Nonlinear Elliptic Equations with Nonlinear Boundary Conditions

Abstract

Publisher Summary This chapter focuses on nonlinear elliptic equations with nonlinear boundary conditions, and discusses mildly nonlinear elliptic boundary value problems (BVPs). For the study of the stability of the solutions of the parabolic initial-boundary value problem, one has to have a good knowledge of the steady states, that is, of the solutions of the elliptic BVP. The most interesting case occurs if the elliptic BVP has several distinct solutions. Some of the tools, namely the theory of increasing, completely continuous maps in ordered Banach spaces are then used to enlarge the domain of applicability of the general existence theorem by deriving simple sufficient criteria for the existence of sub- and supersolutions. In addition, a nonexistence and a general uniqueness theorem are derived. Also, to demonstrate the power of this abstract approach, a multiplicity result, namely a criterion guaranteeing the existence of atleast three distinct solutions, is derived. The chapter also discusses the main results for the nonlinear BVP, a fundamental a priori estimate for the solutions of the linear BVP, the equivalent fixed point equation, and multiplicity theorem.