Generalized anisotropic elliptic Wentzell problems with nonstandard growth conditions

Abstract

Let Ω⊆RN be a bounded Lipschitz domain with the W1,p→(⋅)-extension property, for N≥3. We investigate the solvability and global regularity of a class of quasi-linear elliptic equations involving the anisotropic p→(⋅)-Laplace operator Δp→(⋅), with nonhomogeneous anisotropic Wentzell boundary conditions ∑i=1N|∇u|pi(⋅)−2∂u∂xiνi−Δq→(⋅),Γu+β|u|qM(⋅)−2u∋0onΓ≔∂Ω,for β∈L∞(Γ)+ with a positive lower bound, where p→∈C0,1(Ω¯)N and q→∈C0,1(Γ)N−1 fulfill 1<pm−≤pM+<∞, and 1<qm−≤qM+<∞. Here Δq→(⋅),Γ denotes the anisotropic q→(⋅)-Laplace–Beltrami operator. We first show existence and uniqueness of weak solutions for the elliptic problem, and moreover, we prove that such solutions are globally bounded over Ω¯. Key L∞-type a priori estimates for the difference of weak solutions are provided, as well as Maximum principles and inverse positivity results. At the end, we establish a sort of “nonlinear anisotropic Fredholm alternative” for the corresponding anisotropic Wentzell problem of nonstandard structure.