Existence of Solutions for Nonhomogeneous A-Harmonic Equations with Variable Growth

Abstract

We study the following nonhomogeneous $A$-harmonic equations: $d^{*}A(x,du(x))+B(x,u(x))=0,x\in \mathrm{\Omega },u(x)=0,x\in \partial \mathrm{\Omega }$, where $\mathrm{\Omega }\subset {ℝ}^n$ is a bounded and convex Lipschitz domain, $A(x,du(x))$ and $B(x,u(x))$ satisfy some $p(x)$-growth conditions, respectively. We obtain the existence of weak solutions for the above equations in subspace ${𝔎}_0^{1,p(x)}(\mathrm{\Omega },{\mathrm{\Lambda }}^{l-\mathrm{1}})$ of $W_0^{1,p(x)}(\mathrm{\Omega },{\mathrm{\Lambda }}^{l-\mathrm{1}})$.