Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions

Abstract

We study a boundary value problem of the type in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in (N≥ 3) with smooth boundary and the functions are of the type with , (i = 1, …, N). Combining the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle we show that under suitable conditions the problem has two non-trivial weak solutions.

Keywords:

• non-homogeneous differential operator
• anisotropic Sobolev spaces
• mountain pass theorem
• Ekeland's variational principle

AMS Subject Classifications:

• 35D05
• 35J60
• 35J70
• 58E05

Acknowledgements

M. Mihăilescu has been supported by Grant CNCSIS PNII–79/2007 ‘Degenerate and Singular Nonlinear Processes’.