Results of singular Direchelet problem involving the $p(x)$-laplacian with critical growth
Abstract
In this paper, we study the existence and multiplicity of solutions for Dirichlet singular elliptic problems involving
the $p(x)$-Laplace equation with critical growth. The technical approach is mainly based on the variational method combined with the genus theory.
Downloads
References
C. O. Alves ., Jose L. P. Barreiro , Existence and multiplicity of solutions for a p(x)-Laplacian equation with critical growth J. Math. Anal. Appl. 403 (2013) 143-154 https://doi.org/10.1016/j.jmaa.2013.02.025
S. Antontsev, S. Shmarev, Handbook of Differential Equations, Stationary Partial Differ. Equ. 3 (2006). Chap. 1 https://doi.org/10.1016/S1874-5733(06)80005-7
M. Bocea, M. Mihˇailescu, Γ-convergence of power-law functionals with variable exponents, Nonlinear Anal. 73 (2010) 110-121. https://doi.org/10.1016/j.na.2010.03.004
M. Bocea, M. Mihˇailescu, C. Popovici, On the asymptotic behavior of variable exponent power-law functionals and applications, Ric. Mat. 59 (2010) 207-238. https://doi.org/10.1007/s11587-010-0081-x
J. F. Bonder, A. Silva, Concentration-compactness principle for variable exponent spaces and applications, Electron. J. Differential Equations 2010, No. 141, 1-18. MR2729462
A. Callegari, A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic fuids, SIAM J. Appl. Math. 38 (1980) 275-281 https://doi.org/10.1137/0138024
Y. Chen, S. Levine, R. Rao, Variable exponent, linear growth functionals in image restoration, SIAMJ. Appl. Math. 66 (2006) 1383-1406. https://doi.org/10.1137/050624522
D. Edmunds, J. Rakosnık; Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267-293. MR1815935 (2001m:46072) https://doi.org/10.4064/sm-143-3-267-293
X. Fan, X. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , Nonlinear Anal. 59 (2004) 173-188. https://doi.org/10.1016/S0362-546X(04)00254-8
X. Fan, Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients ,J. Math. Anal. Appl. 312 (2005) 464-477. https://doi.org/10.1016/j.jmaa.2005.03.057
X. Fan, D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω) J. Math. Anal. Appl. 263 (2001) 424-446. https://doi.org/10.1006/jmaa.2000.7617
X. Fan, Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843-1852. https://doi.org/10.1016/S0362-546X(02)00150-5
A. Ferrero, F. Gazzola, Existence of Solutions for Singular Critical Growth Semilinear Elliptic Equations,Journal of Differential Equations 177, 494-522 (2001). https://doi.org/10.1006/jdeq.2000.3999
L. R. d. Freitas, Existencia e multiplicidade de soluc˜oes para uma classe de problemas quasilineares com crescimento crıtico exponencial, in: Tese de doutorado USP - S˜ao Carlos, 2010
Y. Fu, X. Zhang, A multiplicity result for p(x)-Laplacian problem in RN , Nonlinear Anal. 70 (2009) 2261-2269. https://doi.org/10.1016/j.na.2008.03.038
N. Ghoussoub, C. Yuan, Multiple Solutions For Quasi-Linear Pdes Involving The Critical Sobolev And Hardy Exponents, transactions of the american mathematical society, V. 352, Number 12, pp :5703-5743. https://doi.org/10.1090/S0002-9947-00-02560-5
Y. Li, Q. Guo, P. Niu, The existence of solutions for quasilinear elliptic problems with combined critical Sobolev-Hardy terms, J. Math. Anal. Appl. 388 (2012) 525-538. https://doi.org/10.1016/j.jmaa.2011.10.042
R. M. Khanghahia, A. Razania, Solutions for a Singular Elliptic Problem Involving the p(x)-Laplacian Published by Faculty of Sciences and Mathematics,University of Nis, Serbia, Available at: http://www.pmf.ni.ac.rs/filomat Filomat 32:14 (2018), 4841-4850 https://doi.org/10.2298/FIL1814841M
M. Massar, Existence of multiple solutions for a class of nonhomogeneous problems with critical growth, Electronic Journal of Qualitative Theory of Differential Equations 2017, No. 22, 1-18 https://doi.org/10.14232/ejqtde.2017.1.22
M. Mihailescu, On a class of nonlinear problems involving a p(x)-Laplace type operator, Czechoslovak Math. J. 58 (2008) 155--172. https://doi.org/10.1007/s10587-008-0011-1
Y. Mei, F. Yongqiang, L. Wang, Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent, Hindawi Publishing Corporation Abstract and Applied Analysis V. 2012, Article ID 894925, 17 pages. https://doi.org/10.1155/2012/894925
P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. In CBMS Regional Conference Series in Mathemat- ics, Vol. 65, American Mathematical Soceity, Providence (1986). https://doi.org/10.1090/cbms/065
D. Repovs; Stationary waves of Schr¨odinger-type equations with variable exponent, Anal. Appl., 13 (2015) ,645-661. https://doi.org/10.1142/S0219530514500420
A. Kristaly, V. Radulescu, C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, in: Encyclopedia of Mathematics and its Applications, vol. 136, Cambridge University Press, Cambridge, 2010.
W. Zhihui, W. Xinmin, A multiplicity result for quasilinear elliptic equations involving critical sobolev exponents, Nonlinear Anal. TMA 18 (1992) 559-567 https://doi.org/10.1016/0362-546X(92)90210-6
Copyright (c) 2022 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
