On nonlocal image-Laplacian equations Original Research Article

This paper deals with the nonlocal anisotropic View the MathML sourcep⃗(x)-Laplacian Dirichlet problems with non-variational form View the MathML source−∑i=1NAi(u)Di(|Diu|pi(x)−2Diu)=B(u)f(x,u(x))in Ω;u|∂Ω=0, Turn MathJax on and with variational form View the MathML source−∑i=1Nai(∫Ω|Diu|pi(x)pi(x)dx)Di(|Diu|pi(x)−2Diu)=b(∫ΩF(x,u)dx)f(x,u(x))in Ω;u|∂Ω=0, Turn MathJax on where View the MathML sourceF(x,t)=∫0tf(x,s)ds, and aiai is allowed to be singular at zero. Using (S+)(S+) mapping theory and the variational method, some results on existence and multiplicity for the problems are obtained.