Existence of bounded solutions for a class of quasilinear elliptic systems on manifolds with boundary

Abstract

We consider nonlinear elliptic partial differential equations for quasilinear operators of the formA(u)=−div(a(x,u,∇u))+A0(x,u,∇u),x∈Ω, subject to fully nonlinear boundary conditions involving boundary operators of the form, for each β⩾0,Bβ(u)=−βdivΓ(b(x,u,∇Γu))+B0(x,u,∇u,∇Γu),x∈∂Ω. The main goal of this paper is to give, under suitable assumptions on A and Bβ, an explicit L∞ estimate for bounded solutions of these elliptic boundary value problems. Then, we establish the existence of at least one solution to such problems extending the authorʼs previous work. Our methods rely on the definition of approximate problems, deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit. These methods can be applied to a large class of equations involving operators of Leray–Lions type (on suitable Banach spaces) for a general class of boundary operators Bβ which are, possibly, of the same order as A. As examples, these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions, and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea-level temperatures for relatively long time scales.