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- W2012397978 abstract "Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F α defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g α and of coefficients a α on M, we discuss the compactness problem of minimal energy type solutions of the equation $$-frac{1}{p} {rm div}_{g_alpha}left( nabla^xi_{g_alpha}F_alpha(nabla_{g_alpha} u) right) + a_alpha(x) |u|^{p-2} u = |u|^{p^* - 2}uquad {rm in} M.$$ This question is directly connected to the study of the first best constant $${A^alpha_{rm opt}}$$ associated with the Riemannian F α -Sobolev inequality $$left( intlimits_M |u|^{p^*}; dv_{g_alpha} right)^{frac{p}{p^*}}leq A intlimits_M F_alpha(nabla_{g_alpha} u) ; dv_{g_alpha} + B intlimits_M |u|^p ; dv_{g_alpha}.$$ Precisely, we need to know the dependence of $${A^alpha_{rm opt}}$$ under F α and g α . For that, we obtain its value as the supremum on M of best constants associated with certain homogeneous Sobolev inequalities on each tangent space and show that $${A^alpha_{rm opt}}$$ is attained on M. We then establish the continuous dependence of $${A^alpha_{rm opt}}$$ in relation to F α and g α . The tools used here are based on convex analysis, blow-up, and variational approach." @default.
- W2012397978 created "2016-06-24" @default.
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- W2012397978 date "2009-02-06" @default.
- W2012397978 modified "2023-10-04" @default.
- W2012397978 title "Compactness results for divergence type nonlinear elliptic equations" @default.
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- W2012397978 doi "https://doi.org/10.1007/s10231-009-0095-8" @default.
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