Chapter 3 Superlinear elliptic equations and systems

Abstract

This chapter reviews some recent results on superlinear elliptic equations and systems. The chapter focuses on the borderline situations of so-called critical growth. The borderline situation (critical growth) may be defined as the limiting situation in which (Sobolev) space setup works. The chapter discusses various phenomena connected with critical growth. The chapter states that for the functionals associated to systems there is more freedom in the choice of the space, in fact it may choose among a whole continuum of products of Sobolev spaces. Each choice yields different maximal growths for the respective nonlinearities but for a fixed pair of such critical growth nonlinearities there exists a unique choice of a product Sobolev space. The pairs of critical growth nonlinearities form together the so-called critical hyperbola. The chapter focuses on some limiting cases of elliptic systems. Contrary to the situation in scalar equations and in (nonlimiting case) systems, a wide range of (Sobolev) spaces available in which the corresponding functionals may be defined, and the question of the right functional setup becomes quite delicate. In some limiting cases, the various possible choices of Sobolev spaces yield for the same functional, different maximal growths and the more refined Sobolev–Lorentz spaces provide an optimal functional setup.