Abstract
In this chapter, symmetry results in the half-space and in \(\mathbb {R}^N\) will be used towards the characterization of the asymptotic profiles of solutions in the quarter-space and in the half-space, respectively. As we have seen in the previous Chaps. 3 and 4 , here the dimension of the underlying domain plays an important role and to extend the results on the symmetrization and stabilization of solutions of semilinear elliptic equations for dimensions less or equal 3 to the case of dimensions less or equal 4 requires nontrivial arguments and assumptions on the nonlinearities. The goal of this chapter is to extend the results from Chaps. 3 and 4 to the case of dimensions less or equal 5. As we will see below, to this end we need new arguments and we cannot use the techniques from Chaps. 3 and 4 . Similar to the previous chapters, we will apply the trajectory dynamical systems approach in order to study the asymptotic profiles of solutions for this new case of dimension 5 or higher. Moreover, in contrast to the previous chapters, we will also study the case when the asymptotic profile is a constant.
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Efendiev, M. (2018). Symmetry and Attractors. In: Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations. Fields Institute Monographs, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-319-98407-0_5
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