Multiplicity results for a weighted p(x)–Laplacian equation

We prove the existence and multiplicity of nontrivial weak solutions for the following Steklov p(x)– Laplacian equation involving variable exponents:{ div(a(x)|Δu|p(x)-۲Δu) = b(x) lul p(x)-۲u, in Ω, a(x) lΔul p(x)-۲Δu du/dv = λf (x,u) on ∂Ω. Using the variational method and Krasnoselskii,s genus theory, we show the existence and multiplicity of weak solutions for the problem.

Keywords:

p(x)–Laplacian , variational method , genus theory , Sobolev space