Abstract
This chapter focuses on multiplicity results for nonlinear problems with a lack of compactness, which could probably find new and different applications. The corresponding existence results are well known long before but a full understanding of the multiplicity problem has required more specific techniques. This chapter presents techniques by stressing on the geometric ideas underlying them. Two main problems discussed in this chapter are: (1) elliptic problems at critical growth on a bounded domain, and (2) elliptic problems at subcritical growth on the whole domain. For both the problems, there is a lack of compactness which is because of the existence of extremely concentrated solutions in case 1 and to the existence of solutions whose centers of mass escape to infinity in case 2. In both cases the existence results are available thanks to suitable hypotheses on the linear term that make the compactness degeneracy increase the functional which is going to be minimized. Thus, deviation from compactness is possible in principle but it is not advantageous, under different hypotheses on the lower-order terms one would easily show nonexistence results.
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