Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions. (27th June 2017) - Gal, Ciprian G.; Warma, Mahamadi
Abstract : Given a bounded domain Ω ⊂ R N with a Lipschitz boundary ∂ Ω and p, q ∈ ( 1, + ∞ ), we consider the quasilinear elliptic equation - Δ p u + α 1 u = f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form b x ∇ u p - 2 ∂ n u - ρ b x Δ q, Γ u + α 2 u = g on ∂ Ω . In the first part of the article, we give necessary and sufficient conditions in terms of the given functions f, g and the nonlinearities α 1, α 2, for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of "nonlinear Fredholm alternative" for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an L ∞ -setting.