Existence of weak solutions for nonlinear elliptic systems involving the (p(x), q(x))-Laplacian
Abstract
Description
In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -Δ_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{α(x)}|v|^{β(x)} v + f(x) in Ω, Δ_{q(x)}v = c(x) |v|^{q(x)-2}v - d(x)|v|^{β(x)}|u|^{α(x)} u + g(x) in Ω, u = v = 0 \quad on \partialΩ, where $Ω$ is an open bounded domains of $\mathbb{R}^N$ with a smooth boundary $\partialΩ$ and $Δ_{p(x)}$ denotes the $p(x)$-Laplacian.The existence of weak solutions is proved using the theory of monotone operators. Similar result will be proved when $Ω=\mathbb{R}^N$.
15 pages
15 pages