Existence and regularity of positive solutions of a degenerate fourth order elliptic problem
DOI:
https://doi.org/10.12775/TMNA.2021.019Keywords
Embedding theorem, fourth order operator equations, degenerate elliptic problemAbstract
In this paper, we consider existence and regularity of positive solutions of a degenerate fourth order elliptic problem. Firstly, a new Caffarelli-Kohn-Nirenberg type inequality for the fourth order case is established. Then, by the use of the corresponding embedding, we obtain the existence of positive solutions of a degenerate fourth order elliptic problem. Finally, the regularity of the positive solutions is also studied.References
G. Afrouzi, A. Hadjian and N. Zographopoulos, Multiple nonsemitrivial solutions for a class of degenerate quasilinear elliptic systems, Topol. Methods Nonlinear Anal. 45 (2015), 385–397.
A. Alvino and G. Trombetti, A lower bound for the first eigenvalue of an elliptic operator, J. Math. Anal. Appl. 94 (1983), 328–337.
A. Arapostathis, M. Ghosh and S. Marcus, Harnack’s inequality for cooperative weakly coupled elliptic systems, Comm. Partial Differential Equations 24 (1999), 1555–1571.
G. Arioli, F. Gazzola, H. Grunan and E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36 (2005), 1226–1258.
E. Berchio, A. Farina, A. Ferrero and F. Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem, J. Differential Equations 252 (2012), 2596–2612.
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), 259–275.
P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl. 7 (2000), 187–199.
F. Catrina and Z. Wang, On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), 229–258.
H. Chen, M. Fall and B. Zhang, On isolated singularities of Kirchhoff equations, Adv. Nonlinear Anal. 10 (2021), 102–120.
D. Figueiredo, E. Santos and O. Miyagaki, Sobolev spaces of symmetric functions and applications, J. Funct. Anal. 261 (2011), 3735–3770.
X. Guan and L. Mei, On a weighted Lane–Emden system, Appl. Anal. 99 (2020), 814–839.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1998.
H. Guo, Z. Guo and F. Wan, Radial symmetry of non-maximal entire solutions of a biharmonic equation with exponential nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), 1603–1625.
Z. Guo, Further study of entire radial solutions of a biharmonic equation with exponential nonlinearity, Ann. Mat. Pura Appl. 193 (2014), 187–201.
Z. Guo, X. Guan and F. Wan, Sobolev type embedding and weak solutions with a prescribed singular set, Sci. China Math. 59 (2016), 1975–1994.
Z. Guo, X. Guan and F. Wan, Existence and regularity of positive solutions of a degenerate elliptic problem, Math. Nachr. 292 (2019), 56–78.
Z. Guo, X. Huang and F. Zhou, Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity, J. Funct. Anal. 268 (2015), 1972–2004.
Z. Guo, L. Mei, F. Wan and X. Guan, Embeddings of weighted Sobolev spaces and degenerate elliptic problems, Sci. China Math. 60 (2017), 1399–1418.
Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math. 60 (2017) 2391–2406.
Z. Guo, F. Wan and L. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equation, Commun. Contemp. Math. 22 (2020), 1950057.
Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst. 34 (2014), 2561–2580.
Z. Guo and L. Wei, Radial symmetry of entire solutions of a biharmonic equation with supercritical exponent, Adv. Nonlinear Stud. 19 (2019), 291–316.
Z. Guo, J. Wei and W. Yang, On nonradial singular solutions of supercritical biharmonic equations, Pacific J. Math. 284 (2016), 395–430.
P. Gurka and B. Opic, Continuous and compact imbeddings of weighted Sobolev spaces I, Czechoslovak Math J. 38 (1988), 611–617.
J. Kim, Existence of solutions for the semilinear corner degenerate elliptic equations, Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 585–597.
J. Su and Z. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differential Equations 250 (2011), 223–242.
J. Su, Z. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations 238 (2007), 201–219.
X. Wang and J. Zhang, Some remarks on embedding of weighted Sobolev spaces, J. Math. Anal. Appl. 475 (2019), 1276–1292.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Xiaohuan Wang, Jihui Zhang
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
