Existence results for Navier problems with degenerated (p,q)-Laplacian and (p,q)-Biharmonic operators

Abstract

In this article, we prove the existence and uniqueness of solutions for the Navier problem
(P)



∆

ω(x)(|∆u|
p−2∆u + |∆u|
q−2∆u)

− div
ω(x)(|∇u|
p−2∇u + |∇u|
q−2∇u)

= f(x) − div(G(x)), in Ω,
u(x) = ∆u = 0, in ∂Ω,
where Ω is a bounded open set of R
N (N ≥ 2), f
ω
∈L
p
0
(Ω, ω) and G
ω
∈ [L
q
0
(Ω, ω)]N .

Keywords

• Degenerate nonlinear elliptic equations
• weighted Sobolev space
• 2010 MSC: 35J60, 35J70

References

1. {1} A.C.Cavalheiro, Existence and uniqueness ofsolutions for some degenerate nonlinear Dirichlet problems},Journal of Applied Analysis, 19 (2013), 41-54.
2. {2} M. Chipot, Elliptic Equations: An IntroductoryCourse, Birkh\"auser, Berlin (2009).
3. {3} P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations andSingularities, Walter de Gruyter, Berlin (1997).
4. {4} E. Fabes, C. Kenig, R. Serapioni, The localregularity of solutions of degenerate elliptic equations, Comm.PDEs 7 (1982), 77-116.
5. {5} J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-HollandMathematics Studies 116, (1985).
6. {6} D.Gilbarg and N.S. Trudinger, Elliptic PartialEquations of Second Order, 2nd Ed., Springer, New York (1983).
7. {7} J. Heinonen, T. Kilpelainen and O. Martio,\textit{Nonlinear Potential Theory of Degenerate EllipticEquations, Oxford Math. Monographs, Clarendon Press, (1993).
8. {8} B. Muckenhoupt, Weighted norm inequalities for theHardy maximal function, Trans. Am. Math. Soc. 165 (1972),207-226.

9. {9} E. Stein, Harmonic Analysis, PrincentonUniversity Press, New Jersey (1993).
10. {10} A. Torchinsky, Real-Variable Methods in HarmonicAnalysis, Academic Press, San Diego, (1986).
11. {11} B.O. Turesson, Nonlinear Potential Theory andWeighted Sobolev Spaces, Lecture Notes in Mathematics, vol. 1736,Springer-Verlag, (2000).
12. {12} E. Zeidler, Nonlinear Functional Analysis andits Applications, Vol.II/B, Springer-Verlag, New York (1990).