD_jf_j(x), \ \ {\rm in } \ \ {\Omega } \end{align} in the setting of the weighted Sobolev spaces.
LA - eng
KW - Degenerate nonlinear elliptic equations; weighted Sobolev spaces
UR - http://eudml.org/doc/289841
ER -

References

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  1. Cavalheiro, A. C., Existence and uniqueness of solutions for some degenerate nonlinear Dirichlet problems, J. Appl. Anal. 19 (2013), 41-54. 
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  8. Heinonen, J., Kilpelainen, T., Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Inc., New York, 1993. 
  9. Kufner, A., Weighted Sobolev Spaces, John Wiley & Sons, 1985. 
  10. Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. 
  11. Talbi, M., Tsouli, N., On the spectrum of the weighted p-Biharmonic operator with weight, Mediterr. J. Math. 4 (2007), 73-86. 
  12. Torchinsky, A., Real-Variable Methods in Harmonic Analysis, Academic Press, San Diego, 1986. 
  13. Turesson, B. O., Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 2000. 
  14. Zeidler, E., Nonlinear Functional Analysis and Its Applications. Vol. I, Springer-Verlag, New York, 1990. 
  15. Zeidler, E., Nonlinear Functional Analysis and Its Applications. Vol. II/B, Springer-Verlag, New York, 1990. 

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