Electron. J. Differential Equations, Vol. 2017 (2017), No. 223, pp. 1-12.

Bifurcation analysis of elliptic equations described by nonhomogeneous differential operators

Habib Maagli, Ramzi Alsaedi, Noureddine Zeddini

In this article, we are concerned with a class of nonlinear partial differential elliptic equations with Dirichlet boundary data. The key feature of this paper consists in competition effects of two generalized differential operators, which extend the standard operators with variable exponent. This class of problems is motivated by phenomena arising in non-Newtonian fluids or image reconstruction, which deal with operators and nonlinearities with variable exponents. We establish an existence property in the framework of small perturbations of the reaction term with indefinite potential. The mathematical analysis developed in this paper is based on the theory of anisotropic function spaces. Our analysis combines variational arguments with energy estimates.

Submitted June 10, 2017. Published September 19, 2017.
Math Subject Classifications: 35J20, 35P30, 46E35.
Key Words: Variable exponent; nonhomogeneous differential operator; Ekeland variational principle; energy estimates.

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Habib Mâagli
Department of Mathematics, College of Sciences and Arts
King Abdulaziz University, Rabigh Campus
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: maaglihabib@gmail.com
Ramzi Alsaedi
Department of Mathematics
Faculty of Sciences, King Abdulaziz University
P.O. Box 80203, Jeddah 21589, Saudi Arabia
email: ramzialsaedi@yahoo.co.uk
Noureddine Zeddini
Department of Mathematics, Faculty of Sciences
Taibah University
Medina, Saudi Arabia
email: noureddinezeddini@yahoo.fr

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