Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev-Hardy nonlinearities

Abstract

The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with the nonlocal problems. We study the question of existence and multiplicity of weak solutions to nonlocal problems involving a combined Sobolev and Hardy nonlinearities. We also discuss the regularity of the weak solutions. We treat the question of multiplicity of weak solutions using different variational methods. One of the advantages of these methods compared to other methods such as fixed point methods for example is to obtain the existence of solutions under relatively weak conditions of regularity of the functional energy and the space of the solutions. In this master's thesis we have limited our presentation to nonlocal problems involving critical Hardy-Sobolev exponents (chapters 2 to 5). Precisely, in Chapter 2, we investigate the question of existence of weak solutions. In Chapter 3, we investigate the validity of C1 versus Ws,p energy minimizers for a quasilinear elliptic nonlocal problem. In Chapter 4, we establish new results concerning the existence of multiple weak solutions to nonlocal problems. In Chapter 5, we present global regularity results of weak solutions. The approach used to prove our main results could be consideredfor more general quasilinear operators. It will be interesting to get similar results for anisotropic operators as the fractional p(x)-laplacian operator. The results established was also been pulished in journal of the Korean Mathematical Society.