On a Picone's identity for the $\mathcal{A}_{p(x)}$-Laplacian and its applications

Document Type : Research Paper


Department of Mathematics‎, ‎Faculty of Basic Sciences‎, ‎Babol Noshirvani University of Technology‎, ‎Babol‎, ‎Iran.


‎We present a Picone's identity for the‎ ‎$\mathcal{A}_{p(x)}$-Laplacian‎, ‎which is an extension of the classic‎ ‎identity for the ordinary Laplace‎. ‎Also‎, ‎some applications of our‎ ‎results in Sobolev spaces with variable exponent are suggested.


Main Subjects

W. Allegretto and Y. X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Anal. 32 (1998), no.7, 819--830.
A. Alvino, V. Ferone and G. Trombetti, On the properties of some nonlinear eigenvalues,  SIAM J. Math. Anal. 29 (1998) 437--451.
K. Bal, Picone's identity and its applications, Electron. J. Differential Equations, 2012 (2012), no. 1, 9 pages.
M. Bocea and M. Mihailescu, Γ-convergence of power-law functionals with variable exponents, Nonlinear Anal. 73 (2010) 110--121.
M. Bocea, M. Mihailescu and C. Popovici, On the asymptotic behavior of variable exponent power-law functionals and applications, Ric. Mat. 59 (2010) 207--238.
Y. Chen, S. Levine and M. Rao, Variable exponent linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006) 1383--1406.
O. Dossly, The Picone identity for a class of partial differential equations, Math. Bohem. 127 (2002), no. 4, 581--589.
A. El Khalil, P. Lindqvist and A. Touzani, On the stability of the first eigenvalue of the problem: Apu + λg(x)jujp􀀀2u = 0 with varying p, Rend. Mat. 24 (2004) 321--336.
X.L. Fan, J.S. Shen and D. Zhao, Sobolev embedding theorems for spaces Wk;p(x)(Ω), J. Math. Anal. Appl. 262 (2001) 749--760.
X.L. Fan and D. Zhao, On the spaces Lp(x) and Wk;p(x), J. Math. Anal. Appl. 263 (2001) 424--446.
G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367 (2010) 204--228.
P. Harjulehto, P. Hasto, U.V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010) 4551--4574.
T. Kusano, J. Jaros and N. Yoshida, A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonlinear Anal. 40 (2000), no. 1-8, 381--395.
M. Mihailescu and D. Repovs, On a PDE involving the Ap(:)-Laplace Operator, Non-linear Anal. 75 (2012) 975--981.
M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare ordinaria del second'ordine, Ann. Scuola Norm. Sup. Pisa 11(1909) 1--141.
K.R. Rajagopal and M. Ruzicka, Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn. 13 (2001) 59--78.
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer-Verlag, Berlin, 2000.
J. Tyagi, A nonlinear picone's identity and its applications, Appl. Math. Lett. 26 (2013) 624--626.
N. Yoshida, Picone identities for half-linear elliptic operators with p(x)-Laplacians and applications to Sturmian comparison theory, Nonlinear Anal. 74 (2011), no. 16, 5631--5642.
N. Yoshida, Picone identity for quasilinear elliptic equations with p(x)-Laplacians and Sturmian comparison theory, Appl. Math. Comput. 225 (2013) 79--91.