Browse > Article
http://dx.doi.org/10.22771/nfaa.2020.27.02.10

EXISTENCE AND MULTIPLICITY OF SOLUTIONS OF p(x)-TRIHARMONIC PROBLEM  

Belakhdar, Adnane (Laboratory Nonlinear Analysis, Department of Mathematics, Faculty of Science University Mohammed 1st)
Belaouidel, Hassan (Laboratory Nonlinear Analysis, Department of Mathematics, Faculty of Science University Mohammed 1st)
Filali, Mohammed (Laboratory Nonlinear Analysis, Department of Mathematics, Faculty of Science University Mohammed 1st)
Tsouli, Najib (Laboratory Nonlinear Analysis, Department of Mathematics, Faculty of Science University Mohammed 1st)
Publication Information
Nonlinear Functional Analysis and Applications / v.27, no.2, 2022 , pp. 349-361 More about this Journal
Abstract
In this paper, we study the following nonlinear problem: $$\{-\Delta_{p}^{3}(x)u\;=\;{\lambda}V_{1}(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u\;=\;{\Delta}u\;{\Delta}^{2}u\;=\;0\;on\;{\partial}\Omega, $$ under adequate conditions on the exponent functions p, q and the weight function V1. We prove the existence and nonexistence of eigenvalues for p(x)-triharmonic problem with Navier boundary value conditions on a bounded domain in ℝN. Our technique is based on variational approaches and the theory of variable exponent Lebesgue spaces.
Keywords
Eigenvalues; p(x)-triharmonic operator;
Citations & Related Records
연도 인용수 순위
  • Reference
1 V.V. Zhikov, On some variational problems, Russ. J. Math. Phys, 5 (1997), 105-116.
2 L. Diening, P. Hasto, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces,, Proc. Milovy, Czech Republic, (2004), 38-58.
3 A. El Amrouss and A. Ourraoui, Existence Of Solutions For A Boundary Problem Involving p(x)-Biharmonic Operator, Bol. Soc. Paran. Math., 31(1) (2013), 179-192 .   DOI
4 X.L. Fan and D. Zhao, On the space Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424-446.   DOI
5 O. Kavian, Introduction a la theorie des points critiques et Applications, Springer-Verlag, 13 (1993).
6 K.R. Rajagopal and M. Ruzicka, Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn., 13 (2001), 59-78.   DOI
7 M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin 1748 (2000).
8 D.E. Edmunds, J. Rakosnk, Sobolev embedding with variable exponent, Studia Math., 143(3) (2000), 267-293.   DOI
9 P. Harjulehto, P. Hasto, An overview of variable exponent Lebesgue and Sobolev spaces, in: D. Herron (Ed.), Future Trends in Geometric Function Theory, RNC Workshop, Jyvaskyla, (2003), 85-93.
10 A. Ayoujil, A.R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlinear Anal. T.M.A., 71(10) (2009), 4916-4926.   DOI
11 B. Ge, Q.M. Zhou and Y.H. Wu, Eigenvalues of the p(x)-biharmonic operator with indefinite weight, Zeitschrift fr angewandte Mathematik und Physik, 66(3) (2015), 1007-1021.   DOI
12 O. Kovacik, J.Rakosnik, On spaces Lp(x) and W1,p(x), Czechoslovak Math., J.41 (1991), 592-618.   DOI
13 M. Dammak, R. Jaidane and C. Jerb, Positive solutions for asymptotically linear biharmonic problems, Nonlinear Funct. Anal. Appl., 22(1) (2017), 59-76   DOI
14 A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent LebesgueSobolev spaces , Nonlinear Anal T.M.A., 69(10) (2008), 3629-3636.   DOI
15 A. Ayoujil, Existence and Nonexistence Results for Weighted Fourth Order Eigenvalue Problems With Variable Exponent, Bol. Soc. Paran. Mat,, 37(3) (2019), 55-66.   DOI