Dirichlet problems for stationary von Neumann-Landau wave equations

In this article, we are concerned with the Dirichlet problem of the stationary von Neumann–Landau wave equation: {(-Δx+Δy)Φ(x,y)=0,   x,y∈Ω,Φ|ϑΩ×ϑΩ=f,{(-Δx+Δy)Φ(x,y)=0,   x,y∈Ω,Φ|ϑΩ×ϑΩ=f,where Ω is a bounded domain in Rn. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods.