Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent

• Published in: Journal of mathematical analysis and applications Vol. 340; no. 1; pp. 687 - 698
• Main Authors: Mihăilescu, Mihai, Pucci, Patrizia, Rădulescu, Vicenţiu
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.04.2008; Elsevier
• Subjects: Anisotropic Sobolev spaces; Continuous spectrum; Exact sciences and technology; Functional analysis; Mathematical analysis; Mathematics; Nonhomogeneous differential operators; Operator theory; Ordinary differential equations; Partial differential equations; Sciences and techniques of general use; Variational methods; Nonhomogeneous differential operators; Anisotropic Sobolev spaces; Variational methods; Continuous spectrum; Growth rate; Sobolev space; Eigenvalue; Anisotropic equation; Functional; Elliptic equation; Mathematical analysis; Differential operator; Eigenvalue problem; Application; Non existence
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2007.09.015

In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term. One of the main features of our work is that the differential operator involves partial derivatives with different variable exponents, so that the functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations.