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A non-linear problem involving a critical Sobolev exponent
Bae, S.,Hadiji, R.,Vigneron, F.,Yazidi, H. Academic Press 2012 Journal of mathematical analysis and applications Vol.396 No.1
We study a non-linear minimization problem on H<SUB>0</SUB><SUP>1</SUP>(Ω)@?L<SUP>q</SUP> with q=2nn-2: inf@?u@?<SUB>L^q</SUB>=1∫<SUB>Ω</SUB>(1+|x|<SUP>β</SUP>|u|<SUP>k</SUP>)|@?u|<SUP>2</SUP>. We show that minimizers exist only for the range β<kn/q which corresponds to a dominant non-linear term. However, for β≥kn/q the influence of linearity prevents their existence.