Semilinear elliptic equations with critical nonlinearities

Abstract

where A is the Laplacian operator and A is a real parameter. Different domains of definition have invoked interest and various boundary conditions have been considered. In this thesis, the unbounded domain R is analysed. Solutions are sought in the Sobolev space D'(R). In the thesis, issues concerning existence are investigated, and this class of problems acts as a test bed for new methods. The majority of the thesis is concerned with semilinear equations, denoted so because nonlinearities are confined to functions of u in the /(x,u) term. In the final chapter, a quasilinear equation is investigated, which involves an inherently nonlinear operator, the 7V-Laplacian. The primary analysis technique corresponds with the modern paradigm for problems in a variational setting. Once the Euler-Lagrange equations have been converted to a critical point problem concerning the C functional / , the following two step method is applied: (1) develop a Palais-Smale sequence (/(un) —)• c and I'(un) -* 0) from the geometry of the functional and (2) from the topology of level sets, confirm that Palais-Smale sequences at the appropriate levels converge. This second step is termed the Palais-Smale condition at a level c, and initially appears restrictive. The pioneering work of Brezis and Nirenberg in their seminal paper [2] confirmed that this methodology was indeed adequate, and appropriate for dissemination of problems of a wide variety. This paper treated problems involving the critical Sobolev exponent, 2*. In accordance with the Sobolev embedding theorem, the inclusion of a critical exponent means that the variational formulation is at its very limit.