Nonlinear problems related to the Thomas-Fermi equation

Most of the results in this work were obtained over the period 1975-77 and were announced at various meetings (see e.g. items [3], [4], [5] under Brezis [16]). This paper has a rather unusual history. Around 1972 I became interested in nonlinear elliptic equations of the form $$ - \Delta u + |u{|^{{p - 1}}}u = f in a domain \Omega \subset {\mathbb{R}^{N}}, $$ (P.1) with zero Dirichlet condition, where 0 < p < ∞ and f ∈ L 1. The motivation came from the study of the porous medium equation $$ \frac{{\partial \upsilon }}{{\partial t}} - \Delta (|\upsilon {|^{{m - 1}}}\upsilon ) = 0, $$ (P.2) with 0 < m < ∞.

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